| Initial Value | +Δ | $- \Delta$ | Final Range |
| Object size x (m) | 0.15 | 0.01 | -0.01 | $\left\lbrack {{0.14},{0.16}}\right\rbrack$ |
| Object size $\mathrm{z}\left( \mathrm{m}\right)$ | 0.05 | 0.01 | -0.01 | $\left\lbrack {{0.04},{0.06}}\right\rbrack$ |
| Table friction | 0.3 | 0.1 | -0.1 | $\left\lbrack {{0.1},{0.5}}\right\rbrack$ |
| Gripper friction | 3 | / | -1 | $\left\lbrack {2,3}\right\rbrack$ |
| Object Density $\left( {g/{m}^{3}}\right)$ | 86 | 86 | 43 | $\left\lbrack {{43},{172}}\right\rbrack$ |
| Action translation scale (m) | 0.03 | / | -0.005 | $\left\lbrack {{0.02},{0.03}}\right\rbrack$ |
| Action rotation scale (rad) | 0.2 | / | -0.05 | $\left\lbrack {{0.1},{0.2}}\right\rbrack$ |
| Initial distance to wall ( $\mathrm{m}$ ) | 0 | 0.01 | / | $\left\lbrack {0,{0.02}}\right\rbrack$ |
| Table offset $\mathrm{x}\left( \mathrm{m}\right)$ | 0.5 | 0.01 | -0.01 | $\left\lbrack {{0.48},{0.52}}\right\rbrack$ |
| Table offset $\mathrm{z}\left( \mathrm{m}\right)$ | 0.07 | 0.01 | 0.01 | $\left\lbrack {{0.055},{0.075}}\right\rbrack$ |
+
+TABLE II: Simulation parameters in Automatic Domain Randomization
+
+## Appendix III EXPERIMENT SETUP
+
+Simulation: We build the simulation environment with Robosuite [30] in the MuJoCo simulator [31]. We use a box-shaped object in this task with a default grasp location shown in Figure 1. The object is placed in a bin in front of the robot. We use single grasp training by default; the results related to multi-grasp can be found in Appendix VII. Each episode has a length of 40 timesteps which corresponds to 20 seconds of real time execution. The initial joint configuration of the robot is randomized with a Gaussian of 0.02 rad.
+
+Real robot experiment: The policy is trained in the simulator and zero-shot transferred on a physical Franka Emika Panda robot. The code for controlling the robot is built on top of FrankaPy [32]. For real robot experiments, we use Iterative Closest Point (ICP) for pose estimation of the object which matches a template point cloud of the object to the current point cloud [33]. An example of ICP result is shown in Figure 7.
+
+
+
+Fig. 6: Emergent behavior of the policy for the occluded grasping task involves multiple stages of contact mode transitions among the gripper, the object and the bin. The figure shows the corresponding stages in simulation versus the real robot execution of the policy.
+
+
+
+Fig. 7: Illustration of object pose estimation with ICP at three different timesteps of an episode. The blue points are observed point cloud which includes both the gripper and the object. The red points are the template model of the object.
+
+Evaluation metrics: We compare the policies across 5 random seeds of each method and plot the average performance with standard deviation across seeds. Our main evaluation metric is the success rate at the final step of the episode computed as $\mathbb{1}\left( {{\Delta T} < 3\mathrm{\;{cm}}}\right) \cdot \mathbb{1}\left( {{\Delta \theta } < {10}\mathrm{{deg}}}\right)$ (See Section III for definitions). We use 10 episodes for each evaluation setting.
+
+Implementation details: We use Soft Actor Critic [34] to train the RL policy with the impementation from rlkit. Both the policy network and the Q-function are parameterized as a multi-layer perceptron (MLP) with 3 layers of 512 neurons.
+
+## Appendix IV ADDITIONAL RESULTS ON EMERGENT BEHAVIORS
+
+In Section V-B, we discuss a typical emergent strategy of solving this task as a result of the design of the full system. Figure 6 includes a more detailed view of this strategy across multiple stages in simulation and on the real robot.
+
+One of the key decisions in this strategy is to use the left finger to rotate the object instead of the right finger. One might suppose an alternative approach which is to use the right finger to scoop the object against the wall and then directly roll the finger underneath the object to reach the grasp. However, this strategy is not physically feasible on the parallel gripper due to the limited degree of freedom of the finger. We observe that the policies that follow this strategy during exploration usually get stuck at a local optima without successfully reaching the grasp (Figure 8a).
+
+
+
+(a) Local optima: The gripper uses the right finger to lift the object and get stuck at a local optima.
+
+
+
+(b) Standing object: One of the successful strategies is to flip the object until it stands on the side and then reach the grasp.
+
+Fig. 8: More visualizations on the emergent behavior of the policies.
+
+Another type of successful strategy from some of the seeds is to flip the object to stand on its side and then move to the grasp (Figure 8b). This strategy overfits to the box object because it relies on the fact that the object remains stable after the flip. If the agent is trained on a more diverse set of objects without such stable poses, it might learn to avoid this strategy; however, for a box object, this is also a viable approach.
+
+
+
+Fig. 9: Ablations on the choice of controller.
+
+
+
+Fig. 10: Evaluation on the generalization of the policies by changing one physical parameter at a time.
+
+## Appendix V ABLATIONS ON LOW-LEVEL CONTROLLER
+
+We compare our method to different types of controllers to demonstrate that the choice of Operational Space Controller (OSC) is critical for extrinsic dexterity. From Figure 9, both joint torque and joint position control lead to worse performance which indicates the importance of using end-effector coordinates for the action space. We also try increasing the gain of OSC so that it becomes roughly equivalent to position control. The success rate becomes lower which demonstrates that being compliant is important for the success of contact-rich tasks in addition to the importance of compliance for safety considerations.
+
+## Appendix VI MORE RESULTS ON POLICY GENERALIZATION
+
+To further analyze the robustness of the policy across environment variations, we modify the important physical parameters one at a time to understand the sensitivity of the policies to these parameters. Following Section V-C, we include the comparison of open loop trajectories (Open Loop), policies trained over a fixed environment (Fixed Env) and policies trained with ADR (With ADR). The closed-loop policies with ADR can deal with much wider variations of physical parameters than open loop trajectories.
+
+
+
+(b) MultiGrasp-Side:The policy can use another side of the wall to rotate the object and reach the desired grasp.
+
+Fig. 11: Visualizations of the multi-grasp policies.
+
+## Appendix VII MULTIGRASP TRAINING AND SELECTION
+
+In previous sections, we only consider the scenario when a single grasp is given for each episode. In this section, we consider the scenario in which a set of desired grasp configurations $G = {g}_{i}$ are given. We will first discuss the method for multi-grasp training and selection and then provide the experimental results.
+
+MultiGrasp Training with Curriculum: During training, we aim at covering as many grasp configurations from ${G}_{\text{train }}$ as possible. The straight-forward approach is to uniformly sample a goal $g \sim {G}_{\text{train }}$ for each episode. However, previous work has shown that learning directly over such a diverse set of goals might create a difficulty for policy learning [35], [36]. Instead, we use an automatic curriculum following [8] to gradually expand the set of grasps to be trained with. We start the training with just a single fixed grasp; after the policy has achieved a success rate larger than a threshold, it will be trained on a slightly larger set with grasps close to the initial grasp location.
+
+MultiGrasp Selection: During testing, a set of grasps ${G}_{\text{test }}$ is provided. Our method selects the best grasp within the set that maximizes the learned Q-function for the current observation: ${g}^{ * } = \arg \mathop{\max }\limits_{{g \sim {G}_{\text{test }}}}Q\left( {{s}_{t},{a}_{t},{g}_{t}}\right)$ . Selecting the best grasp from the set (instead of just using a single grasp) can improve the performance of the grasping task, following previous work in integrated grasp and motion planning [20], [21], [6]. The learned Q-function can select the grasp that is most easily reached with the trained policy; which grasp is selected thus depends both on the environmental configuration as well as how well the policy has learned to achieve different grasp configurations.
+
+MultiGrasp Training Results: In this experiment, we train the policy to reach a range of grasp locations with curriculum as described above. Given the box object, we generate the grasp configurations around the box and parameterized the grasps into a continuous scalar grasp ID in the range of $\left\lbrack {0,4}\right\rbrack$ (Figure 12a). Grasp ID 1.5 is the default grasp we use in the single grasp experiments. The policy is trained with an automatic curriculum. When the success rate of policy on a boundary case of the training range is above 0.8 , it will expand the range of grasps by 0.25 . For example, if the policy is currently training with grasps [1,2], and the success rate evaluated at grasp ID 1 is above 0.8 , the new training range will be $\left\lbrack {{0.75},2}\right\rbrack$ . We train two types of multi-grasp policies starting from two different grasp poses: MultiGrasp-Front which starts the training from ID 1.5 and MultiGrasp-Side which starts the training from ID 2.5. As a baseline, we also train a policy by uniformly sampling from the entire set of grasps without using ADR, named All Grasp.
+
+
+
+Fig. 12: Multi-grasp training: Left: Visualization of the range of grasp configurations and the grasp IDs used in multi-grasp training. Right: Performance of the multi-grasp policies across grasp configurations.
+
+Figure 11a and Figure 11b include qualitative examples of the behaviors of MultiGrasp-Front and MultiGrasp-Side. The policy will rotate the object first and then try to match the pose more precisely. MultiGrasp-Side will use a different wall of the bin to rotate the object than MultiGrasp-Front. Figure 12b shows the performance of these policies evaluated over across grasp configuration IDs. We found that both MultiGrasp-Front and MultiGrasp-Side are able to expand from a single grasp to most of the grasps on one side of the object based on the curriculum. The policies have difficulties in reaching other sides potentially due to exploration issues or limited policy capacity. It may require a completely different strategy to reach different grasp configurations (Figure 11) which is difficult to learn with a single policy (related to [35]). In contrast, All Grasp has difficulties in learning any of the grasp configurations, thus showing the importance of using a curriculum for multi-grasp training.
+
+MultiGrasp Selection Results: To compare grasp selection methods, at the beginning of each episode, we sample 50 grasp configurations from the training range of the policy. The grasp selection methods will use it as the set of desired grasps. We evaluate the following grasp selection options:
+
+- Argmax $Q$ : passes all the possible grasp configurations into the Q-function and select the one that corresponds to the highest Q-value.
+
+- PoseDiff: selects the grasp by the closest distance to the current gripper location according to Equation 2 (with the same weights as the reward function).
+
+TABLE III: Comparison of grasp selection methods in two scenarios: front grasps and side grasps. When grasping from the side, the policy achieves better performance when using the $\mathrm{Q}$ -function to select the grasp.
+
+| $\mathbf{{Parameter}}$ | Description |
| ${\Lambda }_{1}$ | The number of susceptible media entering the communication system per unit time. |
| ${\Lambda }_{2}$ | The number of ignorant individuals entering the communication system per unit time. |
| $\lambda$ | The contact rate of susceptible medias with spreaders. |
| $\eta$ | The probability of affected media becoming rumor refuting media. |
| $\alpha$ | Rumor propagation rate of offline personal interaction. |
| $\beta$ | Rumor propagation rate under two-tier network interaction. |
| $\delta$ | The probability of propagating individuals becoming rumor refuting individuals. |
| $\theta$ | The probability of propagating individuals becoming immune individuals. |
| ${\xi }_{1}$ | The rate of ignorant individuals becoming rumor refuting individuals. |
| ${\xi }_{2}$ | The rate of ignorant individuals becoming immune individuals. |
| $\phi$ | The probability of rumor refuting individuals becoming immune individuals. |
| ${\mu }_{1}$ | The rate at which medias in the network move out of the propagation system. |
| ${\mu }_{2}$ | Migration rate of individuals in personal friendship network layer. |
+
+Based on the above analysis, we participated in the construction of an ${XYZ} - {SIDR}$ rumor propagation model. Then,
+
+$$
+\left\{ \begin{array}{l} {X}^{\prime } = {\Lambda }_{1} - {\lambda XI} - {\mu }_{1}X, \\ {Y}^{\prime } = {\lambda XI} - {\eta Y} - {\mu }_{1}Y, \\ {Z}^{\prime } = {\eta Y} - {\mu }_{1}Z, \\ {S}^{\prime } = {\Omega }_{2} - {\alpha SY} - {\beta SI} - \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {I + Y}\right) S - {\mu }_{2}S, \\ {I}^{\prime } = {\alpha SY} + {\beta SI} - \left( {\theta + \delta }\right) I - {\mu }_{2}I, \\ {D}^{\prime } = {\xi }_{1}S\left( {I + Y}\right) + {\delta I} - {\phi D} - {\mu }_{2}D, \\ {B}^{\prime } = {\xi }_{2}S\left( {I + Y}\right) + {\theta I} + {\delta D} - {\mu }_{2}B, \end{array}\right. \tag{1}
+$$
+
+
+
+Fig 1. Schematic representation of the ${XYZ} - {SIDR}$ rumor spreading model
+
+Since the model represents the process of rumor propagation, the parameters involved are non negative, and the initial conditions are met:
+
+$$
+X\left( 0\right) = {X}_{0} \geq 0, Y\left( 0\right) = {Y}_{0} \geq 0, Z\left( 0\right) = {Z}_{0} \geq 0,
+$$
+
+$$
+S\left( 0\right) = {S}_{0} \geq 0, I\left( 0\right) = {I}_{0} \geq 0, D\left( 0\right) = {D}_{0} \geq 0\text{,} \tag{2}
+$$
+
+$$
+R\left( 0\right) = {R}_{0} \geq 0\text{.}
+$$
+
+## III. MODEL ANALYSIS AND CALCULATION
+
+### A.The basic reproduction number ${R}_{0}$
+
+For system (1), the basic regeneration number ${R}_{0}$ is calculated as follows:
+
+Let $\mathcal{X} = {\left( I, Y, R, D, S, X, Z\right) }^{T}$ , equation (1) can be written as $\frac{d\mathcal{X}}{dt} = \mathcal{F}\left( \mathcal{X}\right) - \mathcal{V}\left( \mathcal{X}\right)$ .
+
+$$
+\mathcal{F}\left( \mathcal{X}\right) = \left( \begin{matrix} {\alpha SY} + {\beta SI} \\ {\lambda XI} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}\right) , \tag{3}
+$$
+
+$$
+\mathcal{V}\left( \mathcal{X}\right) = \left( \begin{matrix} {\theta I} + {\delta I} + {\mu }_{2}I \\ {\eta Y} + {\mu }_{1}Y \\ - {\xi }_{2}{SI} - {\xi }_{2}{SY} - {\theta I} - {\phi D} + {\mu }_{2}R \\ - {\xi }_{1}{SI} - {\xi }_{1}{SY} - {\delta I} + {\phi D} + {\mu }_{2}D \\ {H}_{1} \\ - {\Lambda }_{1} + {\lambda SI} + {\mu }_{1}X \\ - {\eta Y} + {\mu }_{1}Z \end{matrix}\right) \tag{4}
+$$
+
+where ${H}_{1} = - {\Lambda }_{2} + {\alpha SY} + {\beta SI} + {\xi }_{1}{SI} + {\xi }_{1}{SY} + {\xi }_{2}{SI} +$ ${\xi }_{2}{SY} + {\mu }_{2}S$ .
+
+Therefore
+
+$$
+F = \left( \begin{matrix} \beta \frac{{\Lambda }_{2}}{{\mu }_{2}} & \alpha \frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 \\ \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}\right) , \tag{5}
+$$
+
+$$
+V = \left( \begin{matrix} \theta + \delta + {\mu }_{2} & 0 & 0 & 0 \\ 0 & \eta + {\mu }_{1} & 0 & 0 \\ - {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} - \theta & - {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} & {\mu }_{2} & - \phi \\ - {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{2}} - \delta & - {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & \phi + {\mu }_{2} \end{matrix}\right) \tag{6}
+$$
+
+By calculation we can get
+
+$$
+F{V}^{-1} = \left( \begin{matrix} \frac{\beta {\Lambda }_{2}}{{\mu }_{2}\left( {\theta + \delta + {\mu }_{2}}\right) } & \frac{\alpha {\Lambda }_{2}}{{\mu }_{2}\left( {\eta + {\mu }_{1}}\right) } & 0 & 0 \\ \frac{\lambda {\Lambda }_{1}}{{\mu }_{1}\left( {\theta + \delta + {\mu }_{2}}\right) } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}\right) \tag{7}
+$$
+
+Hence, according to reference [27], the basic reproduction number of system (1) is the spectral radius of matrix $F{V}^{-1}$ as follows:
+
+$$
+{R}_{0} = \frac{\beta {\Lambda }_{2}}{{\mu }_{2}\left( {\theta + \delta + {\mu }_{2}}\right) } \tag{8}
+$$
+
+## B. Existence of equilibrium
+
+According to the system dynamics equation (1), we can calculate the equilibrium $E = \left( {X, Y, Z, S, I, D, R}\right)$ . It is easy to see that the positive equilibrium points of system (1) are ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ and ${E}^{ * } =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ , and the rumor free equilibrium point ${E}_{0}$ always exists.
+
+Theorem 1 The equilibrium point ${E}^{ * }\; =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ exists if ${R}_{0} > 1$ and $\left( {\theta + \delta + {\mu }_{2}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) > {\beta \lambda }{\Lambda }_{2}.$
+
+Proof The rumors about system (1) have a balance point that satisfies:
+
+$$
+\left\{ \begin{array}{l} {\Lambda }_{1} - {\lambda XI} - {\mu }_{1}X = 0, \\ {\lambda XI} - {\eta Y} - {\mu }_{1}Y = 0, \\ {\eta Y} - {\mu }_{1}Z = 0, \\ {\Lambda }_{2} - {\alpha SY} - {\beta SI} - \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {I + Y}\right) S - {\mu }_{2}S = 0, \\ {\alpha SY} + {\beta SI} - \left( {\theta + \delta }\right) I - {\mu }_{2}I = 0, \\ {\xi }_{1}S\left( {I + Y}\right) + {\delta I} - {\phi D} - {\mu }_{2}D = 0, \\ {\xi }_{2}S\left( {I + Y}\right) + {\theta I} + {\phi D} - {\mu }_{2}R = 0. \end{array}\right. \tag{9}
+$$
+
+According to formula (9), ${X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{D}^{ * },{R}^{ * }$ are represented by ${I}^{ * }$ respectively and brought into the fifth equation to get
+
+$$
+a{I}^{2} + {bI} + c = 0 \tag{10}
+$$
+
+Where
+
+$$
+a = \lambda \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) ,
+$$
+
+$$
+b = \left( {\theta + \delta + {\mu }_{2}}\right) \left\lbrack {\left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack
+$$
+
+$$
++ \lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) - {\beta \lambda }{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \text{,}
+$$
+
+$$
+c = \left( {\eta + {\mu }_{1}}\right) \left\lbrack {{\mu }_{2}\lambda \left( {\theta + \delta + {\mu }_{2}}\right) - {\mu }_{1}\beta {\Lambda }_{2}}\right\rbrack - {\alpha \lambda }{\Lambda }_{1}{\Lambda }_{2}.
+$$
+
+(11)
+
+It can be obtained by calculation that
+
+$$
+\Delta = {b}^{2} - {4ac}
+$$
+
+$$
+= {\left\lbrack \lambda {\Lambda }_{1}\left( \alpha + {\xi }_{2}\right) + \left( \eta + {\mu }_{1}\right) \left( {\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda \right) \right\rbrack }^{2}
+$$
+
+$$
+* {\left( \theta + \delta + {\mu }_{2}\right) }^{2} + {\left\lbrack \beta \lambda {\Lambda }_{2}\left( \eta + {\mu }_{1}\right) \right\rbrack }^{2} + {4\alpha }{\lambda }^{2}{\Lambda }_{1}{\Lambda }_{2}\left( {\beta + {\xi }_{1}}\right)
+$$
+
+$$
+* \left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) - {2\beta }{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right)
+$$
+
+$$
+* \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack
+$$
+
+$$
+- {4\lambda }\left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right)
+$$
+
+$$
+* \left\lbrack {{\mu }_{2}\lambda \left( {\theta + \delta + {\mu }_{2}}\right) - \beta {\mu }_{1}{\Lambda }_{2}}\right\rbrack
+$$
+
+(12)
+
+According to the discriminant calculation, when ${R}_{0} > 1$ and
+
+$\left( {\theta + \delta + {\mu }_{2}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) > {\beta \lambda }{\Lambda }_{2}$ , the negative solution is omitted:
+
+$$
+{I}^{ * } = \frac{{\beta \lambda }{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) - {H}_{2}\left( {\theta + \delta + {\mu }_{2}}\right) + \sqrt{\Delta }}{{2\lambda }\left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) } \tag{13}
+$$
+
+where ${H}_{2} = \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack$ .
+
+Therefore ${E}^{ * } = \left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ , where
+
+$$
+{X}^{ * } = \frac{{\Lambda }_{1}}{\lambda {I}^{ * } + {\mu }_{1}}, \tag{14}
+$$
+
+$$
+{Y}^{ * } = \frac{\lambda {\Lambda }_{1}{I}^{ * }}{\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }, \tag{15}
+$$
+
+$$
+{Z}^{ * } = \frac{{\lambda \eta }{\Lambda }_{1}{I}^{ * }}{{\mu }_{1}\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }, \tag{16}
+$$
+
+$$
+{S}^{ * } = \frac{{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }{T}, \tag{17}
+$$
+
+$$
+{D}^{ * } = \frac{\lambda {\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) {I}^{*2} + \left\lbrack {\lambda {\Lambda }_{1}{\Lambda }_{2} + {\mu }_{1}\left( {\eta + {\mu }_{1}}\right) }\right\rbrack {I}^{ * }}{\left( {\phi + {\mu }_{2}}\right) T}, \tag{18}
+$$
+
+$$
+{R}^{ * } = \frac{{\xi }_{2}{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) {H}_{3} + \theta {H}_{4}}{\left( {{\mu }_{2} - \phi }\right) {H}_{4}} \tag{19}
+$$
+
+where ${H}_{3} = \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) \left\lbrack {\lambda {\Lambda }_{1} + \left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }\right\rbrack ,{H}_{4} =$ $\lambda \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) {I}^{*2} + \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + }\right. }\right.$ $\left. \left. {{\mu }_{2}\lambda }\right) \right\rbrack {I}^{ * } + {\mu }_{2}\lambda \left( {\eta + {\mu }_{1}}\right)$ .
+
+## C. Stability of equilibrium
+
+Theorem 2 The equilibrium point ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is locally asymptotically stable if ${R}_{0} < 1$ . And the equilibrium point ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is unstable if ${R}_{0} > 1$ .
+
+Proof The Jacobian matrix of system (1) at
+
+${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is
+
+$J\left( {E}_{0}\right) =$
+
+$$
+\left( \begin{matrix} - {\mu }_{1} & 0 & 0 & 0 & - \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & - \eta - {\mu }_{1} & 0 & 0 & \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & \eta & - {\mu }_{1} & 0 & 0 & 0 & 0 \\ 0 & {H}_{5} & 0 & - {\mu }_{2} & {H}_{6} & 0 & 0 \\ 0 & \alpha \frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {H}_{7} & 0 & 0 \\ 0 & {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} + \theta & - {\mu }_{2} & - {\mu }_{2} \\ 0 & {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{1}} & 0 & 0 & {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{1}} + \delta & 0 & {H}_{8} \end{matrix}\right)
+$$
+
+where ${H}_{5} = - \left( {\alpha + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}},{H}_{6} = - \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}}$ ,
+
+${H}_{7} = \beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - \left( {\theta + \delta + {\mu }_{2}}\right) ,{H}_{8} = - \left( {\phi + {\mu }_{2}}\right) .$
+
+The characteristic equation of matrix $J\left( {E}_{0}\right)$ is
+
+$\left| {J\left( {E}_{0}\right) - {hE}}\right| =$
+
+$$
+\left. \begin{matrix} {T}_{1} & 0 & 0 & 0 & - \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & {T}_{1} & 0 & 0 & \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & \eta & {T}_{4} & 0 & 0 & 0 & 0 \\ 0 & {T}_{2} & 0 & {T}_{5} & {T}_{3} & 0 & 0 \\ 0 & \alpha \frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {T}_{4} & 0 & 0 \\ 0 & {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {T}_{5} & - {\mu }_{2} - h & - {\mu }_{2} \\ 0 & {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{1}} & 0 & 0 & {T}_{6} & 0 & - \left( {\phi + {\mu }_{2}}\right) - h \end{matrix}\right|
+$$
+
+$= {\left( {\mu }_{1} + h\right) }^{2}{\left( {\mu }_{2} + h\right) }^{2}\left( {\phi + {\mu }_{2} + h}\right) \left( {\eta + {\mu }_{1} + h}\right) \left\lbrack {\beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - (\theta + }\right.$
+
+$$
+\left. {\left. {\delta + {\mu }_{2}}\right) - h}\right\rbrack = 0
+$$
+
+Where ${T}_{1} = - {\mu }_{1} - h,{T}_{2} = - \eta - {\mu }_{1} - h,{T}_{3} =$ $- \left( {\alpha + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}},{T}_{4} = - {\mu }_{1} - h,{T}_{5} = - {\mu }_{2} - h,{T}_{6} =$ $- \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}},{T}_{7} = \beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - \left( {\theta + \delta + {\mu }_{2}}\right) - h,{T}_{8} =$ $\beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - \left( {\theta + \delta + {\mu }_{2}}\right) - h,{T}_{9} = {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{2}} + \delta .$
+
+Therefore, the characteristic root corresponding to the characteristic equation of $J\left( {E}_{0}\right)$ is:
+
+$$
+{h}_{01} = - {\mu }_{1} < 0,{h}_{02} = - {\mu }_{2} < 0,{h}_{03} = - \left( {\phi + {\mu }_{2}}\right) < 0,
+$$
+
+$$
+{h}_{04} = - \left( {\eta + {\mu }_{1}}\right) < 0,{h}_{05} = \frac{\theta + \delta + {\mu }_{2}}{{\mu }_{2}}\left( {{R}_{0} - 1}\right) < 0
+$$
+
+(20)
+
+According to Routh-Hurwitz stability criterion, the equilibrium point
+
+${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is locally asymptotically stable if ${R}_{0} < 1$ .
+
+And the equilibrium point ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is unstable if ${R}_{0} > 1$ .
+
+Theorem 3 The equilibrium point ${E}^{ * }\; =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ is locally asymptotically stable if ${R}_{0} > 1$ and $\beta {\Lambda }_{2} < {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) \left( {\theta + \delta + {\mu }_{2}}\right)$ , otherwise, the equilibrium point ${E}^{ * }$ is unstable.
+
+Proof The Jacobian matrix at ${E}^{ * } =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ is
+
+$J\left( {E}^{ * }\right) =$
+
+$$
+\left( \begin{matrix} {A}_{1} & 0 & 0 & 0 & - \lambda {X}^{ * } & 0 & 0 \\ \lambda {I}^{ * } & {A}_{2} & 0 & 0 & \lambda {X}^{ * } & 0 & 0 \\ 0 & \eta & - {\mu }_{1} & 0 & 0 & 0 & 0 \\ 0 & {A}_{3} & 0 & {A}_{4} & {A}_{8} & 0 & 0 \\ 0 & \alpha {S}^{ * } & 0 & {A}_{5} & {A}_{9} & 0 & 0 \\ 0 & {\xi }_{2}{S}^{ * } & 0 & {A}_{6} & {\xi }_{2}{S}^{ * } + \theta & - {\mu }_{2} & - {\mu }_{2} \\ 0 & {\xi }_{1}{S}^{ * } & 0 & {A}_{7} & {\xi }_{1}{S}^{ * } + \delta & 0 & {A}_{10} \end{matrix}\right)
+$$
+
+Where ${A}_{1} = \lambda {I}^{ * } - {\mu }_{1},{A}_{2} = - \eta - {\mu }_{1},{A}_{3} =$
+
+$$
+- \left( {\alpha + {\xi }_{1} + {\xi }_{2}}\right) {S}^{ * },{A}_{4} = - \alpha {Y}^{ * } - \beta {I}^{ * },{A}_{5} = \alpha {Y}^{ * } + \beta {I}^{ * }\text{,}
+$$
+
+$$
+{A}_{6} = {\xi }_{2}\left( {{I}^{ * } + {Y}^{ * }}\right) ,\;{A}_{7} = {\xi }_{1}\left( {{I}^{ * } + {Y}^{ * }}\right) ,\;{A}_{8} =
+$$
+
+$$
+- \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) {S}^{ * },{A}_{9} = \beta {S}^{ * } - \left( {\theta + \delta + {\mu }_{2}}\right) ,{A}_{10} =
+$$
+
+$- \left( {\phi + {\mu }_{2}}\right)$ .
+
+The characteristic equation of matrix $J\left( {E}^{ * }\right)$ is
+
+$\left| {J\left( {E}^{ * }\right) - {hE}}\right| =$
+
+$$
+\left| \begin{matrix} {B}_{1} & 0 & 0 & 0 & - \lambda {X}^{ * } & 0 & 0 \\ \lambda {I}^{ * } & {B}_{2} & 0 & 0 & \lambda {X}^{ * } & 0 & 0 \\ 0 & \eta & - {\mu }_{1} - h & 0 & 0 & 0 & 0 \\ 0 & {B}_{3} & 0 & {B}_{3} & {B}_{7} & 0 & 0 \\ 0 & \alpha {S}^{ * } & 0 & {B}_{4} & {B}_{8} & 0 & 0 \\ 0 & {\xi }_{2}{S}^{ * } & 0 & {B}_{5} & {\xi }_{2}{S}^{ * } + \theta & {B}_{9} & - {\mu }_{2} \\ 0 & {\xi }_{1}{S}^{ * } & 0 & {B}_{6} & {\xi }_{1}{S}^{ * } + \delta & 0 & {B}_{10} \end{matrix}\right|
+$$
+
+Where ${B}_{1} = \lambda {I}^{ * } - {\mu }_{1} - h,{B}_{2} = - \eta - {\mu }_{1} - h,{B}_{3} =$ $- \alpha {Y}^{ * } - \beta {I}^{ * } - h,{B}_{4} = \alpha {Y}^{ * } + \beta {I}^{ * },{B}_{5} = {\xi }_{2}\left( {{I}^{ * } + }\right.$ $\left. {Y}^{ * }\right) ,{B}_{6} = {\xi }_{1}\left( {{I}^{ * } + {Y}^{ * }}\right) ,{B}_{7} = - \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) {S}^{ * }$ , ${B}_{8} = \beta {S}^{ * } - \left( {\theta + \delta + {\mu }_{2}}\right) - h,{B}_{9} = - {\mu }_{2} - h,{B}_{10} =$ $- \left( {\phi + {\mu }_{2}}\right) - h$ .
+
+Thus, we can obtain
+
+$$
+\left| {J\left( {E}^{ * }\right) - {hE}}\right| = \left( {{\mu }_{1} + h}\right) \left( {{\mu }_{2} + h}\right) \left( {\phi + {\mu }_{2} + h}\right) \left( {\eta + {\mu }_{1} + }\right.
+$$
+
+$\left. h\right) \left( {\lambda {I}^{ * } + {\mu }_{1} + h}\right) G$ .
+
+Where $G = - \left\lbrack {\alpha {Y}^{ * } + \beta {I}^{ * } + \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {{I}^{ * } + {Y}^{ * }}\right) + {\mu }_{2}}\right\rbrack - h$ .
+
+Therefore, the characteristic root corresponding to the characteristic equation of $J\left( {E}^{ * }\right)$ is:
+
+$$
+{h}_{01} = - {\mu }_{1} < 0,{h}_{02} = - {\mu }_{2} < 0, \tag{21}
+$$
+
+$$
+{h}_{03} = - \left( {\phi + {\mu }_{2}}\right) < 0,{h}_{04} = - \left( {\eta + {\mu }_{1}}\right) < 0, \tag{22}
+$$
+
+$$
+{h}_{05} = - \left\lbrack {\alpha {Y}^{ * } + \beta {I}^{ * } + \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {{I}^{ * } + {Y}^{ * }}\right) + {\mu }_{2}}\right\rbrack < 0, \tag{23}
+$$
+
+$$
+{h}_{06} = \beta {S}^{ * } - \left( {\theta + \delta + {\mu }_{2}}\right) . \tag{24}
+$$
+
+Then, we take ${S}^{ * }$ into ${h}_{06}$ ,
+
+${h}_{06} = \frac{\beta {\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }{\lambda \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) {I}^{*2} + {C}_{1} + {\mu }_{2}\lambda \left( {\eta + {\mu }_{1}}\right) } - \left( {\theta + \delta + {\mu }_{2}}\right) ,$
+
+where ${C}_{1} = \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack {I}^{ * }$ .
+
+## IV. NUMERICAL SIMULATION
+
+In this section, we will assign reasonable values to the parameters in system (1) as established in this paper, and verify the results of our theoretical analysis through numerical simulations. On the one hand, we combine some similar examples in reality. On the other hand, the parameter values in relevant literature are referred to.
+
+Order ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\lambda = {0.01},\eta = {0.3},\alpha = {0.01},\beta =$ ${0.01},\theta = {0.2},\delta = {0.2},\phi = {0.15},{\xi }_{1} = {0.1},{\xi }_{2} = {0.1},{\mu }_{1} =$ ${0.2},{\mu }_{2} = {0.2}$ . Calculated ${R}_{0} = {0.8333} < 1$ , then the no rumor propagation equilibrium point ${E}_{0}$ is stable.
+
+
+
+Fig 2. Stability of equilibrium point ${E}_{0}$ .
+
+Fig. 2 shows when ${R}_{0} = {0.0833} < 1$ , the density of each subclass in the model changes with time. At first, the number of unaffected media and unknowns gradually decreased at a similar rate, and finally stabilized. Due to the limited number of moving in and the large number of moving out, the number of affected media and communicators gradually decreases at a similar rate and finally becomes 0 . The number of rumor refuting media and rumor refuters first increased with the increase of the number of affected media and disseminators. It gradually decreases over time and finally becomes 0 . The number of immunized persons increased with the increase of the number of communicators and rumor refuters. The growth rate gradually slowed down and finally stabilized. Namely, the rumor disappears and reaches a stable equilibrium point, and there is no rumor.
+
+Let ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\lambda = {0.2},\eta = {0.3},\alpha = {0.5},\beta =$ ${0.6},\theta = {0.4},\delta = {0.4},\phi = {0.15},{\xi }_{1} = {0.2},{\xi }_{2} = {0.2},{\mu }_{1} =$ ${0.2},{\mu }_{2} = {0.2}$ and calculate ${R}_{0} = 3 > 1$ , the equilibrium point ${E}^{ * }$ is stable, as shown in Fig. 3.
+
+
+
+Fig 3. Stability of equilibrium point ${E}^{ * }$ .
+
+In Fig. 3, considering the media network layer, due to the small number of new media entering the communication system and the transformation of some unaffected media into affected media, the number of unaffected media gradually decreases and tends to stabilize after a period of time. Originally, the number of affected media increased due to the transformation of some unaffected media into affected media. Over time, most of the affected media were transformed into rumor refuting media, so the number of affected media decreased and stabilized. As the affected media changed into rumor refuting media, the number of rumor refuting media increased and gradually stabilized.
+
+Fig. 3 illustrates that, within the individual interpersonal network layer, the number of ignorant individuals begins to decline. Initially, the low influx of new individuals and a fixed rate of departures contribute to this decrease. Additionally, some ignorant individuals transition to become communicators, while others become immune or rumor refuters. Consequently, the number of communicators increases as ignorant individuals transform into communicators. Over time, as communicators transition to immune individuals or rumor refuters, the number of communicators gradually decreases and eventually stabilizes. As more communicators and ignorant individuals become rumor refuters, the number of rumor refuters rises and stabilizes. Simultaneously, with some ignorant individuals, communicators, and rumor refuters becoming immune, the number of immune individuals significantly increases and gradually stabilizes. Ultimately, the model reaches a steady state, with each groups number stabilizing over time.
+
+Fig. 4 to Fig. 7 depict ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\eta = {0.3},\theta =$ ${0.4},\phi = {0.15},{\xi }_{1} = {0.2},{\xi }_{2} = {0.5},{\mu }_{1} = {0.2},{\mu }_{2} = {0.2}$ , the evolution of the density of $X\left( t\right) , Y\left( t\right)$ and $S\left( t\right)$ with different parameters.
+
+Fig. 4 and Fig. 5 describe the effect of parameter $\lambda$ on the density change of $X\left( t\right)$ and $Y\left( t\right)$ respectively. Parameter $\lambda$ represents the probability that the unaffected media will be transformed into the affected media. It can be seen from the figure that the parameter $\lambda$ has a negative correlation with the density of $X\left( t\right)$ and a positive correlation with the density of $Y\left( t\right)$ . That is, with the increase of the parameter $\lambda$ , the rate of transformation from unaffected media to affected media increases. The number of unaffected media decreases gradually, and the number of affected media increases gradually, accelerating the spread of rumors in the media network layer.
+
+
+
+Fig 4. Density of $X\left( t\right)$ under the parameter $\lambda$ .
+
+
+
+Fig 5. Density of $Y\left( t\right)$ under the parameter $\lambda$ .
+
+Fig. 6 and Fig. 7 describe the influence of parameters $\alpha$ and $\beta$ on the density change of $S\left( t\right)$ respectively. Parameter $\alpha$ represents the probability that the unknown person will become a spreader by accessing the affected media, and parameter $\beta$ represents the probability that the unknown person will become a spreader by contacting spreaders. It can be seen from the figure that the density of $S\left( t\right)$ decreases with the increase of parameters $\alpha$ and $\beta$ . Namely, with the increase of the propagation rate of individual network layer and double-layer network interaction, the number of unknowns gradually decreases, which accelerates the spread of rumors in the double-layer network.
+
+
+
+Fig 6. Density of $S\left( t\right)$ under the parameter $\alpha$ .
+
+
+
+Fig 7. Density of $S\left( t\right)$ under the parameter $\beta$ .
+
+Fig. 8 and Fig. 9 describe when ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\eta =$ ${0.3},\theta = {0.4},\phi = {0.15},{\xi }_{1} = {0.2},{\xi }_{2} = {0.5},{\mu }_{1} = {0.2},{\mu }_{2} =$ 0.2, the evolution of the density of $I\left( t\right)$ with different parameters.
+
+Fig. 8 and Fig. 9 describe the influence of parameters $\alpha$ and $\beta$ on the density change of $I\left( t\right)$ respectively. Considering the meaning of parameters $\alpha$ and $\beta$ , it is easy to know that the values of parameters $\alpha$ and $\beta$ are positively correlated with the density of $I\left( t\right)$ . As can be seen from the figure, the density of $I\left( t\right)$ increases with the increase of parameters $\alpha$ and $\beta$ . That is, the increasing number of communicators promotes the expansion of the scale of communication, which is not conducive to the control of rumors.
+
+## V. CONCLUSION
+
+At present, many scholars have separately studied the influence of media refutation or individual refutation on the spread of rumors. We believe that considering these two effects comprehensively is better than considering one of them alone. This paper integrates both media refutation and individual refutation into the analysis and introduces a novel ${XYZ} - {SIDR}$ two-tier rumor propagation model, further demonstrates the existence and stability of equilibrium points within the model. The research results show that this two-layer network model is more effective in controlling the spread of rumors.
+
+
+
+Fig 8. Density of $I\left( t\right)$ under the parameter $\alpha$ .
+
+
+
+Fig 9. Density of $I\left( t\right)$ under the parameter $\beta$ .
+
+Theoretical analysis indicates that integrating both media and individual rumor refutation exerts a more significant and broader impact on rumor propagation. We suggest strengthening the dissemination of rumor refutation information through the official media rather than relying solely on individuals to control the spread of rumor. The research conclusion can help relevant departments formulate effective measures to control the spread of rumors. On the other hand, the model established in this paper can also be analogically applied to the study of infectious disease model.
+
+## REFERENCES
+
+[1] W. Jinling, J. Haijun, H. Cheng, Y. Zhiyong and L. Jiarong,"Stability and Hopf bifurcation analysis of multi-lingual rumor spreading model with nonlinear inhibition mechanism," Chaos, Solitons & Fractals, vol. 153, pp. 111464, December 2021.
+
+[2] L. Qiming, L. Tao and S. Meici, "The analysis of an SEIR rumor propagation model on heterogeneous network," Physica A: Statistical Mechanics and its Applications, vol.469, PP. 372-380, March 2017.
+
+[3] H. Yuhan, P. Qiuhui, H. Wenbing and H. Mingfeng, "Rumor spreading
+
+model considering the proportion of wisemen in the crowd," Physica A: Statistical Mechanics and its Applications, vol. 505, pp. 1084-1094, September 2018.
+
+[4] W. Juan, L. Chao and X. Chengyi, "Improved centrality indicators to characterize the nodal spreading capability in complex networks," Applied Mathematics and Computation, vol. 334, pp. 388-400, October 2018.
+
+[5] K. Eismann, "Diffusion and persistence of false rumors in social media networks: implications of searchability on rumor self-correction on Twitter," Journal of Business Economics, vol. 91, pp. 1299-1329, February 2021.
+
+[6] W. Jinling, J. Haijun, M. ianlong and H. Cheng, "Global dynamics of the multi-lingual SIR rumor spreading model with cross-transmitted mechanism," Chaos, Solitons & Fractals, vol. 126, pp. 148-157, September 2019.
+
+[7] D. Xuefan, L. Yijun, W. Chao, L. Ying and T. Daisheng, "A double-identity rumor spreading model," Physica A: Statistical Mechanics and its Applications, vol. 528, pp. 121479 August 2019.
+
+[8] K. Afassinou, "Analysis of the impact of education rate on the rumor spreading mechanism," Physica A: Statistical Mechanics and Its Applications, vol. 414, pp. 43-52, November 2014.
+
+[9] Z. Linhe, Y. Yang, G. Gui and Z. Zhengdi, "Modeling the dynamics of rumor diffusion over complex networks," Information Sciences, vol. 562, pp. 240-258, July 2021.
+
+[10] Z. Linhe and W. Bingxu, "Stability analysis of a SAIR rumor spreading model with control strategies in online social networks," Information Sciences, vol. 526, pp. 1-19, July 2020.
+
+[11] A. Abta, H. Laarabi, M. Rachik, H. T. Alaoui and S. Boutayeb, "Optimal control of a delayed rumor propagation model with saturated control functions and ${L}^{1}$ -type objectives," Social Network Analysis and Mining, vol. 10, August 2020.
+
+[12] X. Jiuping, T. Weiyao, Z. Yi and W. Fengjuan, "A dynamic dissemination model for recurring online public opinion," Nonlinear Dynamics, vol. 99, pp. 12691293, November 2019.
+
+[13] C. Yingying, H. Liangan and Z. Laijun, "Rumor spreading in complex networks under stochastic node activity," Physica A: Statistical Mechanics and its Applications, vol. 559, pp. 125061, December 2020.
+
+[14] Z. Linhe, L. Wenshan and Z. Zhengdi, "Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function," Applied Mathematics and Computation, vol. 370, pp. 124925, April 2020.
+
+[15] Y. Fulian , Z. Xiaowei, S. Xueying, X. Xinyu, P. Yanyan and W. Jianhong, "Modeling and quantifying the influence of opinion involving opinion leaders on delayed information propagation dynamics," Applied Mathematics Letters, vol. 121, pp. 107356, November 2021.
+
+[16] Z. Hongyong and Z. Linhe, "Dynamic Analysis of a ReactionDiffusion Rumor Propagation Model," International Journal of Bifurcation and Chaos, vol. 26, pp. 1650101, 2016.
+
+[17] Z. Linhe, W. Xuewei, A. Zhengdi and S. Shuling, "Global Stability and Bifurcation Analysis of a Rumor Propagation Model with Two Discrete Delays in Social Networks," International Journal of Bifurcation and Chaos, vol. 30, pp. 2050175, 2020.
+
+[18] Y. Shuzhen, Y. Zhiyong, J. Haijun and Y. Shuai, "The dynamics and control of 2I2SR rumor spreading models in multilingual online social networks," Information Sciences, vol. 581, pp. 18-41, December 2021.
+
+[19] M. Ghosh, S. Das and P. Das , "Dynamics and control of delayed rumor propagation through social networks," Journal of Applied Mathematics and Computing, vol. 68, pp. 1-30, November 2021.
+
+[20] Z. Linhe and H. Le, "Pattern formation in a reactiondiffusion rumor propagation system with Allee effect and time delay," Nonlinear Dynamics, vol. 107, pp. 3041-3063, January 2022.
+
+[21] C. Yingying, H. Liang'an and Z. Laijun, "Dynamical behaviors and control measures of rumor-spreading model in consideration of the infected media and time delay," Information Sciences, vol. 564, pp. 237- 253, July 2021.
+
+[22] L. Jiarong, J. Haijun, Y. Zhiyong and H. Cheng, "Dynamical analysis of rumor spreading model in homogeneous complex networks," Applied Mathematics and Computation, vol. 359, pp. 374-385, October 2019.
+
+[23] J. Wenjun, L. Yi, Z. Xiaoqin, Z. Juping and J. Zhen, "A rumor spreading pairwise model on weighted networks," Physica A: Statistical Mechanics and its Applications, vol. 585, pp. 126451, January 2022.
+
+[24] Y. Lan, L. Zhiwu and A. Giua , "Containment of rumor spread in complex social networks," Information Sciences, vol. 506, pp. 113-130, January 2020.
+
+[25] H. Liangan, D. Fan and C. Yingying, "Dynamic analysis of a S I b I n I u, rumor spreading model in complex social network," Physica A: Statistical Mechanics and its Applications, vol. 523, pp. 924-932, June 2019.
+
+[26] P. van den Driessche and J. Watmough, "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission," Mathematical Biosciences, vol. 180, pp. 2948, December 2002.
+
+[27] J. M. Heffernan, R. J. Smith and L. M. Wahl, "Perspectives on the basic reproductive ratio," Journal of the Royal Society Interface, vol. 2, pp. 281293, 2005.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/Cox7GQmwAI/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/Cox7GQmwAI/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..5656ff95e8e3c7eb084c4d89196b126939cd616f
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/Cox7GQmwAI/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,464 @@
+§ DYNAMICAL ANALYSIS OF RUMOR PROPAGATION MODEL CONSIDERING MEDIA REFUTATION AND INDIVIDUAL REFUTATION*
+
+${1}^{\text{ st }}$ Wenqi Pan
+
+College of Marine Electrical Engineering
+
+Dalian Maritime University
+
+Dalian, China
+
+panwenqi07@163.com
+
+${2}^{\text{ nd }}$ Li-Ying Hao*
+
+College of Marine Electrical Engineering
+
+Dalian Maritime University
+
+Dalian, China
+
+haoliying_0305@163.com
+
+Abstract-The factor of refutation significantly impacts the spread of rumors. Common methods of refuting rumors include media intervention and individual efforts. While many scholars have explored the effects of these factors separately, few studies have comprehensively examined both simultaneously. This model integrates the influence of both media and individual refutation on the rumor propagation process. We propose a novel two-tier network model for rumor spread. We demonstrated the existence and stability of equilibrium points within the model. Theoretical analysis demonstrates that authoritative media refutations exert a broader and more substantial influence on rumor dissemination compared to individual refutations.
+
+Index Terms-rumor propagation, rumor refuting medias, rumor refuters, stability
+
+§ I. INTRODUCTION
+
+Rumor refers to the speech fabricated without corresponding factual basis and with a certain purpose and promoted its dissemination by some means. With the exponential growth of technology and the widespread adoption of internet-based social networks, misinformation and harmful rumors have the potential to swiftly propagate across online platforms, posing significant threats to social cohesion, stability, as well as disrupting people's daily lives and productive activities. For example, the panic of buying salt caused by the Fukushima Daiichi Nuclear Disaster [1], there was a rumor that SHL-C could prevent COVID-19, which caused great harm to the public's psychology and body, and seriously disturbed the normal order of the society.
+
+The propagate of rumors has attracted the attention and research of many scholars. Some scholars compared the disseminate of rumors with the propagate of infectious diseases in humans, and applied the infectious disease model to the disseminate of rumors [2]-[5]. Considering the influence of different propagation mechanisms on rumor propagation, many scholars have studied cross propagation mechanism [7] and education mechanism [6]. Komi [8] established rumor propagation model based on population education and forgetting mechanism, and found that educated ignorant people are less likely to be transformed into disseminators and more likely to be transformed into suppressors than uneducated ignorant people.
+
+At the same time, many scholars considered the influence of different function methods [9]-[11] in the research process. Zhu et al. [14] proposed a rumor propagation model in homogeneous and heterogeneous networks, and comprehensively studied the influence of forced silence function, time delay and network topology on rumor propagation in social networks. The influence of time delay on the propagation process has also been studied by many scholars [15]-[18] on rumor propagation process in existing research. Cheng et al. [21] established an improved ${XY} - {ISR}$ rumor propagation model on the basis of interactive system, comprehensively discussed the influence of different delays on rumor propagation, further proposed control strategies such as deleting posts, popular science education and immunotherapy.
+
+With the complexity of the network environment, some scholars have comprehensively considered the influence of various factors on rumor propagation on the complex network [22]-[24]. Considering the reaction of the ignorant when hearing the rumor for the first time, Huo et al. [25] divided the individuals in the network into four groups: the ignorant, the trustworthy, the spreader and the uninterested, and proposed ${SIbInIu}$ rumor propagation model in the complex network. The theoretical analysis and simulation results show that the loss rate and suppression rate have a negative impact on the final rumor spread scale.
+
+In the existing literature, it is not common to comprehensively consider the impact of media refutation and individual refutation on the two-tier network rumor propagation model. Based on the actual assumptions, we believe that the rumor refutation effect of comprehensive consideration of the two is better than that of single consideration. This paper mainly make a dynamic analysis on the rumor propagation considering the rumor refutation effect of these two factors.
+
+The rest of this paper is distributed as follows. We propose a two-tier network rumor propagation model in section 1. Section 2 describes a two-tier network rumor propagation model considering both rumor refuting media and rumor refuter groups. In section 3, we discuss the existence and stability conditions of the equilibrium points. Finally, the feasibility of the results presented in this paper was confirmed through numerical simulations.
+
+This work was funded by the National Natural Science Foundation of China (51939001, 52171292), Dalian Outstanding Young Talents Program (2022RJ05).
+
+§ II. TWO-TIER NETWORK RUMOR PROPAGATION MODEL
+
+In the two-tier rumor propagation model constructed in this paper, the media network model is composed of networks with $M$ media websites, and the personal friendship network model is composed of networks with $N$ personal friendship websites.
+
+In the network layer of media websites, media can be divided into three states: vulnerable media without rumor information (represented by $X$ ), affected media with rumor information (represented by $Y$ ) and rumor refuting media with rumor refuting information (represented by $Z$ ). When communicators visit the vulnerable media, they will release or leave rumors on the media network, so that the vulnerable media will be affected and become the affected media. When the rumor refuters visit the affected media, they will release or leave rumor refutation information on the media network to make the affected media become rumor refutation media.
+
+In the personal network layer, individuals are categorized into four distinct groups: those who have never heard of the rumors (denoted by $S$ ), those who actively spread rumors (denoted by $I$ ), those who do not believe in the rumors but disseminate refutation information (denoted by $D$ ), and those who neither believe in nor propagate any information (denoted by $R$ ). In the process of network node interaction, after visiting the vulnerable media, the disseminator spreads rumor information on the media website, so that the vulnerable media is infected and evolved into the affected media. When an ignorant person visits the affected media, affected by the rumor information, the ignorant person becomes a disseminator with a certain probability. Thus, rumors can be spread not only between people, but also between individuals and online media. The basic assumptions of this paper are as follows:
+
+Hypothesis 1: In the media network layer, considering that the media website has a certain registration rate and cancellation rate, the number of vulnerable media entering the communication system per unit time is ${\Lambda }_{1}$ . Moreover, there will be benign competition among the media. The three types of media websites $X,Y$ and $Z$ may move out of the communication system with a certain probability ${\mu }_{1}$ . When communicators visit vulnerable media and publish their own views and comments, the rate of conversion to affected media is $\lambda$ . When the rumor refuter visits the affected media and publishes rumor refutation information on it, the affected media will change into rumor refutation media with a certain probability $\eta$ .
+
+Hypothesis 2: In the personal interpersonal network layer, assume that the rate at which individuals who are unaware of rumors enter the communication system is ${\Lambda }_{2}$ . Those who question the rumor but neither spread rumor information nor disseminate refutation will transition to an immune state at a rate of ${\xi }_{2}$ . Individuals who initially spread rumors but later find the information untrue may become rumor disclaimers with probability $\delta$ . If these communicators lose interest in rumors and cease both rumor propagation and refutation, they will transition to an immune state with probability $\theta$ . Rumor disclaimers affected by the environment or who lose interest in refutation will also become immune with probability $\phi$ . Additionally, individual groups may exit the rumor spreading network due to migration at a rate ${\mu }_{2}$ .
+
+Hypothesis 3: In the interaction of offline individuals, the ignorant will become the disseminator at a certain rate $\alpha$ after contacting the disseminator. If ignorant person believe and propagate rumors after visiting the affected media, they will become disseminators at a certain rate $\beta$ . It is assumed that after the unknown person contacts the rumor information (including contact with people and knowing the rumor information from the media), they realize that the rumor information is untrue due to them own experience or discrimination ability. If an individual who is initially unaware of the rumors chooses to disseminate rumor refutation information, they will transition to the status of a rumor refuter at a rate of ${\xi }_{1}$ .
+
+Based on the above analysis, the rumor propagation process of ${XYZ} - {SIDR}$ model established in this paper is shown in Fig. 1.
+
+The meanings of symbols in Fig. 1 are shown in the following table. I.
+
+TABLE I
+
+DESCRIPTION OF PARAMETERS IN THE MODEL
+
+max width=
+
+$\mathbf{{Parameter}}$ Description
+
+1-2
+${\Lambda }_{1}$ The number of susceptible media entering the communication system per unit time.
+
+1-2
+${\Lambda }_{2}$ The number of ignorant individuals entering the communication system per unit time.
+
+1-2
+$\lambda$ The contact rate of susceptible medias with spreaders.
+
+1-2
+$\eta$ The probability of affected media becoming rumor refuting media.
+
+1-2
+$\alpha$ Rumor propagation rate of offline personal interaction.
+
+1-2
+$\beta$ Rumor propagation rate under two-tier network interaction.
+
+1-2
+$\delta$ The probability of propagating individuals becoming rumor refuting individuals.
+
+1-2
+$\theta$ The probability of propagating individuals becoming immune individuals.
+
+1-2
+${\xi }_{1}$ The rate of ignorant individuals becoming rumor refuting individuals.
+
+1-2
+${\xi }_{2}$ The rate of ignorant individuals becoming immune individuals.
+
+1-2
+$\phi$ The probability of rumor refuting individuals becoming immune individuals.
+
+1-2
+${\mu }_{1}$ The rate at which medias in the network move out of the propagation system.
+
+1-2
+${\mu }_{2}$ Migration rate of individuals in personal friendship network layer.
+
+1-2
+
+Based on the above analysis, we participated in the construction of an ${XYZ} - {SIDR}$ rumor propagation model. Then,
+
+$$
+\left\{ \begin{array}{l} {X}^{\prime } = {\Lambda }_{1} - {\lambda XI} - {\mu }_{1}X, \\ {Y}^{\prime } = {\lambda XI} - {\eta Y} - {\mu }_{1}Y, \\ {Z}^{\prime } = {\eta Y} - {\mu }_{1}Z, \\ {S}^{\prime } = {\Omega }_{2} - {\alpha SY} - {\beta SI} - \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {I + Y}\right) S - {\mu }_{2}S, \\ {I}^{\prime } = {\alpha SY} + {\beta SI} - \left( {\theta + \delta }\right) I - {\mu }_{2}I, \\ {D}^{\prime } = {\xi }_{1}S\left( {I + Y}\right) + {\delta I} - {\phi D} - {\mu }_{2}D, \\ {B}^{\prime } = {\xi }_{2}S\left( {I + Y}\right) + {\theta I} + {\delta D} - {\mu }_{2}B, \end{array}\right. \tag{1}
+$$
+
+ < g r a p h i c s >
+
+Fig 1. Schematic representation of the ${XYZ} - {SIDR}$ rumor spreading model
+
+Since the model represents the process of rumor propagation, the parameters involved are non negative, and the initial conditions are met:
+
+$$
+X\left( 0\right) = {X}_{0} \geq 0,Y\left( 0\right) = {Y}_{0} \geq 0,Z\left( 0\right) = {Z}_{0} \geq 0,
+$$
+
+$$
+S\left( 0\right) = {S}_{0} \geq 0,I\left( 0\right) = {I}_{0} \geq 0,D\left( 0\right) = {D}_{0} \geq 0\text{ , } \tag{2}
+$$
+
+$$
+R\left( 0\right) = {R}_{0} \geq 0\text{ . }
+$$
+
+§ III. MODEL ANALYSIS AND CALCULATION
+
+§ A.THE BASIC REPRODUCTION NUMBER ${R}_{0}$
+
+For system (1), the basic regeneration number ${R}_{0}$ is calculated as follows:
+
+Let $\mathcal{X} = {\left( I,Y,R,D,S,X,Z\right) }^{T}$ , equation (1) can be written as $\frac{d\mathcal{X}}{dt} = \mathcal{F}\left( \mathcal{X}\right) - \mathcal{V}\left( \mathcal{X}\right)$ .
+
+$$
+\mathcal{F}\left( \mathcal{X}\right) = \left( \begin{matrix} {\alpha SY} + {\beta SI} \\ {\lambda XI} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}\right) , \tag{3}
+$$
+
+$$
+\mathcal{V}\left( \mathcal{X}\right) = \left( \begin{matrix} {\theta I} + {\delta I} + {\mu }_{2}I \\ {\eta Y} + {\mu }_{1}Y \\ - {\xi }_{2}{SI} - {\xi }_{2}{SY} - {\theta I} - {\phi D} + {\mu }_{2}R \\ - {\xi }_{1}{SI} - {\xi }_{1}{SY} - {\delta I} + {\phi D} + {\mu }_{2}D \\ {H}_{1} \\ - {\Lambda }_{1} + {\lambda SI} + {\mu }_{1}X \\ - {\eta Y} + {\mu }_{1}Z \end{matrix}\right) \tag{4}
+$$
+
+where ${H}_{1} = - {\Lambda }_{2} + {\alpha SY} + {\beta SI} + {\xi }_{1}{SI} + {\xi }_{1}{SY} + {\xi }_{2}{SI} +$ ${\xi }_{2}{SY} + {\mu }_{2}S$ .
+
+Therefore
+
+$$
+F = \left( \begin{matrix} \beta \frac{{\Lambda }_{2}}{{\mu }_{2}} & \alpha \frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 \\ \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}\right) , \tag{5}
+$$
+
+$$
+V = \left( \begin{matrix} \theta + \delta + {\mu }_{2} & 0 & 0 & 0 \\ 0 & \eta + {\mu }_{1} & 0 & 0 \\ - {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} - \theta & - {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} & {\mu }_{2} & - \phi \\ - {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{2}} - \delta & - {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & \phi + {\mu }_{2} \end{matrix}\right) \tag{6}
+$$
+
+By calculation we can get
+
+$$
+F{V}^{-1} = \left( \begin{matrix} \frac{\beta {\Lambda }_{2}}{{\mu }_{2}\left( {\theta + \delta + {\mu }_{2}}\right) } & \frac{\alpha {\Lambda }_{2}}{{\mu }_{2}\left( {\eta + {\mu }_{1}}\right) } & 0 & 0 \\ \frac{\lambda {\Lambda }_{1}}{{\mu }_{1}\left( {\theta + \delta + {\mu }_{2}}\right) } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}\right) \tag{7}
+$$
+
+Hence, according to reference [27], the basic reproduction number of system (1) is the spectral radius of matrix $F{V}^{-1}$ as follows:
+
+$$
+{R}_{0} = \frac{\beta {\Lambda }_{2}}{{\mu }_{2}\left( {\theta + \delta + {\mu }_{2}}\right) } \tag{8}
+$$
+
+§ B. EXISTENCE OF EQUILIBRIUM
+
+According to the system dynamics equation (1), we can calculate the equilibrium $E = \left( {X,Y,Z,S,I,D,R}\right)$ . It is easy to see that the positive equilibrium points of system (1) are ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ and ${E}^{ * } =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ , and the rumor free equilibrium point ${E}_{0}$ always exists.
+
+Theorem 1 The equilibrium point ${E}^{ * }\; =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ exists if ${R}_{0} > 1$ and $\left( {\theta + \delta + {\mu }_{2}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) > {\beta \lambda }{\Lambda }_{2}.$
+
+Proof The rumors about system (1) have a balance point that satisfies:
+
+$$
+\left\{ \begin{array}{l} {\Lambda }_{1} - {\lambda XI} - {\mu }_{1}X = 0, \\ {\lambda XI} - {\eta Y} - {\mu }_{1}Y = 0, \\ {\eta Y} - {\mu }_{1}Z = 0, \\ {\Lambda }_{2} - {\alpha SY} - {\beta SI} - \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {I + Y}\right) S - {\mu }_{2}S = 0, \\ {\alpha SY} + {\beta SI} - \left( {\theta + \delta }\right) I - {\mu }_{2}I = 0, \\ {\xi }_{1}S\left( {I + Y}\right) + {\delta I} - {\phi D} - {\mu }_{2}D = 0, \\ {\xi }_{2}S\left( {I + Y}\right) + {\theta I} + {\phi D} - {\mu }_{2}R = 0. \end{array}\right. \tag{9}
+$$
+
+According to formula (9), ${X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{D}^{ * },{R}^{ * }$ are represented by ${I}^{ * }$ respectively and brought into the fifth equation to get
+
+$$
+a{I}^{2} + {bI} + c = 0 \tag{10}
+$$
+
+Where
+
+$$
+a = \lambda \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) ,
+$$
+
+$$
+b = \left( {\theta + \delta + {\mu }_{2}}\right) \left\lbrack {\left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack
+$$
+
+$$
++ \lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) - {\beta \lambda }{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \text{ , }
+$$
+
+$$
+c = \left( {\eta + {\mu }_{1}}\right) \left\lbrack {{\mu }_{2}\lambda \left( {\theta + \delta + {\mu }_{2}}\right) - {\mu }_{1}\beta {\Lambda }_{2}}\right\rbrack - {\alpha \lambda }{\Lambda }_{1}{\Lambda }_{2}.
+$$
+
+(11)
+
+It can be obtained by calculation that
+
+$$
+\Delta = {b}^{2} - {4ac}
+$$
+
+$$
+= {\left\lbrack \lambda {\Lambda }_{1}\left( \alpha + {\xi }_{2}\right) + \left( \eta + {\mu }_{1}\right) \left( {\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda \right) \right\rbrack }^{2}
+$$
+
+$$
+* {\left( \theta + \delta + {\mu }_{2}\right) }^{2} + {\left\lbrack \beta \lambda {\Lambda }_{2}\left( \eta + {\mu }_{1}\right) \right\rbrack }^{2} + {4\alpha }{\lambda }^{2}{\Lambda }_{1}{\Lambda }_{2}\left( {\beta + {\xi }_{1}}\right)
+$$
+
+$$
+* \left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) - {2\beta }{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right)
+$$
+
+$$
+* \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack
+$$
+
+$$
+- {4\lambda }\left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right)
+$$
+
+$$
+* \left\lbrack {{\mu }_{2}\lambda \left( {\theta + \delta + {\mu }_{2}}\right) - \beta {\mu }_{1}{\Lambda }_{2}}\right\rbrack
+$$
+
+(12)
+
+According to the discriminant calculation, when ${R}_{0} > 1$ and
+
+$\left( {\theta + \delta + {\mu }_{2}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) > {\beta \lambda }{\Lambda }_{2}$ , the negative solution is omitted:
+
+$$
+{I}^{ * } = \frac{{\beta \lambda }{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) - {H}_{2}\left( {\theta + \delta + {\mu }_{2}}\right) + \sqrt{\Delta }}{{2\lambda }\left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) \left( {\theta + \delta + {\mu }_{2}}\right) } \tag{13}
+$$
+
+where ${H}_{2} = \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack$ .
+
+Therefore ${E}^{ * } = \left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ , where
+
+$$
+{X}^{ * } = \frac{{\Lambda }_{1}}{\lambda {I}^{ * } + {\mu }_{1}}, \tag{14}
+$$
+
+$$
+{Y}^{ * } = \frac{\lambda {\Lambda }_{1}{I}^{ * }}{\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }, \tag{15}
+$$
+
+$$
+{Z}^{ * } = \frac{{\lambda \eta }{\Lambda }_{1}{I}^{ * }}{{\mu }_{1}\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }, \tag{16}
+$$
+
+$$
+{S}^{ * } = \frac{{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }{T}, \tag{17}
+$$
+
+$$
+{D}^{ * } = \frac{\lambda {\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) {I}^{*2} + \left\lbrack {\lambda {\Lambda }_{1}{\Lambda }_{2} + {\mu }_{1}\left( {\eta + {\mu }_{1}}\right) }\right\rbrack {I}^{ * }}{\left( {\phi + {\mu }_{2}}\right) T}, \tag{18}
+$$
+
+$$
+{R}^{ * } = \frac{{\xi }_{2}{\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) {H}_{3} + \theta {H}_{4}}{\left( {{\mu }_{2} - \phi }\right) {H}_{4}} \tag{19}
+$$
+
+where ${H}_{3} = \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) \left\lbrack {\lambda {\Lambda }_{1} + \left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }\right\rbrack ,{H}_{4} =$ $\lambda \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) {I}^{*2} + \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + }\right. }\right.$ $\left. \left. {{\mu }_{2}\lambda }\right) \right\rbrack {I}^{ * } + {\mu }_{2}\lambda \left( {\eta + {\mu }_{1}}\right)$ .
+
+§ C. STABILITY OF EQUILIBRIUM
+
+Theorem 2 The equilibrium point ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is locally asymptotically stable if ${R}_{0} < 1$ . And the equilibrium point ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is unstable if ${R}_{0} > 1$ .
+
+Proof The Jacobian matrix of system (1) at
+
+${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is
+
+$J\left( {E}_{0}\right) =$
+
+$$
+\left( \begin{matrix} - {\mu }_{1} & 0 & 0 & 0 & - \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & - \eta - {\mu }_{1} & 0 & 0 & \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & \eta & - {\mu }_{1} & 0 & 0 & 0 & 0 \\ 0 & {H}_{5} & 0 & - {\mu }_{2} & {H}_{6} & 0 & 0 \\ 0 & \alpha \frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {H}_{7} & 0 & 0 \\ 0 & {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} + \theta & - {\mu }_{2} & - {\mu }_{2} \\ 0 & {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{1}} & 0 & 0 & {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{1}} + \delta & 0 & {H}_{8} \end{matrix}\right)
+$$
+
+where ${H}_{5} = - \left( {\alpha + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}},{H}_{6} = - \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}}$ ,
+
+${H}_{7} = \beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - \left( {\theta + \delta + {\mu }_{2}}\right) ,{H}_{8} = - \left( {\phi + {\mu }_{2}}\right) .$
+
+The characteristic equation of matrix $J\left( {E}_{0}\right)$ is
+
+$\left| {J\left( {E}_{0}\right) - {hE}}\right| =$
+
+$$
+\left. \begin{matrix} {T}_{1} & 0 & 0 & 0 & - \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & {T}_{1} & 0 & 0 & \lambda \frac{{\Lambda }_{1}}{{\mu }_{1}} & 0 & 0 \\ 0 & \eta & {T}_{4} & 0 & 0 & 0 & 0 \\ 0 & {T}_{2} & 0 & {T}_{5} & {T}_{3} & 0 & 0 \\ 0 & \alpha \frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {T}_{4} & 0 & 0 \\ 0 & {\xi }_{2}\frac{{\Lambda }_{2}}{{\mu }_{2}} & 0 & 0 & {T}_{5} & - {\mu }_{2} - h & - {\mu }_{2} \\ 0 & {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{1}} & 0 & 0 & {T}_{6} & 0 & - \left( {\phi + {\mu }_{2}}\right) - h \end{matrix}\right|
+$$
+
+$= {\left( {\mu }_{1} + h\right) }^{2}{\left( {\mu }_{2} + h\right) }^{2}\left( {\phi + {\mu }_{2} + h}\right) \left( {\eta + {\mu }_{1} + h}\right) \left\lbrack {\beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - (\theta + }\right.$
+
+$$
+\left. {\left. {\delta + {\mu }_{2}}\right) - h}\right\rbrack = 0
+$$
+
+Where ${T}_{1} = - {\mu }_{1} - h,{T}_{2} = - \eta - {\mu }_{1} - h,{T}_{3} =$ $- \left( {\alpha + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}},{T}_{4} = - {\mu }_{1} - h,{T}_{5} = - {\mu }_{2} - h,{T}_{6} =$ $- \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) \frac{{\Lambda }_{2}}{{\mu }_{2}},{T}_{7} = \beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - \left( {\theta + \delta + {\mu }_{2}}\right) - h,{T}_{8} =$ $\beta \frac{{\Lambda }_{2}}{{\mu }_{2}} - \left( {\theta + \delta + {\mu }_{2}}\right) - h,{T}_{9} = {\xi }_{1}\frac{{\Lambda }_{2}}{{\mu }_{2}} + \delta .$
+
+Therefore, the characteristic root corresponding to the characteristic equation of $J\left( {E}_{0}\right)$ is:
+
+$$
+{h}_{01} = - {\mu }_{1} < 0,{h}_{02} = - {\mu }_{2} < 0,{h}_{03} = - \left( {\phi + {\mu }_{2}}\right) < 0,
+$$
+
+$$
+{h}_{04} = - \left( {\eta + {\mu }_{1}}\right) < 0,{h}_{05} = \frac{\theta + \delta + {\mu }_{2}}{{\mu }_{2}}\left( {{R}_{0} - 1}\right) < 0
+$$
+
+(20)
+
+According to Routh-Hurwitz stability criterion, the equilibrium point
+
+${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is locally asymptotically stable if ${R}_{0} < 1$ .
+
+And the equilibrium point ${E}_{0} = \left( {\frac{{\Lambda }_{1}}{{\mu }_{1}},0,0,\frac{{\Lambda }_{2}}{{\mu }_{2}},0,0,0}\right)$ is unstable if ${R}_{0} > 1$ .
+
+Theorem 3 The equilibrium point ${E}^{ * }\; =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ is locally asymptotically stable if ${R}_{0} > 1$ and $\beta {\Lambda }_{2} < {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) \left( {\theta + \delta + {\mu }_{2}}\right)$ , otherwise, the equilibrium point ${E}^{ * }$ is unstable.
+
+Proof The Jacobian matrix at ${E}^{ * } =$ $\left( {{X}^{ * },{Y}^{ * },{Z}^{ * },{S}^{ * },{I}^{ * },{D}^{ * },{R}^{ * }}\right)$ is
+
+$J\left( {E}^{ * }\right) =$
+
+$$
+\left( \begin{matrix} {A}_{1} & 0 & 0 & 0 & - \lambda {X}^{ * } & 0 & 0 \\ \lambda {I}^{ * } & {A}_{2} & 0 & 0 & \lambda {X}^{ * } & 0 & 0 \\ 0 & \eta & - {\mu }_{1} & 0 & 0 & 0 & 0 \\ 0 & {A}_{3} & 0 & {A}_{4} & {A}_{8} & 0 & 0 \\ 0 & \alpha {S}^{ * } & 0 & {A}_{5} & {A}_{9} & 0 & 0 \\ 0 & {\xi }_{2}{S}^{ * } & 0 & {A}_{6} & {\xi }_{2}{S}^{ * } + \theta & - {\mu }_{2} & - {\mu }_{2} \\ 0 & {\xi }_{1}{S}^{ * } & 0 & {A}_{7} & {\xi }_{1}{S}^{ * } + \delta & 0 & {A}_{10} \end{matrix}\right)
+$$
+
+Where ${A}_{1} = \lambda {I}^{ * } - {\mu }_{1},{A}_{2} = - \eta - {\mu }_{1},{A}_{3} =$
+
+$$
+- \left( {\alpha + {\xi }_{1} + {\xi }_{2}}\right) {S}^{ * },{A}_{4} = - \alpha {Y}^{ * } - \beta {I}^{ * },{A}_{5} = \alpha {Y}^{ * } + \beta {I}^{ * }\text{ , }
+$$
+
+$$
+{A}_{6} = {\xi }_{2}\left( {{I}^{ * } + {Y}^{ * }}\right) ,\;{A}_{7} = {\xi }_{1}\left( {{I}^{ * } + {Y}^{ * }}\right) ,\;{A}_{8} =
+$$
+
+$$
+- \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) {S}^{ * },{A}_{9} = \beta {S}^{ * } - \left( {\theta + \delta + {\mu }_{2}}\right) ,{A}_{10} =
+$$
+
+$- \left( {\phi + {\mu }_{2}}\right)$ .
+
+The characteristic equation of matrix $J\left( {E}^{ * }\right)$ is
+
+$\left| {J\left( {E}^{ * }\right) - {hE}}\right| =$
+
+$$
+\left| \begin{matrix} {B}_{1} & 0 & 0 & 0 & - \lambda {X}^{ * } & 0 & 0 \\ \lambda {I}^{ * } & {B}_{2} & 0 & 0 & \lambda {X}^{ * } & 0 & 0 \\ 0 & \eta & - {\mu }_{1} - h & 0 & 0 & 0 & 0 \\ 0 & {B}_{3} & 0 & {B}_{3} & {B}_{7} & 0 & 0 \\ 0 & \alpha {S}^{ * } & 0 & {B}_{4} & {B}_{8} & 0 & 0 \\ 0 & {\xi }_{2}{S}^{ * } & 0 & {B}_{5} & {\xi }_{2}{S}^{ * } + \theta & {B}_{9} & - {\mu }_{2} \\ 0 & {\xi }_{1}{S}^{ * } & 0 & {B}_{6} & {\xi }_{1}{S}^{ * } + \delta & 0 & {B}_{10} \end{matrix}\right|
+$$
+
+Where ${B}_{1} = \lambda {I}^{ * } - {\mu }_{1} - h,{B}_{2} = - \eta - {\mu }_{1} - h,{B}_{3} =$ $- \alpha {Y}^{ * } - \beta {I}^{ * } - h,{B}_{4} = \alpha {Y}^{ * } + \beta {I}^{ * },{B}_{5} = {\xi }_{2}\left( {{I}^{ * } + }\right.$ $\left. {Y}^{ * }\right) ,{B}_{6} = {\xi }_{1}\left( {{I}^{ * } + {Y}^{ * }}\right) ,{B}_{7} = - \left( {\beta + {\xi }_{1} + {\xi }_{2}}\right) {S}^{ * }$ , ${B}_{8} = \beta {S}^{ * } - \left( {\theta + \delta + {\mu }_{2}}\right) - h,{B}_{9} = - {\mu }_{2} - h,{B}_{10} =$ $- \left( {\phi + {\mu }_{2}}\right) - h$ .
+
+Thus, we can obtain
+
+$$
+\left| {J\left( {E}^{ * }\right) - {hE}}\right| = \left( {{\mu }_{1} + h}\right) \left( {{\mu }_{2} + h}\right) \left( {\phi + {\mu }_{2} + h}\right) \left( {\eta + {\mu }_{1} + }\right.
+$$
+
+$\left. h\right) \left( {\lambda {I}^{ * } + {\mu }_{1} + h}\right) G$ .
+
+Where $G = - \left\lbrack {\alpha {Y}^{ * } + \beta {I}^{ * } + \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {{I}^{ * } + {Y}^{ * }}\right) + {\mu }_{2}}\right\rbrack - h$ .
+
+Therefore, the characteristic root corresponding to the characteristic equation of $J\left( {E}^{ * }\right)$ is:
+
+$$
+{h}_{01} = - {\mu }_{1} < 0,{h}_{02} = - {\mu }_{2} < 0, \tag{21}
+$$
+
+$$
+{h}_{03} = - \left( {\phi + {\mu }_{2}}\right) < 0,{h}_{04} = - \left( {\eta + {\mu }_{1}}\right) < 0, \tag{22}
+$$
+
+$$
+{h}_{05} = - \left\lbrack {\alpha {Y}^{ * } + \beta {I}^{ * } + \left( {{\xi }_{1} + {\xi }_{2}}\right) \left( {{I}^{ * } + {Y}^{ * }}\right) + {\mu }_{2}}\right\rbrack < 0, \tag{23}
+$$
+
+$$
+{h}_{06} = \beta {S}^{ * } - \left( {\theta + \delta + {\mu }_{2}}\right) . \tag{24}
+$$
+
+Then, we take ${S}^{ * }$ into ${h}_{06}$ ,
+
+${h}_{06} = \frac{\beta {\Lambda }_{2}\left( {\eta + {\mu }_{1}}\right) \left( {\lambda {I}^{ * } + {\mu }_{1}}\right) }{\lambda \left( {\beta + {\xi }_{1}}\right) \left( {\eta + {\mu }_{1}}\right) {I}^{*2} + {C}_{1} + {\mu }_{2}\lambda \left( {\eta + {\mu }_{1}}\right) } - \left( {\theta + \delta + {\mu }_{2}}\right) ,$
+
+where ${C}_{1} = \left\lbrack {\lambda {\Lambda }_{1}\left( {\alpha + {\xi }_{2}}\right) + \left( {\eta + {\mu }_{1}}\right) \left( {{\mu }_{1}\beta + {\mu }_{1}{\xi }_{1} + {\mu }_{2}\lambda }\right) }\right\rbrack {I}^{ * }$ .
+
+§ IV. NUMERICAL SIMULATION
+
+In this section, we will assign reasonable values to the parameters in system (1) as established in this paper, and verify the results of our theoretical analysis through numerical simulations. On the one hand, we combine some similar examples in reality. On the other hand, the parameter values in relevant literature are referred to.
+
+Order ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\lambda = {0.01},\eta = {0.3},\alpha = {0.01},\beta =$ ${0.01},\theta = {0.2},\delta = {0.2},\phi = {0.15},{\xi }_{1} = {0.1},{\xi }_{2} = {0.1},{\mu }_{1} =$ ${0.2},{\mu }_{2} = {0.2}$ . Calculated ${R}_{0} = {0.8333} < 1$ , then the no rumor propagation equilibrium point ${E}_{0}$ is stable.
+
+ < g r a p h i c s >
+
+Fig 2. Stability of equilibrium point ${E}_{0}$ .
+
+Fig. 2 shows when ${R}_{0} = {0.0833} < 1$ , the density of each subclass in the model changes with time. At first, the number of unaffected media and unknowns gradually decreased at a similar rate, and finally stabilized. Due to the limited number of moving in and the large number of moving out, the number of affected media and communicators gradually decreases at a similar rate and finally becomes 0 . The number of rumor refuting media and rumor refuters first increased with the increase of the number of affected media and disseminators. It gradually decreases over time and finally becomes 0 . The number of immunized persons increased with the increase of the number of communicators and rumor refuters. The growth rate gradually slowed down and finally stabilized. Namely, the rumor disappears and reaches a stable equilibrium point, and there is no rumor.
+
+Let ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\lambda = {0.2},\eta = {0.3},\alpha = {0.5},\beta =$ ${0.6},\theta = {0.4},\delta = {0.4},\phi = {0.15},{\xi }_{1} = {0.2},{\xi }_{2} = {0.2},{\mu }_{1} =$ ${0.2},{\mu }_{2} = {0.2}$ and calculate ${R}_{0} = 3 > 1$ , the equilibrium point ${E}^{ * }$ is stable, as shown in Fig. 3.
+
+ < g r a p h i c s >
+
+Fig 3. Stability of equilibrium point ${E}^{ * }$ .
+
+In Fig. 3, considering the media network layer, due to the small number of new media entering the communication system and the transformation of some unaffected media into affected media, the number of unaffected media gradually decreases and tends to stabilize after a period of time. Originally, the number of affected media increased due to the transformation of some unaffected media into affected media. Over time, most of the affected media were transformed into rumor refuting media, so the number of affected media decreased and stabilized. As the affected media changed into rumor refuting media, the number of rumor refuting media increased and gradually stabilized.
+
+Fig. 3 illustrates that, within the individual interpersonal network layer, the number of ignorant individuals begins to decline. Initially, the low influx of new individuals and a fixed rate of departures contribute to this decrease. Additionally, some ignorant individuals transition to become communicators, while others become immune or rumor refuters. Consequently, the number of communicators increases as ignorant individuals transform into communicators. Over time, as communicators transition to immune individuals or rumor refuters, the number of communicators gradually decreases and eventually stabilizes. As more communicators and ignorant individuals become rumor refuters, the number of rumor refuters rises and stabilizes. Simultaneously, with some ignorant individuals, communicators, and rumor refuters becoming immune, the number of immune individuals significantly increases and gradually stabilizes. Ultimately, the model reaches a steady state, with each groups number stabilizing over time.
+
+Fig. 4 to Fig. 7 depict ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\eta = {0.3},\theta =$ ${0.4},\phi = {0.15},{\xi }_{1} = {0.2},{\xi }_{2} = {0.5},{\mu }_{1} = {0.2},{\mu }_{2} = {0.2}$ , the evolution of the density of $X\left( t\right) ,Y\left( t\right)$ and $S\left( t\right)$ with different parameters.
+
+Fig. 4 and Fig. 5 describe the effect of parameter $\lambda$ on the density change of $X\left( t\right)$ and $Y\left( t\right)$ respectively. Parameter $\lambda$ represents the probability that the unaffected media will be transformed into the affected media. It can be seen from the figure that the parameter $\lambda$ has a negative correlation with the density of $X\left( t\right)$ and a positive correlation with the density of $Y\left( t\right)$ . That is, with the increase of the parameter $\lambda$ , the rate of transformation from unaffected media to affected media increases. The number of unaffected media decreases gradually, and the number of affected media increases gradually, accelerating the spread of rumors in the media network layer.
+
+ < g r a p h i c s >
+
+Fig 4. Density of $X\left( t\right)$ under the parameter $\lambda$ .
+
+ < g r a p h i c s >
+
+Fig 5. Density of $Y\left( t\right)$ under the parameter $\lambda$ .
+
+Fig. 6 and Fig. 7 describe the influence of parameters $\alpha$ and $\beta$ on the density change of $S\left( t\right)$ respectively. Parameter $\alpha$ represents the probability that the unknown person will become a spreader by accessing the affected media, and parameter $\beta$ represents the probability that the unknown person will become a spreader by contacting spreaders. It can be seen from the figure that the density of $S\left( t\right)$ decreases with the increase of parameters $\alpha$ and $\beta$ . Namely, with the increase of the propagation rate of individual network layer and double-layer network interaction, the number of unknowns gradually decreases, which accelerates the spread of rumors in the double-layer network.
+
+ < g r a p h i c s >
+
+Fig 6. Density of $S\left( t\right)$ under the parameter $\alpha$ .
+
+ < g r a p h i c s >
+
+Fig 7. Density of $S\left( t\right)$ under the parameter $\beta$ .
+
+Fig. 8 and Fig. 9 describe when ${\Lambda }_{1} = 1,{\Lambda }_{2} = 1,\eta =$ ${0.3},\theta = {0.4},\phi = {0.15},{\xi }_{1} = {0.2},{\xi }_{2} = {0.5},{\mu }_{1} = {0.2},{\mu }_{2} =$ 0.2, the evolution of the density of $I\left( t\right)$ with different parameters.
+
+Fig. 8 and Fig. 9 describe the influence of parameters $\alpha$ and $\beta$ on the density change of $I\left( t\right)$ respectively. Considering the meaning of parameters $\alpha$ and $\beta$ , it is easy to know that the values of parameters $\alpha$ and $\beta$ are positively correlated with the density of $I\left( t\right)$ . As can be seen from the figure, the density of $I\left( t\right)$ increases with the increase of parameters $\alpha$ and $\beta$ . That is, the increasing number of communicators promotes the expansion of the scale of communication, which is not conducive to the control of rumors.
+
+§ V. CONCLUSION
+
+At present, many scholars have separately studied the influence of media refutation or individual refutation on the spread of rumors. We believe that considering these two effects comprehensively is better than considering one of them alone. This paper integrates both media refutation and individual refutation into the analysis and introduces a novel ${XYZ} - {SIDR}$ two-tier rumor propagation model, further demonstrates the existence and stability of equilibrium points within the model. The research results show that this two-layer network model is more effective in controlling the spread of rumors.
+
+ < g r a p h i c s >
+
+Fig 8. Density of $I\left( t\right)$ under the parameter $\alpha$ .
+
+ < g r a p h i c s >
+
+Fig 9. Density of $I\left( t\right)$ under the parameter $\beta$ .
+
+Theoretical analysis indicates that integrating both media and individual rumor refutation exerts a more significant and broader impact on rumor propagation. We suggest strengthening the dissemination of rumor refutation information through the official media rather than relying solely on individuals to control the spread of rumor. The research conclusion can help relevant departments formulate effective measures to control the spread of rumors. On the other hand, the model established in this paper can also be analogically applied to the study of infectious disease model.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/CxWEOEhqo6/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/CxWEOEhqo6/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..466ba33b71181f695005b75923e605bfe983d813
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/CxWEOEhqo6/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,427 @@
+# Asynchronous Thruster Fault Detection for Unmanned Marine Vehicles under DoS Attacks
+
+Fuxing Wang
+
+School of Automation Engineering
+
+University of Electronic Science and Technology of China
+
+Chengdu 611731, China
+
+wfx614328@163.com
+
+Yue Long
+
+School of Automation Engineering
+
+University of Electronic Science and Technology of China
+
+Chengdu 611731, China
+
+longyue@uestc.edu.cn
+
+Tieshan Li
+
+School of Automation Engineering University of Electronic Science and Technology of China Chengdu 611731, China
+
+tieshanli@126.com
+
+Abstract-This paper investigates a thruster fault detection strategy for unmanned marine vehicles (UMVs) subjected to external disturbances and aperiodic Denial of Service (DoS) attacks. To address the challenge of timely detection of DoS attacks, the UMV and the corresponding filters are modeled within the framework of an asynchronous switched system. Sufficient conditions ensuring the system's exponential stability and prescribed performance are derived using model-dependent average dwell time and piecewise Lyapunov functions. Additionally, the tolerable lower bound of the sleep interval and the upper bound of the attack interval for DoS attacks are established. Solvable conditions for the designed fault detection filters are obtained by leveraging decoupling techniques. Finally, simulations conducted on a UMV validate the effectiveness of the proposed methods.
+
+Index Terms-Unmanned marine vehicles, asynchronous switched system, DoS attacks, fault detection.
+
+## I. INTRODUCTION
+
+In recent years, unmanned marine vehicles (UMVs) have attracted significant attention in marine science and technology due to their wide-ranging applications in marine exploration, environmental monitoring, and resource development [1]. Nevertheless, the operational environment for UMVs is inherently complex, and their reliance on wireless communication networks for communication with shore-based centers makes them vulnerable to external disturbances, equipment malfunctions, cyber-attacks, and other disruptions [2]. The unpredictable nature of potential harm caused by these disturbances or faults, combined with the inherent vulnerabilities of cyberspace, renders UMV systems particularly susceptible to cyber-attacks. These risks can result in system failures and potentially catastrophic accidents [3]. As a result, improving the reliability and security of UMVs has emerged as a crucial area of research and development.
+
+The unpredictable nature of potential harm caused by disturbances or faults to unmanned marine vehicles (UMVs) underscores the critical need for a real-time fault detection (FD) warning mechanism. The core of fault detection methodology involves comparing system performances to identify fault signals. Current research predominantly focuses on model-based fault detection, which has shown significant success in various systems, including continuous-discrete systems [4], T-S fuzzy systems [5], and Markovian jump systems [6]. The primary approach involves generating residual signals through filters or observers and subsequently establishing a fault warning mechanism. For UMVs, several studies have made noteworthy contributions. [7] has explored the design of controllers and FD filters based on observers for networked UMVs, [8] proposed event-triggered fault detection mechanisms for UMVs in networked environments, and [2] utilized T-S fuzzy systems to model UMV systems, particularly addressing fault detection under replay attacks. Despite these advancements, the scope of fault detection research for UMVs remains relatively narrow and lacks comprehensive coverage [9]. Consequently, further investigation into robust and holistic fault detection strategies for UMVs is imperative to enhance their reliability and operational safety [10].
+
+On the other hand, due to the openness of cyberspace, UMV systems are particularly vulnerable to cyber-attacks. Deception attacks and Denial of Service (DoS) attacks are currently common types of attacks [11]. Deception attacks involve sending incorrect or tampered data to the system [12], including replay attacks [13] and false data injection attacks [14]. Compared to deception attacks, DoS attacks cause signal transmission to be unavailable for a period, leaving the system in an open-loop state, which makes it easier to cause severe disruption in system operations. Consequently, numerous studies on DoS attacks have emerged [15], [16].
+
+However, most existing research assumes that Denial of Service (DoS) attacks can be detected promptly, suggesting that the switching of filters corresponding to each subsystem happens simultaneously with the subsystem switching [10], [17]. However, in practical applications, detecting DoS attacks in a timely manner proves challenging, leading to delays. This delay implies that the filter often takes additional time to adjust to the appropriate control mode based on the subsystem mode, resulting in asynchronous filter/subsystem switching [18]. As a result, filters designed for synchronous switching may not provide optimal detection performance in real-world scenarios [19]. Thus, incorporating asynchronous switching into thruster fault detection for unmanned marine vehicles (UMVs) under DoS attacks is of substantial practical significance.
+
+---
+
+This work is supported in part by the National Natural Science Foundation of China under Grants 62273072, 51939001. (Corresponding author: Yue Long)
+
+---
+
+Inspired by the previous discussion, this paper investigates thruster fault detection (FD) for unmanned marine vehicles (UMVs) under Denial of Service (DoS) attacks using an asynchronous switched method to enhance reliability and security. Addressing the challenge of timely DoS attack detection, the paper proposes an asynchronous switched filter specifically designed for thruster fault detection. Furthermore, leveraging model-dependent average dwell time (MDADT) and piecewise Lyapunov functions (PLF), the paper establishes the tolerable lower bound of the sleep interval and the upper bound of the attack interval for DoS attacks. The filter parameters are determined based on linear solvability conditions. The effectiveness of the proposed method is ultimately validated through simulation.
+
+## II. Problem formulation and modeling
+
+### A.UMV Model
+
+Consider the UMV and the following body-fixed equations of motion
+
+$$
+M\dot{\delta }\left( t\right) + {N\delta }\left( t\right) + {R\psi }\left( t\right) = {E\varphi }\left( t\right) , \tag{1}
+$$
+
+$$
+\dot{\psi }\left( t\right) = J\left( {\eta \left( t\right) }\right) \delta \left( t\right) ,
+$$
+
+where $\delta \left( t\right) = {\left\lbrack {\delta }_{u}\left( t\right) ,{\delta }_{v}\left( t\right) ,{\delta }_{r}\left( t\right) \right\rbrack }^{T}$ with ${\delta }_{u}\left( t\right) ,{\delta }_{v}\left( t\right) ,{\delta }_{r}\left( t\right)$ representing the surge, sway and yaw velocities, respectively. $\psi \left( t\right) = {\left\lbrack {x}_{p}\left( t\right) ,{y}_{p}\left( t\right) ,\eta \left( t\right) \right\rbrack }^{T}$ with ${x}_{p}\left( t\right)$ and ${y}_{p}\left( t\right)$ are positions and $\eta \left( t\right)$ is the yaw angle. $\varphi \left( t\right)$ is the control input. $M, N, R$ and $E$ denote inertia, damping, mooring forces and configuration matrices, and $M$ is a symmetric positive-definite and invertible matrix that satisfies $M = {M}^{T} > 0$ ,
+
+$J\left( {\eta \left( t\right) }\right) = \left\lbrack \begin{matrix} \cos \left( {\eta \left( t\right) }\right) & - \sin \left( {\eta \left( t\right) }\right) & 0 \\ \sin \left( {\eta \left( t\right) }\right) & \cos \left( {\eta \left( t\right) }\right) & 0 \\ 0 & 0 & 1 \end{matrix}\right\rbrack .$
+
+Then, by defining $x\left( t\right) = \delta \left( t\right) - {\delta }_{\text{ref }}, A\left( t\right) =$ $- M{\left( t\right) }^{-1}N\left( t\right) ,{B}_{1}\left( t\right) = M{\left( t\right) }^{-1}R$ and ${B}_{2}\left( t\right) = M{\left( t\right) }^{-1}E$ , and taking into account the unavoidable disturbance $\widetilde{d}\left( t\right)$ caused by wind, wave and current, the system (1) can be expressed as
+
+$$
+\left\{ \begin{array}{l} \dot{x}\left( t\right) = {Ax}\left( t\right) + {B}_{1}d\left( t\right) + {B}_{2}\varphi \left( t\right) , \\ y\left( t\right) = {Cx}\left( t\right) , \end{array}\right. \tag{2}
+$$
+
+where $d\left( t\right) = {B}_{1}{\left( t\right) }^{-1}{d}^{ * }\left( t\right) - \psi \left( t\right) + {B}_{1}{\left( t\right) }^{-1}A{\delta }_{\text{ref }}$ and $C =$ $\left\lbrack \begin{array}{lll} 0 & 0 & 1 \end{array}\right\rbrack$ denotes the output matrix.
+
+Consider thruster fault ${\varphi }^{F}\left( t\right) = {\rho \varphi }\left( t\right) + {\sigma f}\left( t\right)$ and assume control inputs $\varphi \left( t\right) = {Kx}\left( t\right)$ are designed,(2) is represented
+
+as
+
+$$
+\left\{ \begin{array}{l} \dot{x}\left( t\right) = \widehat{A}x\left( t\right) + {B}_{1}d\left( t\right) + {B}_{2}\widehat{f}\left( t\right) , \\ y\left( t\right) = {Cx}\left( t\right) , \end{array}\right. \tag{3}
+$$
+
+where $\widehat{A} = A + {B}_{2}K$ and $\widehat{f}\left( t\right) = - \bar{\rho }\varphi \left( t\right) + {\sigma f}\left( t\right)$ .
+
+### B.DoS Attacks Model
+
+Consider the aperiodic dos attacks as follows:
+
+$$
+{A}_{\text{Dos }} = \left\{ \begin{matrix} 0, & t \in \left\lbrack {{t}_{2l},{t}_{{2l} + 1}}\right) \triangleq {\kappa }_{0,{2l}} \\ 1, & t \in \left\lbrack {{t}_{{2l} + 1},{t}_{2\left( {l + 1}\right) }}\right) \triangleq {\kappa }_{1,{2l}} \end{matrix}\right. \tag{4}
+$$
+
+where $t \in \left\lbrack {{t}_{2l},{t}_{{2l} + 1}}\right) \triangleq {\kappa }_{0,{2l}}\;\left( {l \in \mathrm{N},{t}_{2l} \geq 0}\right)$ indicates the ${l}^{th}$ sleep interval with the length ${s}_{l} = {t}_{{2l} + 1} - {t}_{2l}$ , and $t \in \left\lbrack {{t}_{{2l} + 1},{t}_{2\left( {l + 1}\right) }}\right) \triangleq {\kappa }_{1,{2l}}$ indicates the ${l}^{th}$ DoS attacks interval with the length ${d}_{l} = {t}_{2\left( {l + 1}\right) } - {t}_{{2l} + 1}$ .
+
+Due to the communication disruption caused by DoS attacks, the UMV system (3) can be augmented into the following switched system, which has been discretized. The sleeping interval can be expressed as $k \in \left\lbrack {{k}_{2l},{k}_{{2l} + 1}}\right)$ , and the DoS attacks interval can be expressed as $k \in \left\lbrack {{k}_{{2l} + 1},{k}_{2\left( {l + 1}\right) }}\right)$ .
+
+$$
+\left\{ \begin{array}{l} x\left( {k + 1}\right) = {A}_{id}x\left( k\right) + {B}_{1id}d\left( k\right) + {B}_{2id}\widehat{f}\left( k\right) \\ y\left( k\right) = {C}_{d}x\left( k\right) \end{array}\right. \tag{5}
+$$
+
+## C. Asynchronous Switching Filter
+
+In the case of the DoS attacks and thruster faults, the residual signal produced by the switched filter is as follows:
+
+$$
+\left\{ {\begin{array}{l} {x}_{f}\left( {k + 1}\right) = {A}_{fi}{x}_{f}\left( k\right) + {B}_{fi}y\left( k\right) \\ r\left( k\right) = {C}_{fi}{x}_{f}\left( k\right) + {D}_{fi}y\left( k\right) \end{array}\left( {i = 0,1}\right) }\right. \tag{6}
+$$
+
+where ${x}_{f}\left( k\right)$ is the state of the filters, $r\left( k\right)$ is the residual signal of the switched system (5). Define $\widetilde{x}\left( k\right) =$ ${\left\lbrack \begin{array}{ll} {x}^{T}\left( k\right) & {x}_{f}^{T}\left( k\right) \end{array}\right\rbrack }^{T},\varpi \left( k\right) = {\left\lbrack \begin{array}{ll} {d}^{T}\left( k\right) & {f}^{T}\left( k\right) \end{array}\right\rbrack }^{T}$ and the residual evaluation signal $e\left( k\right) = r\left( k\right) - \widehat{f}\left( k\right) ,\left( 6\right)$ is rewritten as (7)
+
+$$
+{\Phi }_{0} : \left\{ {\begin{array}{l} \widetilde{x}\left( {k + 1}\right) = {\widetilde{A}}_{i}\widetilde{x}\left( k\right) + {\widetilde{B}}_{i}\varpi \left( k\right) \\ e\left( k\right) = {\widetilde{C}}_{i}\widetilde{x}\left( k\right) + {\widetilde{D}}_{i}\varpi \left( k\right) \end{array}, k \in \left\lbrack {{k}_{l} + {\varepsilon }_{l},{k}_{l + 1}}\right) }\right.
+$$
+
+$$
+{\Phi }_{1} : \left\{ {\begin{array}{l} \widetilde{x}\left( {k + 1}\right) = {\widetilde{A}}_{ij}\widetilde{x}\left( k\right) + {\widetilde{B}}_{ij}\varpi \left( k\right) \\ e\left( k\right) = {\widetilde{C}}_{ij}\widetilde{x}\left( k\right) + {\widetilde{D}}_{ij}\varpi \left( k\right) \end{array}, k \in \left\lbrack {{k}_{l},{k}_{l} + {\varepsilon }_{l}}\right) }\right.
+$$
+
+where $i \neq j, i \in \{ 0,1\} , j \in \{ 0,1\} ,{\widetilde{A}}_{ij} = \left\lbrack \begin{matrix} {A}_{id} & 0 \\ {B}_{fj}{C}_{d} & {A}_{fj} \end{matrix}\right\rbrack$ , ${\widetilde{B}}_{ij} = \left\lbrack \begin{matrix} {B}_{1i} & {B}_{2i} \\ 0 & 0 \end{matrix}\right\rbrack ,{\widetilde{C}}_{ij} = \left\lbrack \begin{array}{ll} {D}_{fj}{C}_{d} & {C}_{fj} \end{array}\right\rbrack$ and ${\widetilde{D}}_{ij} =$ $\left\lbrack \begin{array}{ll} 0 & - \bar{I} \end{array}\right\rbrack$ .
+
+To better set the stage for the next section, the following definitions are presented.
+
+Definition 1: For any switching signal $\tau \left( k\right)$ and $0 < {k}_{0} \leq$ $k$ , let ${\mathcal{M}}_{\tau , l}\left( {{k}_{0}, k}\right)$ indicate the number of switching times that the ${l}_{th}$ subsystem is activated over $\left\lbrack {{k}_{0}, k}\right)$ . If
+
+$$
+{M}_{\tau , l}\left( {{k}_{0}, k}\right) \leq {N}_{{\mathcal{M}}_{0, l}} + \frac{{N}_{l}\left( {{k}_{0}, k}\right) }{{\lambda }_{l}}
+$$
+
+holds for scalar ${\lambda }_{l} > 0$ and integer ${N}_{{M}_{0, l}} \geq 0$ , then ${\lambda }_{l}$ is called model-dependent average dwell time. ${N}_{l}\left( {{k}_{0}, k}\right)$ is the total running time of the ${l}_{th}$ subsystem over $\left\lbrack {{k}_{0}, k}\right)$ .
+
+Definition 2: Consider asynchronous switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ , and given scalar $\alpha ,\beta$ , and $\gamma$ satisfying $0 < \alpha < 1$ , $\beta > 0$ and $\gamma > 0$ . Under zero initial condition, if the asynchronous switched system is exponentially stable and satisfies $\mathop{\sum }\limits_{{s = {k}_{0}}}^{\infty }{\left( 1 - \alpha \right) }^{s}{e}^{\mathrm{T}}\left( s\right) e\left( s\right) \leq {\gamma }^{2}\mathop{\sum }\limits_{{s = {k}_{0}}}^{\infty }{\varpi }^{\mathrm{T}}\left( s\right) \varpi \left( s\right)$ , it is said that the system exhibits exponential stability and has exponential ${H}_{\infty }$ index $\gamma$ .
+
+## III. Main Results
+
+In this section, the stability and ${H}_{\infty }$ performance of asynchronous switched systems (7) will be analyzed, and the sufficient and linearly solvable conditions for the designed switched FD filters are given.
+
+Theorem 1: Consider the switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ under DoS attacks, scalars ${\alpha }_{i},{\beta }_{i},\gamma ,{\mu }_{0}$ and ${\mu }_{1}$ satisfying $0 < {\alpha }_{i} < 1,{\beta }_{i} > 0,\gamma > 0,{\mu }_{0} > 1$ and $0 < {\mu }_{1} < 1$ , if there exist symmetric positive-definite matrices ${\mathcal{P}}_{i}$ satisfying the following conditions
+
+$$
+{\widetilde{A}}_{i}^{T}{\mathcal{P}}_{i}{\widetilde{A}}_{i} - {\mathcal{P}}_{i} + {\alpha }_{i}{\mathcal{P}}_{i} < 0, \tag{8}
+$$
+
+$$
+{\widetilde{A}}_{ij}^{T}{\mathcal{P}}_{i}{\widetilde{A}}_{ij} - {\mathcal{P}}_{i} - {\beta }_{i}{\mathcal{P}}_{i} < 0, \tag{9}
+$$
+
+$$
+{\mathcal{P}}_{i} \leq {\mu }_{i}{\mathcal{P}}_{j} \tag{10}
+$$
+
+$$
+{\tau }_{D} < \frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{\ln {\widetilde{\alpha }}_{1}},{\tau }_{F} > - \frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{\ln {\widetilde{\alpha }}_{0}}, \tag{11}
+$$
+
+the switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are exponentially asymptotically stable with the exponential ${H}_{\infty }$ performance, where $i \neq j,{\widetilde{\alpha }}_{i} = 1 - {\alpha }_{i},{\widetilde{\beta }}_{i} = 1 + {\beta }_{i},{\phi }_{i} = \frac{{\widetilde{\beta }}_{i}}{{\widetilde{\alpha }}_{i}}$ and ${\varepsilon }_{M}$ denotes the maximum time that the filter lags the subsystem.
+
+Proof: The piecewise Lyapunov function for the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are given as follows
+
+$$
+{\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) = {\widetilde{x}}^{T}\left( k\right) {\mathcal{P}}_{i}\widetilde{x}\left( k\right) . \tag{12}
+$$
+
+When $\varpi \left( k\right) = 0$ and $k \in \left\lbrack {{k}_{2l},{k}_{{2l} + 1}}\right)$ , it can be obtained
+
+$$
+\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \leq \left\{ \begin{array}{l} {\widetilde{\alpha }}_{i}^{k - {k}_{2l} - {\varepsilon }_{2l}}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( {{k}_{2l} + {\varepsilon }_{2l}}\right) }\right) , k \in {\Gamma }^{ + } \\ {\widetilde{\beta }}_{i}^{k - {k}_{2l}}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( {k}_{2l}\right) }\right) , k \in {\Gamma }^{ - } \end{array}\right. \tag{13}
+$$
+
+where ${\widetilde{\alpha }}_{i} = 1 - {\alpha }_{i}$ and ${\widetilde{\beta }}_{i} = 1 + {\beta }_{i}$ . And when $k \in {\mathcal{T}}^{ + }\left( {{k}_{2l},{k}_{{2l} + 1}}\right)$ , from (8) and (11), it can be derived
+
+$$
+\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \leq {\widetilde{\alpha }}_{0}^{k - {k}_{2l} - {\varepsilon }_{2l}}{\mathcal{V}}_{0}\left( {\widetilde{x}\left( {{k}_{2l} + {\varepsilon }_{2l}}\right) }\right)
+$$
+
+$$
+\leq {\widetilde{\alpha }}_{0}^{k - {k}_{2l} - {\varepsilon }_{2l}} \cdot {\widetilde{\beta }}_{0}^{{\varepsilon }_{2l}} \cdot {\mathcal{V}}_{0}\left( {\widetilde{x}\left( {k}_{2l}\right) }\right)
+$$
+
+$$
+< \cdots
+$$
+
+$$
+\leq \theta \exp \left\{ {\max \left( {\frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{{\tau }_{F}} + {v}_{0}, - \frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{{\tau }_{D}} + {v}_{1}}\right) }\right.
+$$
+
+$$
+\left. \left( {{\Xi }_{F}\left( {{k}_{0}, k}\right) + {\Xi }_{D}\left( {{k}_{0}, k}\right) }\right) \right\} \mathcal{V}\left( {\widetilde{x}\left( {k}_{0}\right) }\right)
+$$
+
+(14)
+
+where $\theta = \exp \left\lbrack {\left( {{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}\right) {\xi }_{F} - \left( {{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}\right) {\xi }_{D}}\right\rbrack$ , $\omega = \max \left\{ {-\frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{{\tau }_{F}} - \ln {\widetilde{\alpha }}_{0},\frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{{\tau }_{D}} - \ln {\widetilde{\alpha }}_{1}}\right\} ,$ ${\chi }_{0} = {\theta }_{0}^{{\varepsilon }_{M}}{\mu }_{0},{\chi }_{1} = {\theta }_{1}^{{\varepsilon }_{M}}{\mu }_{1},{v}_{i} = \ln {\widetilde{\alpha }}_{i}.$
+
+From (11), it has $\omega > 0$ . Then, it is clear that $\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right)$ converges to zero when $k \rightarrow \infty$ . Therefore, the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are exponentially asymptotically stable when (8) and (11) hold.
+
+Next, if $\varpi \left( k\right) \neq 0$ for $k \in \left\lbrack {{k}_{2l},{k}_{{2l} + 1}}\right)$ and zero initial conditions, (??) is derived as follows
+
+$$
+\Delta {\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) < \left\{ \begin{array}{l} - {\alpha }_{i}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) - \Upsilon \left( k\right) , k \in {\Gamma }^{ + } \\ {\beta }_{i}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) - \Upsilon \left( k\right) , k \in {\Gamma }^{ - } \end{array}\right. \tag{15}
+$$
+
+where $i = 0,1,\Upsilon \left( k\right) = {e}^{T}\left( k\right) e\left( k\right) - {\gamma }^{2}{\varpi }^{T}\left( k\right) \varpi \left( k\right)$ . When $k \in {\mathcal{T}}^{ + }\left( {{k}_{2l},{k}_{{2l} + 1}}\right)$ , it can have the following inequality in the similar way from (10) and (15)
+
+$$
+\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \leq {\widetilde{\alpha }}_{0}^{k - {k}_{2l}}{\widetilde{\alpha }}_{0}^{{k}_{{2l} - 1} - {k}_{{2l} - 2}}\cdots {\widetilde{\alpha }}_{0}^{{k}_{1} - {k}_{0}}{\phi }_{0}^{{\varepsilon }_{2l}}{\phi }_{0}^{{\varepsilon }_{{2l} - 2}}\cdots {\phi }_{0}^{{\varepsilon }_{0}}.
+$$
+
+$$
+{\mu }_{0}^{{\mathrm{M}}_{F}\left( {{k}_{0}, k}\right) }{\widetilde{\alpha }}_{1}^{{k}_{2l} - {k}_{{2l} - 1}}\cdots {\widetilde{\alpha }}_{1}^{{k}_{2} - {k}_{1}}{\phi }_{1}^{{\varepsilon }_{{2l} - 1}}\cdots {\phi }_{1}^{{\varepsilon }_{1}}.
+$$
+
+$$
+{\mu }_{1}^{{\mathrm{M}}_{D}\left( {{k}_{0}, k}\right) }\mathcal{V}\left( {\widetilde{x}\left( {k}_{0}\right) }\right) - {\widetilde{\alpha }}_{0}^{k - {k}_{2l}}{\widetilde{\alpha }}_{0}^{{k}_{{2l} - 1} - {k}_{{2l} - 2}}\ldots
+$$
+
+$$
+{\widetilde{\alpha }}_{0}^{{k}_{1} - {k}_{0}}{\phi }_{0}^{{\varepsilon }_{2l}}{\phi }_{0}^{{\varepsilon }_{{2l} - 2}}\cdots {\phi }_{0}^{{\varepsilon }_{0}}{\mu }_{0}^{{\mathrm{M}}_{F}\left( {{k}_{0}, k}\right) }{\widetilde{\alpha }}_{1}^{{k}_{2l} - {k}_{{2l} - 1}}\cdots
+$$
+
+$$
+{\widetilde{\alpha }}_{1}^{{k}_{2} - {k}_{1}}{\phi }_{1}^{{\varepsilon }_{{2l} - 1}}\cdots {\phi }_{1}^{{\varepsilon }_{1}}{\mu }_{1}^{{\mathrm{M}}_{D}\left( {{k}_{0}, k}\right) }\mathop{\sum }\limits_{{s = {k}_{0} + {\Delta }_{0}}}^{{{k}_{1} - 1}}{\widetilde{\alpha }}_{0}^{{k}_{1} - s - 1}\Upsilon \left( s\right)
+$$
+
+$$
+- {\widetilde{\alpha }}_{0}^{k - {k}_{2l}}{\widetilde{\alpha }}_{0}^{{k}_{{2l} - 1} - {k}_{{2l} - 2}}\cdots {\widetilde{\alpha }}_{0}^{{k}_{1} - {k}_{0}}{\phi }_{0}^{{\varepsilon }_{2l}}{\phi }_{0}^{{\varepsilon }_{{2l} - 2}}\cdots {\phi }_{0}^{{\varepsilon }_{0}}.
+$$
+
+$$
+{\mu }_{0}^{{\mathrm{M}}_{F}\left( {{k}_{0}, k}\right) }{\widetilde{\alpha }}_{1}^{{k}_{2l} - {k}_{{2l} - 1}}\cdots {\widetilde{\alpha }}_{1}^{{k}_{2} - {k}_{1}}{\phi }_{1}^{{\varepsilon }_{{2l} - 1}}\cdots {\phi }_{1}^{{\varepsilon }_{1}}{\mu }_{1}^{{\mathrm{M}}_{D}\left( {{k}_{0}, k}\right) }
+$$
+
+$$
+\mathop{\sum }\limits_{{s = {k}_{0}}}^{{{\hslash }_{0} - 1}}\left( {{\widetilde{\alpha }}^{{k}_{1} - {\hslash }_{0}}{\phi }_{0}^{{\hslash }_{0} - s - 1}\Upsilon \left( s\right) }\right) - \mathop{\sum }\limits_{{s = {\hslash }_{2l}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}\Upsilon \left( s\right)
+$$
+
+$$
+- \mathop{\sum }\limits_{{s = {k}_{2l}}}^{{{\hslash }_{2l} - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\phi }_{0}^{{\hslash }_{2l} - s - 1}\Upsilon \left( s\right)
+$$
+
+(16)
+
+Since ${\varepsilon }_{M} = \max \left\{ {\varepsilon }_{i}\right\}$ and $1 < {\phi }_{0}^{{k}_{2l} + {\varepsilon }_{2l} - s - 1} < {\phi }_{0}^{{\varepsilon }_{M} - 1}$ , under zero initial conditions $\mathcal{V}\left( {\widetilde{x}\left( {k}_{0}\right) }\right) = 0$ and $\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \geq 0$ and according to the Definition 1, it can get
+
+$$
+\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\widetilde{\alpha }}_{0}^{{\Xi }_{F}\left( {{k}_{0}, s}\right) }{\widetilde{\alpha }}_{1}^{{\Xi }_{D}\left( {{k}_{0}, s}\right) }{e}^{T}\left( s\right) e\left( s\right) \leq \tag{17}
+$$
+
+$$
+{\chi }_{0}^{{\xi }_{F}}{\chi }_{1}^{{\xi }_{D}}{\gamma }^{2}\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\theta }_{0}^{{\varepsilon }_{M} - 1}{\varpi }^{T}\left( s\right) \varpi \left( s\right) .
+$$
+
+The accumulated sum of (17) over $\lbrack k,\infty )$ is given by
+
+$$
+\mathop{\sum }\limits_{{k = {k}_{0}}}^{\infty }\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\widetilde{\alpha }}^{s - {k}_{0}}{e}^{T}\left( s\right) e\left( s\right) \leq {\chi }_{0}^{{\xi }_{F}}{\chi }_{1}^{{\xi }_{D}} \tag{18}
+$$
+
+$$
+{\gamma }^{2}\mathop{\sum }\limits_{{k = {k}_{0}}}^{\infty }\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\theta }_{0}^{{\varepsilon }_{M} - 1}{\varpi }^{T}\left( s\right) \varpi \left( s\right)
+$$
+
+which is equivalent to
+
+$$
+\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}^{s - {k}_{0}}{e}^{T}\left( s\right) e\left( s\right) \leq {\chi }_{0}^{{\xi }_{F}}{\chi }_{1}^{{\xi }_{D}} \tag{19}
+$$
+
+$$
+{\theta }_{0}^{{\varepsilon }_{M} - 1}{\gamma }^{2}\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\varpi }^{T}\left( s\right) \varpi \left( s\right) .
+$$
+
+Thus, the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are finally shown to be exponentially asymptotically stable and satisfy the exponential ${H}_{\infty }$ performance index ${\gamma }_{s} =$
+
+$\max \left\{ {\sqrt{{\left( {\theta }_{0}^{{\varepsilon }_{M}}{\mu }_{0}\right) }^{{\xi }_{F}}{\left( {\theta }_{1}^{{\varepsilon }_{M}}{\mu }_{1}\right) }^{{\xi }_{D}}{\theta }_{0}^{{\varepsilon }_{M} - 1}} \cdot \gamma }\right\}$ , which completes the proof.
+
+Due to the presence of numerous unknown matrix couplings, it is typically difficult to obtain filter gains from Theorem 1. Then, the linear solvability conditions of the designed filters are proposed in Theorem 2.
+
+Theorem 2: Consider the switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ , under DoS attacks with ${\tau }_{F}$ and ${\tau }_{D}$ , scalar ${\alpha }_{i},{\beta }_{i},\gamma ,{\mu }_{0}$ and ${\mu }_{1}$ satisfying $0 < {\alpha }_{i} < 1,{\beta }_{i} > 0,\gamma > 0,{\mu }_{0} > 1$ and $0 < {\mu }_{1} < 1$ . If there exist symmetric positive-definite matrices ${\mathcal{P}}_{i1},{\mathcal{P}}_{i3}$ , matrices ${\mathcal{P}}_{i2},{\mathcal{G}}_{i},{\mathcal{Q}}_{i},{\mathcal{R}}_{i},{\mathcal{A}}_{Fi},{\mathcal{B}}_{Fi},{\mathcal{C}}_{Fi},{\mathcal{D}}_{Fi}$ , scalar $\gamma , i, j, i \neq j$ satisfying the following conditions
+
+$$
+\left\lbrack \begin{matrix} {\Pi }_{i}^{11} & {\Pi }_{i}^{12} & 0 & {\Pi }_{i}^{14} & {\mathcal{A}}_{Fi} & {\Pi }_{i}^{16} & {\Pi }_{i}^{17} \\ * & {\Pi }_{i}^{22} & 0 & {\Pi }_{i}^{24} & {\mathcal{A}}_{Fi} & {\Pi }_{i}^{26} & {\Pi }_{i}^{27} \\ * & * & - I & {\Pi }_{i}^{34} & {\mathcal{C}}_{Fi} & 0 & - I \\ * & * & * & - {\widetilde{\alpha }}_{i}{\mathcal{P}}_{i1} & - {\widetilde{\alpha }}_{i}{\mathcal{P}}_{i2} & 0 & 0 \\ * & * & * & * & - {\widetilde{\alpha }}_{i}{\mathcal{P}}_{i3} & 0 & 0 \\ * & * & * & * & * & - {\gamma }^{2}I & 0 \\ * & * & * & * & * & * & - {\gamma }^{2}I \end{matrix}\right\rbrack < 0,
+$$
+
+(20)
+
+$$
+\left\lbrack \begin{matrix} {\Pi }_{ij}^{11} & {\Pi }_{ij}^{12} & 0 & {\Pi }_{ij}^{14} & {\mathcal{A}}_{Fj} & {\Pi }_{ij}^{16} & {\Pi }_{ij}^{17} \\ * & {\Pi }_{i}^{22} & 0 & {\Pi }_{ij}^{24} & {\mathcal{A}}_{Fj} & {\Pi }_{ij}^{26} & {\Pi }_{ij}^{27} \\ * & * & - I & {\Pi }_{ij}^{34} & {\mathcal{C}}_{Fj} & 0 & - I \\ * & * & * & - {\widetilde{\beta }}_{i}{\mathcal{P}}_{i1} & - {\widetilde{\beta }}_{i}{\mathcal{P}}_{i2} & 0 & 0 \\ * & * & * & * & - {\widetilde{\beta }}_{i}{\mathcal{P}}_{i3} & 0 & 0 \\ * & * & * & * & * & - {\gamma }^{2}I & 0 \\ * & * & * & * & * & * & - {\gamma }^{2}I \end{matrix}\right\rbrack < 0
+$$
+
+(21)
+
+$$
+\left\lbrack \begin{matrix} {\Omega }^{11} & {\Omega }^{12} & {\mathcal{G}}_{i}^{T} & {\mathcal{R}}_{i} \\ & {\Omega }^{22} & {\mathcal{Q}}_{i}^{T} & {\mathcal{R}}_{i} \\ & * & - {\mu }_{i}{\mathcal{P}}_{j1} & - {\mu }_{i}{\mathcal{P}}_{j2} \\ & * & * & - {\mu }_{i}{\mathcal{P}}_{j3} \end{matrix}\right\rbrack \leq 0 \tag{22}
+$$
+
+$$
+{\tau }_{D} < \frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{\ln {\widetilde{\alpha }}_{1}},{\tau }_{F} > - \frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{\ln {\widetilde{\alpha }}_{0}}, \tag{23}
+$$
+
+the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are exponentially asymptotically stable and and satisfy the exponential ${H}_{\infty }$ performance index ${\gamma }_{s} = \max \left\{ {\sqrt{{\left( {\theta }_{0}^{{\varepsilon }_{M}}{\mu }_{0}\right) }^{{\xi }_{F}}{\left( {\theta }_{1}^{{\varepsilon }_{M}}{\mu }_{1}\right) }^{{\xi }_{D}}{\theta }_{0}^{{\varepsilon }_{M} - 1}} \cdot \gamma }\right\}$ , where $\widetilde{\alpha } = 1 - \alpha ,\widetilde{\beta } = 1 + \beta .{\Pi }_{i}^{11} = {\mathcal{P}}_{i1} - {\dot{\mathcal{G}}}_{i} - {\mathcal{G}}_{i} - {\mathcal{G}}_{i}^{T},$ ${\Pi }_{i}^{12} = {\mathcal{P}}_{i2} - {\mathcal{Q}}_{i} - {\mathcal{R}}_{i},{\Pi }_{i}^{14} = {\mathcal{G}}_{i}{}^{T}{A}_{id} + {\mathcal{B}}_{Fi}{C}_{d},$ ${\Pi }_{i}^{16} = {\mathcal{G}}_{i}{}^{T}{B}_{1i},{\Pi }_{i}^{17} = {\mathcal{G}}_{i}{}^{T}{B}_{2i},{\Pi }_{i}^{22} = {\mathcal{P}}_{i3} - {\mathcal{R}}_{i} - {\mathcal{R}}_{i}{}^{T},$ ${\Pi }_{i}^{24} = {\mathcal{Q}}_{i}{}^{T}{A}_{id} + {\mathcal{B}}_{Fi}{C}_{d},{\Pi }_{i}^{26} = {\mathcal{Q}}_{i}{}^{T}{B}_{1i},{\Pi }_{i}^{27} = {\mathcal{Q}}_{i}{}^{T}{B}_{2i},$ ${\Pi }_{i}^{34} = {\mathcal{D}}_{Fi}{C}_{d},{\Pi }_{ij}^{11} = {\mathcal{P}}_{i1} - {\mathcal{G}}_{j} - {\mathcal{G}}_{j}{}^{T},{\Pi }_{ij}^{12} = {\mathcal{P}}_{i2} - {\mathcal{Q}}_{j} - {\mathcal{R}}_{j},$ ${\Pi }_{ij}^{14} = {\mathcal{G}}_{j}{}^{T}{A}_{id} + {\mathcal{B}}_{Fj}{C}_{d},{\Pi }_{ij}^{16} = {\mathcal{G}}_{j}{}^{T}{B}_{1i},{\Pi }_{ij}^{17} = {\mathcal{G}}_{j}{}^{T}{B}_{2i},$ ${\Pi }_{ij}^{22} = {\mathcal{P}}_{i3} - {\mathcal{R}}_{j} - {\mathcal{R}}_{j}{}^{T},{\Pi }_{ij}^{24} = {\mathcal{Q}}_{j}{}^{T}{A}_{id} + {\mathcal{B}}_{Fj}{C}_{d},$ ${\Pi }_{ij}^{26} = {Q}_{j}{}^{T}{B}_{1i},{\Pi }_{ij}^{27} = {Q}_{j}{}^{T}{B}_{2i},{\Pi }_{ij}^{34} = {D}_{Fj}{C}_{d},$ ${\Omega }^{11} = {\mathcal{P}}_{i1} - {\mu }_{i}\left( {{\mathcal{G}}_{i} + {\mathcal{G}}_{i}{}^{T}}\right) ,{\Omega }^{12} = {\mathcal{P}}_{i2} - {\mu }_{i}{\mathcal{Q}}_{i} -$ ${\mu }_{i}{\mathcal{R}}_{i}{}^{T},{\Omega }^{22} = {\mathcal{P}}_{i3} - {\mu }_{i}\left( {{\mathcal{R}}_{i} + {\mathcal{R}}_{i}{}^{T}}\right) .$
+
+In addition, if there is a solution to (20)-(23), then the filter gain can be obtained
+
+$$
+\left\lbrack \begin{matrix} {\mathcal{A}}_{fi} & {\mathcal{B}}_{fi} \\ {\mathcal{C}}_{fi} & {\mathcal{D}}_{fi} \end{matrix}\right\rbrack = \left\lbrack \begin{matrix} {\mathcal{R}}_{i}{}^{-1} & 0 \\ 0 & I \end{matrix}\right\rbrack \left\lbrack \begin{matrix} {\mathcal{A}}_{Fi} & {\mathcal{B}}_{Fi} \\ {\mathcal{C}}_{Fi} & {\mathcal{D}}_{Fi} \end{matrix}\right\rbrack . \tag{24}
+$$
+
+Proof: Based on the Project Lemma and the Schur complement Lemma, pre- and post-multiplying (8), one can deduce that (8) and (20) are equivalent. Similarly, pre- and postmultiplying (9) implies that (9) and (21) are equivalent. Theorem 2 is proved.
+
+For the purpose of fault detection, the residual is obtained from the difference between the measured value and its estimated value. Design the following residual estimation function
+
+$$
+{\mathcal{J}}_{r}\left( k\right) = \sqrt{\frac{1}{k}\mathop{\sum }\limits_{{s = 1}}^{k}{r}^{T}\left( s\right) r\left( s\right) }. \tag{25}
+$$
+
+And select threshold value of (25) as
+
+$$
+{\mathcal{J}}_{th} = \mathop{\sup }\limits_{\substack{{d\left( k\right) \in {l}_{2}} \\ {f\left( k\right) = 0} }}{\mathcal{J}}_{r}\left( k\right) . \tag{26}
+$$
+
+Therefore, the fault detection logical relationship is
+
+$$
+\left\{ \begin{matrix} \begin{Vmatrix}{{\mathcal{J}}_{r}\left( k\right) }\end{Vmatrix} > {\mathcal{J}}_{th} & \text{ Alarm } \\ \begin{Vmatrix}{{\mathcal{J}}_{r}\left( k\right) }\end{Vmatrix} \leq {\mathcal{J}}_{th} & \text{ No-alarm. } \end{matrix}\right. \tag{27}
+$$
+
+## IV. Simulation
+
+This section intends to demonstrate the effectiveness of asynchronous FD strategy for networked UMV under DoS attacks. By choosing matrices $M, N$ and $R$ in system (1) as [20]. Let ${\alpha }_{0} = {0.09},{\beta }_{0} = {0.05},{\alpha }_{1} = {0.11},{\beta }_{1} = {0.03}$ , ${\mu }_{0} = {1.4},{\mu }_{1} = {0.45},{\varepsilon }_{M} = 2,\sigma = 1$ and $\gamma = {44}$ . Then, from (11) the MDADT satisfies ${\tau }_{D} < {4.34}$ and ${\tau }_{F} > {6.60}$ . The UMV fault detection filter gain under DoS attacks can be calculated by Theorem 2.
+
+To demonstrate the practicability of FD filters designed for networked UMV under DoS attacks, the following simulations are performed to verify it. Firstly, UMV are suffered from thruster faults, external disturbances and DoS attacks. One possible sequences of DoS attacks are depicted in Fig. 1, where 1 denotes that attacks have occurred and 0 denotes the sleep state with no attack. Because of the existence of DoS attacks, which in turn leads to asynchronous switching between the filter and the primary system, then the switching sequence between the filter and the subsystem is shown in Fig. 2.
+
+
+
+Fig. 1. DoS attacks sequences.
+
+
+
+Fig. 2. Switching sequences.
+
+The external disturbance $d\left( k\right)$ is given as the following form
+
+$$
+d\left( k\right) = \left\{ {\begin{array}{l} {d}_{1}\left( k\right) = {12}\sin \left( k\right) \exp \left( {-{0.15k}}\right) \\ {d}_{2}\left( k\right) = {15}\sin \left( {0.73k}\right) , k \in \left\lbrack {5,{37}}\right\rbrack \\ {d}_{3}\left( k\right) = 9\sin \left( {0.2k}\right) , k \in \left\lbrack {{11},{45}}\right\rbrack \end{array}.}\right.
+$$
+
+Case 1: Use DoS attacks sequence 1, and the fault signals ${f}^{1}\left( k\right)$ takes the following form
+
+$$
+{f}^{1}\left( k\right) = \left\{ {\begin{array}{l} {f}_{1}\left( k\right) = 2\sin \left( {0.2k}\right) \\ {f}_{2}\left( k\right) = \cos \left( {0.1k}\right) \\ {f}_{3}\left( k\right) = {0.8}\sin \left( {0.15k}\right) \end{array}, k \in \left\lbrack {{25},{35}}\right\rbrack .}\right.
+$$
+
+Under the DoS attack sequence and the faults ${f}^{1}\left( k\right)$ , the curves of the residual signal $\parallel r\left( k\right) {\parallel }_{2}$ and the REF signal are depicted in Fig. 3 and Fig. 4, respectively. In the absence of faults, the threshold value is chosen depending on the maximum value of the REF signal: ${\mathcal{J}}_{th} = {0.215}$ . When $t$ $= {25.11}\mathrm{\;s}$ , the fault signal is detected in time.
+
+
+
+Fig. 3. The residual signal $\parallel r\left( k\right) {\parallel }_{2}$ in Case 1.
+
+
+
+Fig. 4. The REF signal in Case 1.
+
+Case 2: In order to further verify the sensitivity of the FD filter to the faults, a fault with a smaller amplitude than case 1 but with the same frequency is selected for verification, and the DoS attack sequence is still used. The fault form of ${f}^{2}\left( k\right)$ is shown as follows
+
+$$
+{f}^{2}\left( k\right) = \left\{ {\begin{array}{l} {f}_{1}\left( k\right) = {0.4}\sin \left( {0.2k}\right) \\ {f}_{2}\left( k\right) = {0.2}\cos \left( {0.1k}\right) \\ {f}_{3}\left( k\right) = {0.16}\sin \left( {0.15k}\right) \end{array}, k \in \left\lbrack {{25},{35}}\right\rbrack .}\right.
+$$
+
+Under the DoS attack sequence and the faults ${f}^{2}\left( k\right)$ , the curves of the residual signal $\parallel r\left( k\right) {\parallel }_{2}$ and the REF signal are depicted in Fig. 5 and Fig. 6, respectively. Fig. 6 indicates that the threshold for fault detection becomes smaller than in Case 1: ${\mathcal{J}}_{th} = {0.067}$ . And when $t = {25.27s}$ , the fault signal is detected in time. In contrast to Case 1, the residual amplitude and the REF signal are significantly reduced. This shows that the fault amplitude has a non-negligible effect on the system.
+
+
+
+Fig. 5. The residual signal $\parallel r\left( k\right) {\parallel }_{2}$ in Case 2.
+
+
+
+Fig. 6. The REF signal in Case 2.
+
+## V. CONCLUSION
+
+To solve the problem that DoS attacks cannot be detected in time, this paper designs an exponential convergent ${H}_{\infty }$ filters based on an asynchronous switched method for UMVs under DoS attacks, which solves the issue that the filters' switching frequently lags behind subsystems in practical applications. On the basis of the MDADT and the PLF, one criterion on the tolerability of the MDADT is derived to maintain exponential ${H}_{\infty }$ performance. Sufficient conditions for the designed FD filter to exist are described by LMIs, and the filter gain and the related parameters of MDADT can be derived by solving these LMIs. Finally, the effectiveness of the designed filter is verified by numerical simulation.
+
+## REFERENCES
+
+[1] L. Ma, Y.-L. Wang, and Q.-L. Han, "Event-triggered dynamic positioning for mass-switched unmanned marin vehicles in network environments," IEEE Transactions on Cybernetics, no. 5, pp. 3159-3171, MAY 2022.
+
+[2] Q. Liu, Y. Long, T. Li, J. H. Park, and C. P. Chen, "Fault detection for unmanned marine vehicles under replay attack," IEEE Transactions on Fuzzy Sysems, vol. 31, no. 5, pp. 1716-1728, MAY 2023.
+
+[3] B. S. Park and S. J. Yoo, "Fault detection and accommodation of saturated actuators for underactuated surface vessels in the presence of nonlinear uncertainties," Nonlinear Dynamics, vol. 85, no. 2, pp. 1067- 1077, JUL 2016.
+
+[4] Z. Duan, F. Ding, J. Liang, and Z. Xiang, "Observer-based fault detection for continuous-discrete systems in T-S fuzzy model," Nonlinear Analysis-Hybrid Systems, vol. 50, p. 101379, NOV 2023.
+
+[5] X.-L. Wang, G.-H. Yang, and D. Zhang, "Event-triggered fault detection observer design for T-S fuzzy systems," IEEE Transactions on Fuzzy Systems, vol. 29, no. 9, pp. 2532-2542, SEP 2021.
+
+[6] X. Yao, L. Wu, and W. X. Zheng, "Fault detection filter design for Markovian jump singular systems with intermittent measurements," IEEE Transactions on Signal Processing, vol. 59, no. 7, pp. 3099-3109, JUL 2011.
+
+[7] Y.-L. Wang and Q.-L. Han, "Network-based fault detection filter and controller coordinated design for unmanned surface vehicles in network environments," IEEE Transactions on Industrial Informatics, vol. 12, no. 5, pp. 1753-1765, OCT 2016.
+
+[8] X. Wang, Z. Fei, H. Gao, and J. Yu, "Integral-based event-triggered fault detection filter design for unmanned surface vehicles," IEEE Transactions on Industrial Informatics, vol. 15, no. 10, pp. 5626-5636, OCT 2019.
+
+[9] X.-N. Yu, L.-Y. Hao, and X.-L. Wang, "Fault tolerant control for an unmanned surface vessel based on integral sliding mode state feedback control," International Journal of Control Automation and Systems, vol. 20, no. 8, pp. 2514-2522, AUG 2022.
+
+[10] N. Wang, H. He, Y. Hou, and B. Han, "Model-free visual servo swarming of manned-unmanned surface vehicles with visibility maintenance and collision avoidance," IEEE Transactions on Intelligent Transportation Systems, SEP 2023.
+
+[11] S. Chen, Y. Chen, C. Pan, I. Ali, J. Pan, and W. He, "Distributed adaptive platoon secure control on unmanned vehicles system for lane change under compound attacks," IEEE Transactions on Intelligent Transportation Systems, vol. 24, no. 11, pp. 12637-12647, NOV 2023.
+
+[12] D. Ding, Z. Wang, Q.-L. Han, and G. Wei, "Security control for discrete-time stochastic nonlinear systems subject to deception attacks," IEEE Transactions on Systems Man and Cybernetics-Systems, vol. 48, no. 5, pp. 779-789, MAY 2018.
+
+[13] L. Zhao and G.-H. Yang, "Cooperative adaptive fault-tolerant control for multi-agent systems with deception attacks," Journal of the Franklin Institute-Engineering and Applied Mathematics, vol. 357, no. 6, pp. 3419-3433, APR 2020.
+
+[14] Y. Zhao, Z. Chen, C. Zhou, Y.-C. Tian, and Y. Qin, "Passivity-based robust control against quantified false data injection attacks in cyber-physical systems," IEEE-CAA Journal of Automatica Sinica, vol. 8, no. 8, pp. 1440-1450, AUG 2021.
+
+[15] Z. Ye, D. Zhang, and Z.-G. Wu, "Adaptive event-based tracking control of unmanned marine vehicle systems with DoS attack," Journal of the Franklin Institute-Engineering and Applied Mathematics, vol. 358, no. 3, pp. 1915-1939, FEB 2021.
+
+[16] X. Sun, G. Wang, Y. Fan, D. Mu, and B. Qiu, "A formation autonomous navigation system for unmanned surface vehicles with distributed control strategy," IEEE Transactions on Intelligent Transportation Systems, vol. 22, no. 5, pp. 2834-2845, MAY 2021.
+
+[17] D. Zhang, Z. Ye, P. Chen, and Q.-G. Wang, "Intelligent event-based output feedback control with Q-learning for unmanned marine vehicle systems," Control Engineering Practice, vol. 105, p. 104616, Dec. 2020.
+
+[18] M. Liu, J. Yu, and Y. Liu, "Dynamic event-triggered asynchronous fault detection for markov jump systems with partially accessible hidden information and subject to aperiodic DoS attacks," Applied Mathematics and Computation, vol. 431, p. 127317, OCT 15 2022.
+
+[19] D. Du, B. Jiang, P. Shi, and H. R. Karimi, "Fault detection for continuous-time switched systems under asynchronous switching," International Journal of Robust and Nonlinear Control, vol. 24, no. 11, pp. 1694-1706, JUL 2014.
+
+[20] N. E. Kahveci and P. A. Ioannou, "Adaptive steering control for uncertain ship dynamics and stability analysis," Automatica, vol. 49, no. 3, pp. 685-697, Mar. 2013.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/CxWEOEhqo6/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/CxWEOEhqo6/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..bd1030de100ef5d76f8563546c45ecd34cf82375
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/CxWEOEhqo6/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,381 @@
+§ ASYNCHRONOUS THRUSTER FAULT DETECTION FOR UNMANNED MARINE VEHICLES UNDER DOS ATTACKS
+
+Fuxing Wang
+
+School of Automation Engineering
+
+University of Electronic Science and Technology of China
+
+Chengdu 611731, China
+
+wfx614328@163.com
+
+Yue Long
+
+School of Automation Engineering
+
+University of Electronic Science and Technology of China
+
+Chengdu 611731, China
+
+longyue@uestc.edu.cn
+
+Tieshan Li
+
+School of Automation Engineering University of Electronic Science and Technology of China Chengdu 611731, China
+
+tieshanli@126.com
+
+Abstract-This paper investigates a thruster fault detection strategy for unmanned marine vehicles (UMVs) subjected to external disturbances and aperiodic Denial of Service (DoS) attacks. To address the challenge of timely detection of DoS attacks, the UMV and the corresponding filters are modeled within the framework of an asynchronous switched system. Sufficient conditions ensuring the system's exponential stability and prescribed performance are derived using model-dependent average dwell time and piecewise Lyapunov functions. Additionally, the tolerable lower bound of the sleep interval and the upper bound of the attack interval for DoS attacks are established. Solvable conditions for the designed fault detection filters are obtained by leveraging decoupling techniques. Finally, simulations conducted on a UMV validate the effectiveness of the proposed methods.
+
+Index Terms-Unmanned marine vehicles, asynchronous switched system, DoS attacks, fault detection.
+
+§ I. INTRODUCTION
+
+In recent years, unmanned marine vehicles (UMVs) have attracted significant attention in marine science and technology due to their wide-ranging applications in marine exploration, environmental monitoring, and resource development [1]. Nevertheless, the operational environment for UMVs is inherently complex, and their reliance on wireless communication networks for communication with shore-based centers makes them vulnerable to external disturbances, equipment malfunctions, cyber-attacks, and other disruptions [2]. The unpredictable nature of potential harm caused by these disturbances or faults, combined with the inherent vulnerabilities of cyberspace, renders UMV systems particularly susceptible to cyber-attacks. These risks can result in system failures and potentially catastrophic accidents [3]. As a result, improving the reliability and security of UMVs has emerged as a crucial area of research and development.
+
+The unpredictable nature of potential harm caused by disturbances or faults to unmanned marine vehicles (UMVs) underscores the critical need for a real-time fault detection (FD) warning mechanism. The core of fault detection methodology involves comparing system performances to identify fault signals. Current research predominantly focuses on model-based fault detection, which has shown significant success in various systems, including continuous-discrete systems [4], T-S fuzzy systems [5], and Markovian jump systems [6]. The primary approach involves generating residual signals through filters or observers and subsequently establishing a fault warning mechanism. For UMVs, several studies have made noteworthy contributions. [7] has explored the design of controllers and FD filters based on observers for networked UMVs, [8] proposed event-triggered fault detection mechanisms for UMVs in networked environments, and [2] utilized T-S fuzzy systems to model UMV systems, particularly addressing fault detection under replay attacks. Despite these advancements, the scope of fault detection research for UMVs remains relatively narrow and lacks comprehensive coverage [9]. Consequently, further investigation into robust and holistic fault detection strategies for UMVs is imperative to enhance their reliability and operational safety [10].
+
+On the other hand, due to the openness of cyberspace, UMV systems are particularly vulnerable to cyber-attacks. Deception attacks and Denial of Service (DoS) attacks are currently common types of attacks [11]. Deception attacks involve sending incorrect or tampered data to the system [12], including replay attacks [13] and false data injection attacks [14]. Compared to deception attacks, DoS attacks cause signal transmission to be unavailable for a period, leaving the system in an open-loop state, which makes it easier to cause severe disruption in system operations. Consequently, numerous studies on DoS attacks have emerged [15], [16].
+
+However, most existing research assumes that Denial of Service (DoS) attacks can be detected promptly, suggesting that the switching of filters corresponding to each subsystem happens simultaneously with the subsystem switching [10], [17]. However, in practical applications, detecting DoS attacks in a timely manner proves challenging, leading to delays. This delay implies that the filter often takes additional time to adjust to the appropriate control mode based on the subsystem mode, resulting in asynchronous filter/subsystem switching [18]. As a result, filters designed for synchronous switching may not provide optimal detection performance in real-world scenarios [19]. Thus, incorporating asynchronous switching into thruster fault detection for unmanned marine vehicles (UMVs) under DoS attacks is of substantial practical significance.
+
+This work is supported in part by the National Natural Science Foundation of China under Grants 62273072, 51939001. (Corresponding author: Yue Long)
+
+Inspired by the previous discussion, this paper investigates thruster fault detection (FD) for unmanned marine vehicles (UMVs) under Denial of Service (DoS) attacks using an asynchronous switched method to enhance reliability and security. Addressing the challenge of timely DoS attack detection, the paper proposes an asynchronous switched filter specifically designed for thruster fault detection. Furthermore, leveraging model-dependent average dwell time (MDADT) and piecewise Lyapunov functions (PLF), the paper establishes the tolerable lower bound of the sleep interval and the upper bound of the attack interval for DoS attacks. The filter parameters are determined based on linear solvability conditions. The effectiveness of the proposed method is ultimately validated through simulation.
+
+§ II. PROBLEM FORMULATION AND MODELING
+
+§ A.UMV MODEL
+
+Consider the UMV and the following body-fixed equations of motion
+
+$$
+M\dot{\delta }\left( t\right) + {N\delta }\left( t\right) + {R\psi }\left( t\right) = {E\varphi }\left( t\right) , \tag{1}
+$$
+
+$$
+\dot{\psi }\left( t\right) = J\left( {\eta \left( t\right) }\right) \delta \left( t\right) ,
+$$
+
+where $\delta \left( t\right) = {\left\lbrack {\delta }_{u}\left( t\right) ,{\delta }_{v}\left( t\right) ,{\delta }_{r}\left( t\right) \right\rbrack }^{T}$ with ${\delta }_{u}\left( t\right) ,{\delta }_{v}\left( t\right) ,{\delta }_{r}\left( t\right)$ representing the surge, sway and yaw velocities, respectively. $\psi \left( t\right) = {\left\lbrack {x}_{p}\left( t\right) ,{y}_{p}\left( t\right) ,\eta \left( t\right) \right\rbrack }^{T}$ with ${x}_{p}\left( t\right)$ and ${y}_{p}\left( t\right)$ are positions and $\eta \left( t\right)$ is the yaw angle. $\varphi \left( t\right)$ is the control input. $M,N,R$ and $E$ denote inertia, damping, mooring forces and configuration matrices, and $M$ is a symmetric positive-definite and invertible matrix that satisfies $M = {M}^{T} > 0$ ,
+
+$J\left( {\eta \left( t\right) }\right) = \left\lbrack \begin{matrix} \cos \left( {\eta \left( t\right) }\right) & - \sin \left( {\eta \left( t\right) }\right) & 0 \\ \sin \left( {\eta \left( t\right) }\right) & \cos \left( {\eta \left( t\right) }\right) & 0 \\ 0 & 0 & 1 \end{matrix}\right\rbrack .$
+
+Then, by defining $x\left( t\right) = \delta \left( t\right) - {\delta }_{\text{ ref }},A\left( t\right) =$ $- M{\left( t\right) }^{-1}N\left( t\right) ,{B}_{1}\left( t\right) = M{\left( t\right) }^{-1}R$ and ${B}_{2}\left( t\right) = M{\left( t\right) }^{-1}E$ , and taking into account the unavoidable disturbance $\widetilde{d}\left( t\right)$ caused by wind, wave and current, the system (1) can be expressed as
+
+$$
+\left\{ \begin{array}{l} \dot{x}\left( t\right) = {Ax}\left( t\right) + {B}_{1}d\left( t\right) + {B}_{2}\varphi \left( t\right) , \\ y\left( t\right) = {Cx}\left( t\right) , \end{array}\right. \tag{2}
+$$
+
+where $d\left( t\right) = {B}_{1}{\left( t\right) }^{-1}{d}^{ * }\left( t\right) - \psi \left( t\right) + {B}_{1}{\left( t\right) }^{-1}A{\delta }_{\text{ ref }}$ and $C =$ $\left\lbrack \begin{array}{lll} 0 & 0 & 1 \end{array}\right\rbrack$ denotes the output matrix.
+
+Consider thruster fault ${\varphi }^{F}\left( t\right) = {\rho \varphi }\left( t\right) + {\sigma f}\left( t\right)$ and assume control inputs $\varphi \left( t\right) = {Kx}\left( t\right)$ are designed,(2) is represented
+
+as
+
+$$
+\left\{ \begin{array}{l} \dot{x}\left( t\right) = \widehat{A}x\left( t\right) + {B}_{1}d\left( t\right) + {B}_{2}\widehat{f}\left( t\right) , \\ y\left( t\right) = {Cx}\left( t\right) , \end{array}\right. \tag{3}
+$$
+
+where $\widehat{A} = A + {B}_{2}K$ and $\widehat{f}\left( t\right) = - \bar{\rho }\varphi \left( t\right) + {\sigma f}\left( t\right)$ .
+
+§ B.DOS ATTACKS MODEL
+
+Consider the aperiodic dos attacks as follows:
+
+$$
+{A}_{\text{ Dos }} = \left\{ \begin{matrix} 0, & t \in \left\lbrack {{t}_{2l},{t}_{{2l} + 1}}\right) \triangleq {\kappa }_{0,{2l}} \\ 1, & t \in \left\lbrack {{t}_{{2l} + 1},{t}_{2\left( {l + 1}\right) }}\right) \triangleq {\kappa }_{1,{2l}} \end{matrix}\right. \tag{4}
+$$
+
+where $t \in \left\lbrack {{t}_{2l},{t}_{{2l} + 1}}\right) \triangleq {\kappa }_{0,{2l}}\;\left( {l \in \mathrm{N},{t}_{2l} \geq 0}\right)$ indicates the ${l}^{th}$ sleep interval with the length ${s}_{l} = {t}_{{2l} + 1} - {t}_{2l}$ , and $t \in \left\lbrack {{t}_{{2l} + 1},{t}_{2\left( {l + 1}\right) }}\right) \triangleq {\kappa }_{1,{2l}}$ indicates the ${l}^{th}$ DoS attacks interval with the length ${d}_{l} = {t}_{2\left( {l + 1}\right) } - {t}_{{2l} + 1}$ .
+
+Due to the communication disruption caused by DoS attacks, the UMV system (3) can be augmented into the following switched system, which has been discretized. The sleeping interval can be expressed as $k \in \left\lbrack {{k}_{2l},{k}_{{2l} + 1}}\right)$ , and the DoS attacks interval can be expressed as $k \in \left\lbrack {{k}_{{2l} + 1},{k}_{2\left( {l + 1}\right) }}\right)$ .
+
+$$
+\left\{ \begin{array}{l} x\left( {k + 1}\right) = {A}_{id}x\left( k\right) + {B}_{1id}d\left( k\right) + {B}_{2id}\widehat{f}\left( k\right) \\ y\left( k\right) = {C}_{d}x\left( k\right) \end{array}\right. \tag{5}
+$$
+
+§ C. ASYNCHRONOUS SWITCHING FILTER
+
+In the case of the DoS attacks and thruster faults, the residual signal produced by the switched filter is as follows:
+
+$$
+\left\{ {\begin{array}{l} {x}_{f}\left( {k + 1}\right) = {A}_{fi}{x}_{f}\left( k\right) + {B}_{fi}y\left( k\right) \\ r\left( k\right) = {C}_{fi}{x}_{f}\left( k\right) + {D}_{fi}y\left( k\right) \end{array}\left( {i = 0,1}\right) }\right. \tag{6}
+$$
+
+where ${x}_{f}\left( k\right)$ is the state of the filters, $r\left( k\right)$ is the residual signal of the switched system (5). Define $\widetilde{x}\left( k\right) =$ ${\left\lbrack \begin{array}{ll} {x}^{T}\left( k\right) & {x}_{f}^{T}\left( k\right) \end{array}\right\rbrack }^{T},\varpi \left( k\right) = {\left\lbrack \begin{array}{ll} {d}^{T}\left( k\right) & {f}^{T}\left( k\right) \end{array}\right\rbrack }^{T}$ and the residual evaluation signal $e\left( k\right) = r\left( k\right) - \widehat{f}\left( k\right) ,\left( 6\right)$ is rewritten as (7)
+
+$$
+{\Phi }_{0} : \left\{ {\begin{array}{l} \widetilde{x}\left( {k + 1}\right) = {\widetilde{A}}_{i}\widetilde{x}\left( k\right) + {\widetilde{B}}_{i}\varpi \left( k\right) \\ e\left( k\right) = {\widetilde{C}}_{i}\widetilde{x}\left( k\right) + {\widetilde{D}}_{i}\varpi \left( k\right) \end{array},k \in \left\lbrack {{k}_{l} + {\varepsilon }_{l},{k}_{l + 1}}\right) }\right.
+$$
+
+$$
+{\Phi }_{1} : \left\{ {\begin{array}{l} \widetilde{x}\left( {k + 1}\right) = {\widetilde{A}}_{ij}\widetilde{x}\left( k\right) + {\widetilde{B}}_{ij}\varpi \left( k\right) \\ e\left( k\right) = {\widetilde{C}}_{ij}\widetilde{x}\left( k\right) + {\widetilde{D}}_{ij}\varpi \left( k\right) \end{array},k \in \left\lbrack {{k}_{l},{k}_{l} + {\varepsilon }_{l}}\right) }\right.
+$$
+
+where $i \neq j,i \in \{ 0,1\} ,j \in \{ 0,1\} ,{\widetilde{A}}_{ij} = \left\lbrack \begin{matrix} {A}_{id} & 0 \\ {B}_{fj}{C}_{d} & {A}_{fj} \end{matrix}\right\rbrack$ , ${\widetilde{B}}_{ij} = \left\lbrack \begin{matrix} {B}_{1i} & {B}_{2i} \\ 0 & 0 \end{matrix}\right\rbrack ,{\widetilde{C}}_{ij} = \left\lbrack \begin{array}{ll} {D}_{fj}{C}_{d} & {C}_{fj} \end{array}\right\rbrack$ and ${\widetilde{D}}_{ij} =$ $\left\lbrack \begin{array}{ll} 0 & - \bar{I} \end{array}\right\rbrack$ .
+
+To better set the stage for the next section, the following definitions are presented.
+
+Definition 1: For any switching signal $\tau \left( k\right)$ and $0 < {k}_{0} \leq$ $k$ , let ${\mathcal{M}}_{\tau ,l}\left( {{k}_{0},k}\right)$ indicate the number of switching times that the ${l}_{th}$ subsystem is activated over $\left\lbrack {{k}_{0},k}\right)$ . If
+
+$$
+{M}_{\tau ,l}\left( {{k}_{0},k}\right) \leq {N}_{{\mathcal{M}}_{0,l}} + \frac{{N}_{l}\left( {{k}_{0},k}\right) }{{\lambda }_{l}}
+$$
+
+holds for scalar ${\lambda }_{l} > 0$ and integer ${N}_{{M}_{0,l}} \geq 0$ , then ${\lambda }_{l}$ is called model-dependent average dwell time. ${N}_{l}\left( {{k}_{0},k}\right)$ is the total running time of the ${l}_{th}$ subsystem over $\left\lbrack {{k}_{0},k}\right)$ .
+
+Definition 2: Consider asynchronous switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ , and given scalar $\alpha ,\beta$ , and $\gamma$ satisfying $0 < \alpha < 1$ , $\beta > 0$ and $\gamma > 0$ . Under zero initial condition, if the asynchronous switched system is exponentially stable and satisfies $\mathop{\sum }\limits_{{s = {k}_{0}}}^{\infty }{\left( 1 - \alpha \right) }^{s}{e}^{\mathrm{T}}\left( s\right) e\left( s\right) \leq {\gamma }^{2}\mathop{\sum }\limits_{{s = {k}_{0}}}^{\infty }{\varpi }^{\mathrm{T}}\left( s\right) \varpi \left( s\right)$ , it is said that the system exhibits exponential stability and has exponential ${H}_{\infty }$ index $\gamma$ .
+
+§ III. MAIN RESULTS
+
+In this section, the stability and ${H}_{\infty }$ performance of asynchronous switched systems (7) will be analyzed, and the sufficient and linearly solvable conditions for the designed switched FD filters are given.
+
+Theorem 1: Consider the switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ under DoS attacks, scalars ${\alpha }_{i},{\beta }_{i},\gamma ,{\mu }_{0}$ and ${\mu }_{1}$ satisfying $0 < {\alpha }_{i} < 1,{\beta }_{i} > 0,\gamma > 0,{\mu }_{0} > 1$ and $0 < {\mu }_{1} < 1$ , if there exist symmetric positive-definite matrices ${\mathcal{P}}_{i}$ satisfying the following conditions
+
+$$
+{\widetilde{A}}_{i}^{T}{\mathcal{P}}_{i}{\widetilde{A}}_{i} - {\mathcal{P}}_{i} + {\alpha }_{i}{\mathcal{P}}_{i} < 0, \tag{8}
+$$
+
+$$
+{\widetilde{A}}_{ij}^{T}{\mathcal{P}}_{i}{\widetilde{A}}_{ij} - {\mathcal{P}}_{i} - {\beta }_{i}{\mathcal{P}}_{i} < 0, \tag{9}
+$$
+
+$$
+{\mathcal{P}}_{i} \leq {\mu }_{i}{\mathcal{P}}_{j} \tag{10}
+$$
+
+$$
+{\tau }_{D} < \frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{\ln {\widetilde{\alpha }}_{1}},{\tau }_{F} > - \frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{\ln {\widetilde{\alpha }}_{0}}, \tag{11}
+$$
+
+the switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are exponentially asymptotically stable with the exponential ${H}_{\infty }$ performance, where $i \neq j,{\widetilde{\alpha }}_{i} = 1 - {\alpha }_{i},{\widetilde{\beta }}_{i} = 1 + {\beta }_{i},{\phi }_{i} = \frac{{\widetilde{\beta }}_{i}}{{\widetilde{\alpha }}_{i}}$ and ${\varepsilon }_{M}$ denotes the maximum time that the filter lags the subsystem.
+
+Proof: The piecewise Lyapunov function for the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are given as follows
+
+$$
+{\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) = {\widetilde{x}}^{T}\left( k\right) {\mathcal{P}}_{i}\widetilde{x}\left( k\right) . \tag{12}
+$$
+
+When $\varpi \left( k\right) = 0$ and $k \in \left\lbrack {{k}_{2l},{k}_{{2l} + 1}}\right)$ , it can be obtained
+
+$$
+\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \leq \left\{ \begin{array}{l} {\widetilde{\alpha }}_{i}^{k - {k}_{2l} - {\varepsilon }_{2l}}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( {{k}_{2l} + {\varepsilon }_{2l}}\right) }\right) ,k \in {\Gamma }^{ + } \\ {\widetilde{\beta }}_{i}^{k - {k}_{2l}}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( {k}_{2l}\right) }\right) ,k \in {\Gamma }^{ - } \end{array}\right. \tag{13}
+$$
+
+where ${\widetilde{\alpha }}_{i} = 1 - {\alpha }_{i}$ and ${\widetilde{\beta }}_{i} = 1 + {\beta }_{i}$ . And when $k \in {\mathcal{T}}^{ + }\left( {{k}_{2l},{k}_{{2l} + 1}}\right)$ , from (8) and (11), it can be derived
+
+$$
+\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \leq {\widetilde{\alpha }}_{0}^{k - {k}_{2l} - {\varepsilon }_{2l}}{\mathcal{V}}_{0}\left( {\widetilde{x}\left( {{k}_{2l} + {\varepsilon }_{2l}}\right) }\right)
+$$
+
+$$
+\leq {\widetilde{\alpha }}_{0}^{k - {k}_{2l} - {\varepsilon }_{2l}} \cdot {\widetilde{\beta }}_{0}^{{\varepsilon }_{2l}} \cdot {\mathcal{V}}_{0}\left( {\widetilde{x}\left( {k}_{2l}\right) }\right)
+$$
+
+$$
+< \cdots
+$$
+
+$$
+\leq \theta \exp \left\{ {\max \left( {\frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{{\tau }_{F}} + {v}_{0}, - \frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{{\tau }_{D}} + {v}_{1}}\right) }\right.
+$$
+
+$$
+\left. \left( {{\Xi }_{F}\left( {{k}_{0},k}\right) + {\Xi }_{D}\left( {{k}_{0},k}\right) }\right) \right\} \mathcal{V}\left( {\widetilde{x}\left( {k}_{0}\right) }\right)
+$$
+
+(14)
+
+where $\theta = \exp \left\lbrack {\left( {{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}\right) {\xi }_{F} - \left( {{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}\right) {\xi }_{D}}\right\rbrack$ , $\omega = \max \left\{ {-\frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{{\tau }_{F}} - \ln {\widetilde{\alpha }}_{0},\frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{{\tau }_{D}} - \ln {\widetilde{\alpha }}_{1}}\right\} ,$ ${\chi }_{0} = {\theta }_{0}^{{\varepsilon }_{M}}{\mu }_{0},{\chi }_{1} = {\theta }_{1}^{{\varepsilon }_{M}}{\mu }_{1},{v}_{i} = \ln {\widetilde{\alpha }}_{i}.$
+
+From (11), it has $\omega > 0$ . Then, it is clear that $\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right)$ converges to zero when $k \rightarrow \infty$ . Therefore, the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are exponentially asymptotically stable when (8) and (11) hold.
+
+Next, if $\varpi \left( k\right) \neq 0$ for $k \in \left\lbrack {{k}_{2l},{k}_{{2l} + 1}}\right)$ and zero initial conditions, (??) is derived as follows
+
+$$
+\Delta {\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) < \left\{ \begin{array}{l} - {\alpha }_{i}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) - \Upsilon \left( k\right) ,k \in {\Gamma }^{ + } \\ {\beta }_{i}{\mathcal{V}}_{i}\left( {\widetilde{x}\left( k\right) }\right) - \Upsilon \left( k\right) ,k \in {\Gamma }^{ - } \end{array}\right. \tag{15}
+$$
+
+where $i = 0,1,\Upsilon \left( k\right) = {e}^{T}\left( k\right) e\left( k\right) - {\gamma }^{2}{\varpi }^{T}\left( k\right) \varpi \left( k\right)$ . When $k \in {\mathcal{T}}^{ + }\left( {{k}_{2l},{k}_{{2l} + 1}}\right)$ , it can have the following inequality in the similar way from (10) and (15)
+
+$$
+\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \leq {\widetilde{\alpha }}_{0}^{k - {k}_{2l}}{\widetilde{\alpha }}_{0}^{{k}_{{2l} - 1} - {k}_{{2l} - 2}}\cdots {\widetilde{\alpha }}_{0}^{{k}_{1} - {k}_{0}}{\phi }_{0}^{{\varepsilon }_{2l}}{\phi }_{0}^{{\varepsilon }_{{2l} - 2}}\cdots {\phi }_{0}^{{\varepsilon }_{0}}.
+$$
+
+$$
+{\mu }_{0}^{{\mathrm{M}}_{F}\left( {{k}_{0},k}\right) }{\widetilde{\alpha }}_{1}^{{k}_{2l} - {k}_{{2l} - 1}}\cdots {\widetilde{\alpha }}_{1}^{{k}_{2} - {k}_{1}}{\phi }_{1}^{{\varepsilon }_{{2l} - 1}}\cdots {\phi }_{1}^{{\varepsilon }_{1}}.
+$$
+
+$$
+{\mu }_{1}^{{\mathrm{M}}_{D}\left( {{k}_{0},k}\right) }\mathcal{V}\left( {\widetilde{x}\left( {k}_{0}\right) }\right) - {\widetilde{\alpha }}_{0}^{k - {k}_{2l}}{\widetilde{\alpha }}_{0}^{{k}_{{2l} - 1} - {k}_{{2l} - 2}}\ldots
+$$
+
+$$
+{\widetilde{\alpha }}_{0}^{{k}_{1} - {k}_{0}}{\phi }_{0}^{{\varepsilon }_{2l}}{\phi }_{0}^{{\varepsilon }_{{2l} - 2}}\cdots {\phi }_{0}^{{\varepsilon }_{0}}{\mu }_{0}^{{\mathrm{M}}_{F}\left( {{k}_{0},k}\right) }{\widetilde{\alpha }}_{1}^{{k}_{2l} - {k}_{{2l} - 1}}\cdots
+$$
+
+$$
+{\widetilde{\alpha }}_{1}^{{k}_{2} - {k}_{1}}{\phi }_{1}^{{\varepsilon }_{{2l} - 1}}\cdots {\phi }_{1}^{{\varepsilon }_{1}}{\mu }_{1}^{{\mathrm{M}}_{D}\left( {{k}_{0},k}\right) }\mathop{\sum }\limits_{{s = {k}_{0} + {\Delta }_{0}}}^{{{k}_{1} - 1}}{\widetilde{\alpha }}_{0}^{{k}_{1} - s - 1}\Upsilon \left( s\right)
+$$
+
+$$
+- {\widetilde{\alpha }}_{0}^{k - {k}_{2l}}{\widetilde{\alpha }}_{0}^{{k}_{{2l} - 1} - {k}_{{2l} - 2}}\cdots {\widetilde{\alpha }}_{0}^{{k}_{1} - {k}_{0}}{\phi }_{0}^{{\varepsilon }_{2l}}{\phi }_{0}^{{\varepsilon }_{{2l} - 2}}\cdots {\phi }_{0}^{{\varepsilon }_{0}}.
+$$
+
+$$
+{\mu }_{0}^{{\mathrm{M}}_{F}\left( {{k}_{0},k}\right) }{\widetilde{\alpha }}_{1}^{{k}_{2l} - {k}_{{2l} - 1}}\cdots {\widetilde{\alpha }}_{1}^{{k}_{2} - {k}_{1}}{\phi }_{1}^{{\varepsilon }_{{2l} - 1}}\cdots {\phi }_{1}^{{\varepsilon }_{1}}{\mu }_{1}^{{\mathrm{M}}_{D}\left( {{k}_{0},k}\right) }
+$$
+
+$$
+\mathop{\sum }\limits_{{s = {k}_{0}}}^{{{\hslash }_{0} - 1}}\left( {{\widetilde{\alpha }}^{{k}_{1} - {\hslash }_{0}}{\phi }_{0}^{{\hslash }_{0} - s - 1}\Upsilon \left( s\right) }\right) - \mathop{\sum }\limits_{{s = {\hslash }_{2l}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}\Upsilon \left( s\right)
+$$
+
+$$
+- \mathop{\sum }\limits_{{s = {k}_{2l}}}^{{{\hslash }_{2l} - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\phi }_{0}^{{\hslash }_{2l} - s - 1}\Upsilon \left( s\right)
+$$
+
+(16)
+
+Since ${\varepsilon }_{M} = \max \left\{ {\varepsilon }_{i}\right\}$ and $1 < {\phi }_{0}^{{k}_{2l} + {\varepsilon }_{2l} - s - 1} < {\phi }_{0}^{{\varepsilon }_{M} - 1}$ , under zero initial conditions $\mathcal{V}\left( {\widetilde{x}\left( {k}_{0}\right) }\right) = 0$ and $\mathcal{V}\left( {\widetilde{x}\left( k\right) }\right) \geq 0$ and according to the Definition 1, it can get
+
+$$
+\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\widetilde{\alpha }}_{0}^{{\Xi }_{F}\left( {{k}_{0},s}\right) }{\widetilde{\alpha }}_{1}^{{\Xi }_{D}\left( {{k}_{0},s}\right) }{e}^{T}\left( s\right) e\left( s\right) \leq \tag{17}
+$$
+
+$$
+{\chi }_{0}^{{\xi }_{F}}{\chi }_{1}^{{\xi }_{D}}{\gamma }^{2}\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\theta }_{0}^{{\varepsilon }_{M} - 1}{\varpi }^{T}\left( s\right) \varpi \left( s\right) .
+$$
+
+The accumulated sum of (17) over $\lbrack k,\infty )$ is given by
+
+$$
+\mathop{\sum }\limits_{{k = {k}_{0}}}^{\infty }\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\widetilde{\alpha }}^{s - {k}_{0}}{e}^{T}\left( s\right) e\left( s\right) \leq {\chi }_{0}^{{\xi }_{F}}{\chi }_{1}^{{\xi }_{D}} \tag{18}
+$$
+
+$$
+{\gamma }^{2}\mathop{\sum }\limits_{{k = {k}_{0}}}^{\infty }\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}_{0}^{k - s - 1}{\theta }_{0}^{{\varepsilon }_{M} - 1}{\varpi }^{T}\left( s\right) \varpi \left( s\right)
+$$
+
+which is equivalent to
+
+$$
+\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\widetilde{\alpha }}^{s - {k}_{0}}{e}^{T}\left( s\right) e\left( s\right) \leq {\chi }_{0}^{{\xi }_{F}}{\chi }_{1}^{{\xi }_{D}} \tag{19}
+$$
+
+$$
+{\theta }_{0}^{{\varepsilon }_{M} - 1}{\gamma }^{2}\mathop{\sum }\limits_{{s = {k}_{0}}}^{{k - 1}}{\varpi }^{T}\left( s\right) \varpi \left( s\right) .
+$$
+
+Thus, the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are finally shown to be exponentially asymptotically stable and satisfy the exponential ${H}_{\infty }$ performance index ${\gamma }_{s} =$
+
+$\max \left\{ {\sqrt{{\left( {\theta }_{0}^{{\varepsilon }_{M}}{\mu }_{0}\right) }^{{\xi }_{F}}{\left( {\theta }_{1}^{{\varepsilon }_{M}}{\mu }_{1}\right) }^{{\xi }_{D}}{\theta }_{0}^{{\varepsilon }_{M} - 1}} \cdot \gamma }\right\}$ , which completes the proof.
+
+Due to the presence of numerous unknown matrix couplings, it is typically difficult to obtain filter gains from Theorem 1. Then, the linear solvability conditions of the designed filters are proposed in Theorem 2.
+
+Theorem 2: Consider the switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ , under DoS attacks with ${\tau }_{F}$ and ${\tau }_{D}$ , scalar ${\alpha }_{i},{\beta }_{i},\gamma ,{\mu }_{0}$ and ${\mu }_{1}$ satisfying $0 < {\alpha }_{i} < 1,{\beta }_{i} > 0,\gamma > 0,{\mu }_{0} > 1$ and $0 < {\mu }_{1} < 1$ . If there exist symmetric positive-definite matrices ${\mathcal{P}}_{i1},{\mathcal{P}}_{i3}$ , matrices ${\mathcal{P}}_{i2},{\mathcal{G}}_{i},{\mathcal{Q}}_{i},{\mathcal{R}}_{i},{\mathcal{A}}_{Fi},{\mathcal{B}}_{Fi},{\mathcal{C}}_{Fi},{\mathcal{D}}_{Fi}$ , scalar $\gamma ,i,j,i \neq j$ satisfying the following conditions
+
+$$
+\left\lbrack \begin{matrix} {\Pi }_{i}^{11} & {\Pi }_{i}^{12} & 0 & {\Pi }_{i}^{14} & {\mathcal{A}}_{Fi} & {\Pi }_{i}^{16} & {\Pi }_{i}^{17} \\ * & {\Pi }_{i}^{22} & 0 & {\Pi }_{i}^{24} & {\mathcal{A}}_{Fi} & {\Pi }_{i}^{26} & {\Pi }_{i}^{27} \\ * & * & - I & {\Pi }_{i}^{34} & {\mathcal{C}}_{Fi} & 0 & - I \\ * & * & * & - {\widetilde{\alpha }}_{i}{\mathcal{P}}_{i1} & - {\widetilde{\alpha }}_{i}{\mathcal{P}}_{i2} & 0 & 0 \\ * & * & * & * & - {\widetilde{\alpha }}_{i}{\mathcal{P}}_{i3} & 0 & 0 \\ * & * & * & * & * & - {\gamma }^{2}I & 0 \\ * & * & * & * & * & * & - {\gamma }^{2}I \end{matrix}\right\rbrack < 0,
+$$
+
+(20)
+
+$$
+\left\lbrack \begin{matrix} {\Pi }_{ij}^{11} & {\Pi }_{ij}^{12} & 0 & {\Pi }_{ij}^{14} & {\mathcal{A}}_{Fj} & {\Pi }_{ij}^{16} & {\Pi }_{ij}^{17} \\ * & {\Pi }_{i}^{22} & 0 & {\Pi }_{ij}^{24} & {\mathcal{A}}_{Fj} & {\Pi }_{ij}^{26} & {\Pi }_{ij}^{27} \\ * & * & - I & {\Pi }_{ij}^{34} & {\mathcal{C}}_{Fj} & 0 & - I \\ * & * & * & - {\widetilde{\beta }}_{i}{\mathcal{P}}_{i1} & - {\widetilde{\beta }}_{i}{\mathcal{P}}_{i2} & 0 & 0 \\ * & * & * & * & - {\widetilde{\beta }}_{i}{\mathcal{P}}_{i3} & 0 & 0 \\ * & * & * & * & * & - {\gamma }^{2}I & 0 \\ * & * & * & * & * & * & - {\gamma }^{2}I \end{matrix}\right\rbrack < 0
+$$
+
+(21)
+
+$$
+\left\lbrack \begin{matrix} {\Omega }^{11} & {\Omega }^{12} & {\mathcal{G}}_{i}^{T} & {\mathcal{R}}_{i} \\ & {\Omega }^{22} & {\mathcal{Q}}_{i}^{T} & {\mathcal{R}}_{i} \\ & * & - {\mu }_{i}{\mathcal{P}}_{j1} & - {\mu }_{i}{\mathcal{P}}_{j2} \\ & * & * & - {\mu }_{i}{\mathcal{P}}_{j3} \end{matrix}\right\rbrack \leq 0 \tag{22}
+$$
+
+$$
+{\tau }_{D} < \frac{{\varepsilon }_{M}\ln {\phi }_{1} + \ln {\mu }_{1}}{\ln {\widetilde{\alpha }}_{1}},{\tau }_{F} > - \frac{{\varepsilon }_{M}\ln {\phi }_{0} + \ln {\mu }_{0}}{\ln {\widetilde{\alpha }}_{0}}, \tag{23}
+$$
+
+the closed-loop switched subsystems ${\Phi }_{0}$ and ${\Phi }_{1}$ are exponentially asymptotically stable and and satisfy the exponential ${H}_{\infty }$ performance index ${\gamma }_{s} = \max \left\{ {\sqrt{{\left( {\theta }_{0}^{{\varepsilon }_{M}}{\mu }_{0}\right) }^{{\xi }_{F}}{\left( {\theta }_{1}^{{\varepsilon }_{M}}{\mu }_{1}\right) }^{{\xi }_{D}}{\theta }_{0}^{{\varepsilon }_{M} - 1}} \cdot \gamma }\right\}$ , where $\widetilde{\alpha } = 1 - \alpha ,\widetilde{\beta } = 1 + \beta .{\Pi }_{i}^{11} = {\mathcal{P}}_{i1} - {\dot{\mathcal{G}}}_{i} - {\mathcal{G}}_{i} - {\mathcal{G}}_{i}^{T},$ ${\Pi }_{i}^{12} = {\mathcal{P}}_{i2} - {\mathcal{Q}}_{i} - {\mathcal{R}}_{i},{\Pi }_{i}^{14} = {\mathcal{G}}_{i}{}^{T}{A}_{id} + {\mathcal{B}}_{Fi}{C}_{d},$ ${\Pi }_{i}^{16} = {\mathcal{G}}_{i}{}^{T}{B}_{1i},{\Pi }_{i}^{17} = {\mathcal{G}}_{i}{}^{T}{B}_{2i},{\Pi }_{i}^{22} = {\mathcal{P}}_{i3} - {\mathcal{R}}_{i} - {\mathcal{R}}_{i}{}^{T},$ ${\Pi }_{i}^{24} = {\mathcal{Q}}_{i}{}^{T}{A}_{id} + {\mathcal{B}}_{Fi}{C}_{d},{\Pi }_{i}^{26} = {\mathcal{Q}}_{i}{}^{T}{B}_{1i},{\Pi }_{i}^{27} = {\mathcal{Q}}_{i}{}^{T}{B}_{2i},$ ${\Pi }_{i}^{34} = {\mathcal{D}}_{Fi}{C}_{d},{\Pi }_{ij}^{11} = {\mathcal{P}}_{i1} - {\mathcal{G}}_{j} - {\mathcal{G}}_{j}{}^{T},{\Pi }_{ij}^{12} = {\mathcal{P}}_{i2} - {\mathcal{Q}}_{j} - {\mathcal{R}}_{j},$ ${\Pi }_{ij}^{14} = {\mathcal{G}}_{j}{}^{T}{A}_{id} + {\mathcal{B}}_{Fj}{C}_{d},{\Pi }_{ij}^{16} = {\mathcal{G}}_{j}{}^{T}{B}_{1i},{\Pi }_{ij}^{17} = {\mathcal{G}}_{j}{}^{T}{B}_{2i},$ ${\Pi }_{ij}^{22} = {\mathcal{P}}_{i3} - {\mathcal{R}}_{j} - {\mathcal{R}}_{j}{}^{T},{\Pi }_{ij}^{24} = {\mathcal{Q}}_{j}{}^{T}{A}_{id} + {\mathcal{B}}_{Fj}{C}_{d},$ ${\Pi }_{ij}^{26} = {Q}_{j}{}^{T}{B}_{1i},{\Pi }_{ij}^{27} = {Q}_{j}{}^{T}{B}_{2i},{\Pi }_{ij}^{34} = {D}_{Fj}{C}_{d},$ ${\Omega }^{11} = {\mathcal{P}}_{i1} - {\mu }_{i}\left( {{\mathcal{G}}_{i} + {\mathcal{G}}_{i}{}^{T}}\right) ,{\Omega }^{12} = {\mathcal{P}}_{i2} - {\mu }_{i}{\mathcal{Q}}_{i} -$ ${\mu }_{i}{\mathcal{R}}_{i}{}^{T},{\Omega }^{22} = {\mathcal{P}}_{i3} - {\mu }_{i}\left( {{\mathcal{R}}_{i} + {\mathcal{R}}_{i}{}^{T}}\right) .$
+
+In addition, if there is a solution to (20)-(23), then the filter gain can be obtained
+
+$$
+\left\lbrack \begin{matrix} {\mathcal{A}}_{fi} & {\mathcal{B}}_{fi} \\ {\mathcal{C}}_{fi} & {\mathcal{D}}_{fi} \end{matrix}\right\rbrack = \left\lbrack \begin{matrix} {\mathcal{R}}_{i}{}^{-1} & 0 \\ 0 & I \end{matrix}\right\rbrack \left\lbrack \begin{matrix} {\mathcal{A}}_{Fi} & {\mathcal{B}}_{Fi} \\ {\mathcal{C}}_{Fi} & {\mathcal{D}}_{Fi} \end{matrix}\right\rbrack . \tag{24}
+$$
+
+Proof: Based on the Project Lemma and the Schur complement Lemma, pre- and post-multiplying (8), one can deduce that (8) and (20) are equivalent. Similarly, pre- and postmultiplying (9) implies that (9) and (21) are equivalent. Theorem 2 is proved.
+
+For the purpose of fault detection, the residual is obtained from the difference between the measured value and its estimated value. Design the following residual estimation function
+
+$$
+{\mathcal{J}}_{r}\left( k\right) = \sqrt{\frac{1}{k}\mathop{\sum }\limits_{{s = 1}}^{k}{r}^{T}\left( s\right) r\left( s\right) }. \tag{25}
+$$
+
+And select threshold value of (25) as
+
+$$
+{\mathcal{J}}_{th} = \mathop{\sup }\limits_{\substack{{d\left( k\right) \in {l}_{2}} \\ {f\left( k\right) = 0} }}{\mathcal{J}}_{r}\left( k\right) . \tag{26}
+$$
+
+Therefore, the fault detection logical relationship is
+
+$$
+\left\{ \begin{matrix} \begin{Vmatrix}{{\mathcal{J}}_{r}\left( k\right) }\end{Vmatrix} > {\mathcal{J}}_{th} & \text{ Alarm } \\ \begin{Vmatrix}{{\mathcal{J}}_{r}\left( k\right) }\end{Vmatrix} \leq {\mathcal{J}}_{th} & \text{ No-alarm. } \end{matrix}\right. \tag{27}
+$$
+
+§ IV. SIMULATION
+
+This section intends to demonstrate the effectiveness of asynchronous FD strategy for networked UMV under DoS attacks. By choosing matrices $M,N$ and $R$ in system (1) as [20]. Let ${\alpha }_{0} = {0.09},{\beta }_{0} = {0.05},{\alpha }_{1} = {0.11},{\beta }_{1} = {0.03}$ , ${\mu }_{0} = {1.4},{\mu }_{1} = {0.45},{\varepsilon }_{M} = 2,\sigma = 1$ and $\gamma = {44}$ . Then, from (11) the MDADT satisfies ${\tau }_{D} < {4.34}$ and ${\tau }_{F} > {6.60}$ . The UMV fault detection filter gain under DoS attacks can be calculated by Theorem 2.
+
+To demonstrate the practicability of FD filters designed for networked UMV under DoS attacks, the following simulations are performed to verify it. Firstly, UMV are suffered from thruster faults, external disturbances and DoS attacks. One possible sequences of DoS attacks are depicted in Fig. 1, where 1 denotes that attacks have occurred and 0 denotes the sleep state with no attack. Because of the existence of DoS attacks, which in turn leads to asynchronous switching between the filter and the primary system, then the switching sequence between the filter and the subsystem is shown in Fig. 2.
+
+ < g r a p h i c s >
+
+Fig. 1. DoS attacks sequences.
+
+ < g r a p h i c s >
+
+Fig. 2. Switching sequences.
+
+The external disturbance $d\left( k\right)$ is given as the following form
+
+$$
+d\left( k\right) = \left\{ {\begin{array}{l} {d}_{1}\left( k\right) = {12}\sin \left( k\right) \exp \left( {-{0.15k}}\right) \\ {d}_{2}\left( k\right) = {15}\sin \left( {0.73k}\right) ,k \in \left\lbrack {5,{37}}\right\rbrack \\ {d}_{3}\left( k\right) = 9\sin \left( {0.2k}\right) ,k \in \left\lbrack {{11},{45}}\right\rbrack \end{array}.}\right.
+$$
+
+Case 1: Use DoS attacks sequence 1, and the fault signals ${f}^{1}\left( k\right)$ takes the following form
+
+$$
+{f}^{1}\left( k\right) = \left\{ {\begin{array}{l} {f}_{1}\left( k\right) = 2\sin \left( {0.2k}\right) \\ {f}_{2}\left( k\right) = \cos \left( {0.1k}\right) \\ {f}_{3}\left( k\right) = {0.8}\sin \left( {0.15k}\right) \end{array},k \in \left\lbrack {{25},{35}}\right\rbrack .}\right.
+$$
+
+Under the DoS attack sequence and the faults ${f}^{1}\left( k\right)$ , the curves of the residual signal $\parallel r\left( k\right) {\parallel }_{2}$ and the REF signal are depicted in Fig. 3 and Fig. 4, respectively. In the absence of faults, the threshold value is chosen depending on the maximum value of the REF signal: ${\mathcal{J}}_{th} = {0.215}$ . When $t$ $= {25.11}\mathrm{\;s}$ , the fault signal is detected in time.
+
+ < g r a p h i c s >
+
+Fig. 3. The residual signal $\parallel r\left( k\right) {\parallel }_{2}$ in Case 1.
+
+ < g r a p h i c s >
+
+Fig. 4. The REF signal in Case 1.
+
+Case 2: In order to further verify the sensitivity of the FD filter to the faults, a fault with a smaller amplitude than case 1 but with the same frequency is selected for verification, and the DoS attack sequence is still used. The fault form of ${f}^{2}\left( k\right)$ is shown as follows
+
+$$
+{f}^{2}\left( k\right) = \left\{ {\begin{array}{l} {f}_{1}\left( k\right) = {0.4}\sin \left( {0.2k}\right) \\ {f}_{2}\left( k\right) = {0.2}\cos \left( {0.1k}\right) \\ {f}_{3}\left( k\right) = {0.16}\sin \left( {0.15k}\right) \end{array},k \in \left\lbrack {{25},{35}}\right\rbrack .}\right.
+$$
+
+Under the DoS attack sequence and the faults ${f}^{2}\left( k\right)$ , the curves of the residual signal $\parallel r\left( k\right) {\parallel }_{2}$ and the REF signal are depicted in Fig. 5 and Fig. 6, respectively. Fig. 6 indicates that the threshold for fault detection becomes smaller than in Case 1: ${\mathcal{J}}_{th} = {0.067}$ . And when $t = {25.27s}$ , the fault signal is detected in time. In contrast to Case 1, the residual amplitude and the REF signal are significantly reduced. This shows that the fault amplitude has a non-negligible effect on the system.
+
+ < g r a p h i c s >
+
+Fig. 5. The residual signal $\parallel r\left( k\right) {\parallel }_{2}$ in Case 2.
+
+ < g r a p h i c s >
+
+Fig. 6. The REF signal in Case 2.
+
+§ V. CONCLUSION
+
+To solve the problem that DoS attacks cannot be detected in time, this paper designs an exponential convergent ${H}_{\infty }$ filters based on an asynchronous switched method for UMVs under DoS attacks, which solves the issue that the filters' switching frequently lags behind subsystems in practical applications. On the basis of the MDADT and the PLF, one criterion on the tolerability of the MDADT is derived to maintain exponential ${H}_{\infty }$ performance. Sufficient conditions for the designed FD filter to exist are described by LMIs, and the filter gain and the related parameters of MDADT can be derived by solving these LMIs. Finally, the effectiveness of the designed filter is verified by numerical simulation.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/DuY2U9TNuJ/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/DuY2U9TNuJ/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..5793e9b3cd59e845e57a66a55434ff9e6a49fa6b
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/DuY2U9TNuJ/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,97 @@
+# Research on battery SOC estimation method by combining optimization algorithm and multi-model Kalman filtering
+
+Zhi Ming Chen
+
+College of Science
+
+Liaoning University of Technology
+
+Jinzhou, China
+
+chenzhiminglab@163.com
+
+Chang Qi Zhu
+
+Navigation College
+
+Dalian Maritime University
+
+Dalian, China
+
+zhuchangqi_work@163.com
+
+Lei Liu
+
+College of Science
+
+Liaoning University of Technology
+
+Jinzhou, China
+
+liuleill@live.cn
+
+${Abstract}$ -With the rapid growth of electric vehicles and energy storage systems, accurate state of charge (SOC) estimation has become a critical component of battery management systems (BMS), essential for preventing overcharging and over-discharging, enhancing operational safety, and extending battery life. This paper proposes a novel SOC estimation method based on an enhanced self-correcting (ESC) model incorporating a second-order RC circuit, enabling a more accurate simulation of battery response time and dynamic behavior. To improve model reliability, a genetic algorithm-particle swarm optimization (GA-PSO) approach is employed for parameter identification. Additionally, a multi-model adaptive extended Kalman filter (AEKF) algorithm is introduced to achieve precise SOC estimation. MATLAB simulations using constant current discharge and automotive driving cycle data demonstrate that the proposed method outperforms traditional AEKF algorithms, with faster convergence and higher estimation accuracy, particularly in scenarios with varying initial estimation accuracies. The results highlight the potential of this approach to significantly enhance SOC estimation in BMS, contributing to safer operation and prolonged battery life in electric vehicles and energy storage systems.
+
+Keywords—SOC estimation, Enhanced Self-Correcting model, parameter identification, GA-PSO, multi-model AEKF.
+
+## I. INTRODUCTION
+
+As electric vehicles and energy storage systems continue to develop rapidly, the application of batteries as key energy storage devices has become increasingly widespread, highlighting the growing importance of battery management and control [1]. Within battery management systems (BMS), accurately estimating the state of charge (SOC) is a critical task [2]. Precise SOC estimation not only enables more reliable predictions of vehicle range but also improves battery utilization and helps prevent significant reductions in battery lifespan caused by overcharging or deep discharging [3]. However, the nonlinear characteristics, time-varying behavior, and electrochemical reactions of batteries make it impossible to measure their SOC directly with sensors [4]. Instead, SOC must be estimated using indirect measurements such as voltage, current, and temperature. Common SOC estimation methods include approaches based on open-circuit voltage, coulomb counting, data-driven techniques, and model-based estimation methods [5]. Each of these approaches presents distinct advantages and disadvantages [6].
+
+Among these methods, model-based estimation achieves a reasonable balance between accuracy, real-time performance, and computational cost by integrating the battery equivalent circuit model (ECM) with state estimation algorithms. The ECM is a key component in this approach. Previous studies have advanced SOC estimation using various models and algorithms. Li et al. [7] utilized a second-order RC model with a stochastic gradient algorithm for parameter identification and developed a multi-innovation extended Kalman filter, validated experimentally. Shi et al. [8] employed Bayesian belief networks and adaptive extended Kalman particle filtering, demonstrating enhanced convergence and accuracy.
+
+However, these studies largely overlook the hysteresis effect in battery charging and discharging. Gregory L. Plett [9] addressed this by introducing an Enhanced Self-Correcting (ESC) model that incorporates hysteresis into the ECM. Sk Bittu et al. [10] simulated a first-order RC ESC model with an EKF algorithm for SOC estimation but found that the model struggles with complex polarization dynamics, and the EKF's performance deteriorates with significant measurement errors.
+
+Accurate SOC estimation requires precise circuit modeling and effective algorithms. This study incorporates the hysteresis phenomenon using an ESC model with second-order RC characteristics. The GA-PSO algorithm is applied for precise identification of battery model parameters via an optimized fitness function. Additionally, a multi-model AEKF is developed, integrating an adaptive factor into the EKF to refine the gain matrix, thereby improving the capture of the model's dynamic properties. This multi-model approach reduces estimation errors and enhances the robustness, accuracy, and stability of SOC estimation. The main contributions of this paper are as follows:
+
+1) Battery parameter estimation: An ESC model with second-order RC characteristics is used for accurate characterization, with GA-PSO employed for parameter identification, validated through model testing.
+
+2) Multi-model AEKF algorithm for SOC estimation: A multi-model AEKF algorithm is designed, combining adaptive parameters for process noise with a multi-model approach to improve SOC estimation accuracy.
+
+3) Simulation comparative analysis: SOC estimation is analyzed using constant current discharge and automotive driving cycle scenarios, comparing the multi-model AEKF with the traditional AEKF, demonstrating enhanced convergence and accuracy..
+
+## II. LITHIUM-ION BATTERY SOC ESTIMATION METHOD
+
+This paper presents an ESC model based on a second-order RC equivalent circuit, incorporating the hysteresis phenomenon observed during battery charging and discharging. The model captures the battery's dynamic behavior, static characteristics, and hysteresis effects, as shown in Fig. 1.
+
+
+
+Fig. 1. ESC model of second-order RC
+
+To identify the unknown parameters in the ESC model, the GA-PSO algorithm, an integration of Genetic Algorithm and Particle Swarm Optimization, is utilized. The process initiates with GA generating an initial population of parameter sets, which are subsequently evaluated by comparing the model's predictions with experimental battery data. GA operations, including selection, crossover, and mutation, are employed to refine these parameters, while PSO dynamically adjusts their search direction. After several iterations, the algorithm converges on the optimal parameter set, facilitating precise SOC estimation.
+
+Building on the ESC model and parameter identification, a multi-model AEKF framework is developed to enhance SOC estimation. This framework employs multi-model fusion, integrating the estimates from several models to improve filter performance and robustness. The battery SOC is quantized into discrete sets, with $\mathrm{n}$ AEKF models constructed. The conditional probability of each SOC is calculated using Bayesian rules, and the SOC with the highest probability is selected for each time step. By using conditional probability as the switching rule, the multi-model AEKF adapts to varying operating conditions and improves SOC estimation accuracy and stability. The Bayesian rule used to compute these conditional probabilities is given by the following formula:
+
+$$
+p\left( {{s}_{i} \mid {Y}_{k}}\right) = \frac{p\left( {{y}_{k} \mid {Y}_{k - 1},{s}_{i}}\right) p\left( {{Y}_{k - 1} \mid {s}_{i}}\right) p\left( {s}_{i}\right) }{\mathop{\sum }\limits_{{i = 1}}^{N}p\left( {{y}_{k} \mid {Y}_{k - 1},{s}_{i}}\right) p\left( {{Y}_{k - 1} \mid {s}_{i}}\right) p\left( {s}_{i}\right) } \tag{1}
+$$
+
+where $p\left( {s}_{i}\right)$ denotes the prior probability, reflecting the initial estimate of the state ${s}_{i}$ in the absence of any measurement information. The entire expression delineates the posterior probability of each potential state given all previous measurements ${Y}_{k - 1}$ and the current measurement ${s}_{i}$ .
+
+MATLAB simulations were conducted to model constant current discharge and automotive driving cycle discharge scenarios. A comparative experiment was set up between the traditional AEKF and the multi-model AEKF, focusing on evaluating their convergence performance and accuracy under conditions of unstable initial parameters and complex variations in discharge current.
+
+## III. CONCLUSION
+
+This paper focuses on the estimation performance of SOC in lithium-ion batteries. A second-order RC ESC model is considered, and the battery parameters are identified using the GA-PSO algorithm. Additionally, accurate estimation of battery SOC is achieved through the implementation of a multi-model Adaptive Kalman Filter. To validate the effectiveness of the proposed method, a series of simulation comparisons are conducted. The simulation results demonstrate that the proposed multi-model AEKF algorithm exhibits fast convergence and high estimation accuracy in predicting battery SOC, showcasing its superior performance in SOC estimation.
+
+## REFERENCES
+
+[1] J. S. Goud, K. R and B. Singh, "An online method of estimating state of health of a Li-ion battery," IEEE Transactions on Energy Conversion, vol. 36, no. 1, pp. 111-119, Mar. 2021.
+
+[2] X. Fan, W. Zhang, C. Zhang, A. Chen, and F. An, "SOC estimation of Li-ion battery using convolutional neural network with U-Net architecture". Energy, vol. 256, p. 124612, Oct. 2022.
+
+[3] A. Tang, Y. Huang, S. Liu, Q. Yu, W. Shen, Q. Yu, W. Shen, and R. Xiong, "A novel lithium-ion battery state of charge estimation method based on the fusion of neural network and equivalent circuit models," Applied Energy, vol. 348, p. 121578, Oct. 2023.
+
+[4] F. Li, W. Zuo, K. Zhou, Q. Li, Y. Huang, and G. Zhang, "State-of-charge estimation of lithium-ion battery based on second order resistor-capacitance circuit-PSO-TCN model," Energy, vol. 289, p.130025, Feb. 2024.
+
+[5] H. Yu, L. Zhang, W. Wang, S. Li, S. Chen, S. Yang, J. Li, and X. Liu, "State of charge estimation method by using a simplified electrochemical model in deep learning framework for lithium-ion batteries," Energy, vol. 278, p. 127846, Sep. 2023.
+
+[6] H. Zhang, J. Xiong, S. Li, L. Sun, and Y. Zhang, A, "review on estimation strategies of lithium-ion battery state of charge and health for electric vehicle applications," Journal of Power Sources, vol. 356, pp. 11-26, 2017.
+
+[7] W. Li, Y. Yang, D. Wang, and S. Yin, "The multi-innovation extended Kalman filter algorithm for battery SOC estimation," Ionics, vol. 26, pp. 6145-6156. Dec. 2020.
+
+[8] Q. Shi, Z. Jiang, Z. Wang, and L. He, "State of charge estimation by joint approach with model-based and data-driven algorithm for lithium-ion battery," IEEE Transactions on Instrumentation and Measurement, vol. 71, pp. 1-10, Aug. 2022.
+
+[9] G.L. Plett, Battery management systems, Volume I: Battery modeling, Artech House, ch.2, sec.8, p. 44, 2015.
+
+[10] S. Bittu, S. Halder, S. Kumar, N. Das, S. Bhattacharjee, and M. Ghosh, "Battery SOC Estimation Using Enhanced Self-Correcting Model-Based Extended Kalman Filter," in 2023 7th International Conference on Computer Applications in Electrical Engineering-Recent Advances (CERA). IEEE. pp. 1-6, 2023.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/DuY2U9TNuJ/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/DuY2U9TNuJ/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..475cfb16ce64d95e9e41be1e3d65f1f5505b3e7c
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/DuY2U9TNuJ/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,75 @@
+§ RESEARCH ON BATTERY SOC ESTIMATION METHOD BY COMBINING OPTIMIZATION ALGORITHM AND MULTI-MODEL KALMAN FILTERING
+
+Zhi Ming Chen
+
+College of Science
+
+Liaoning University of Technology
+
+Jinzhou, China
+
+chenzhiminglab@163.com
+
+Chang Qi Zhu
+
+Navigation College
+
+Dalian Maritime University
+
+Dalian, China
+
+zhuchangqi_work@163.com
+
+Lei Liu
+
+College of Science
+
+Liaoning University of Technology
+
+Jinzhou, China
+
+liuleill@live.cn
+
+${Abstract}$ -With the rapid growth of electric vehicles and energy storage systems, accurate state of charge (SOC) estimation has become a critical component of battery management systems (BMS), essential for preventing overcharging and over-discharging, enhancing operational safety, and extending battery life. This paper proposes a novel SOC estimation method based on an enhanced self-correcting (ESC) model incorporating a second-order RC circuit, enabling a more accurate simulation of battery response time and dynamic behavior. To improve model reliability, a genetic algorithm-particle swarm optimization (GA-PSO) approach is employed for parameter identification. Additionally, a multi-model adaptive extended Kalman filter (AEKF) algorithm is introduced to achieve precise SOC estimation. MATLAB simulations using constant current discharge and automotive driving cycle data demonstrate that the proposed method outperforms traditional AEKF algorithms, with faster convergence and higher estimation accuracy, particularly in scenarios with varying initial estimation accuracies. The results highlight the potential of this approach to significantly enhance SOC estimation in BMS, contributing to safer operation and prolonged battery life in electric vehicles and energy storage systems.
+
+Keywords—SOC estimation, Enhanced Self-Correcting model, parameter identification, GA-PSO, multi-model AEKF.
+
+§ I. INTRODUCTION
+
+As electric vehicles and energy storage systems continue to develop rapidly, the application of batteries as key energy storage devices has become increasingly widespread, highlighting the growing importance of battery management and control [1]. Within battery management systems (BMS), accurately estimating the state of charge (SOC) is a critical task [2]. Precise SOC estimation not only enables more reliable predictions of vehicle range but also improves battery utilization and helps prevent significant reductions in battery lifespan caused by overcharging or deep discharging [3]. However, the nonlinear characteristics, time-varying behavior, and electrochemical reactions of batteries make it impossible to measure their SOC directly with sensors [4]. Instead, SOC must be estimated using indirect measurements such as voltage, current, and temperature. Common SOC estimation methods include approaches based on open-circuit voltage, coulomb counting, data-driven techniques, and model-based estimation methods [5]. Each of these approaches presents distinct advantages and disadvantages [6].
+
+Among these methods, model-based estimation achieves a reasonable balance between accuracy, real-time performance, and computational cost by integrating the battery equivalent circuit model (ECM) with state estimation algorithms. The ECM is a key component in this approach. Previous studies have advanced SOC estimation using various models and algorithms. Li et al. [7] utilized a second-order RC model with a stochastic gradient algorithm for parameter identification and developed a multi-innovation extended Kalman filter, validated experimentally. Shi et al. [8] employed Bayesian belief networks and adaptive extended Kalman particle filtering, demonstrating enhanced convergence and accuracy.
+
+However, these studies largely overlook the hysteresis effect in battery charging and discharging. Gregory L. Plett [9] addressed this by introducing an Enhanced Self-Correcting (ESC) model that incorporates hysteresis into the ECM. Sk Bittu et al. [10] simulated a first-order RC ESC model with an EKF algorithm for SOC estimation but found that the model struggles with complex polarization dynamics, and the EKF's performance deteriorates with significant measurement errors.
+
+Accurate SOC estimation requires precise circuit modeling and effective algorithms. This study incorporates the hysteresis phenomenon using an ESC model with second-order RC characteristics. The GA-PSO algorithm is applied for precise identification of battery model parameters via an optimized fitness function. Additionally, a multi-model AEKF is developed, integrating an adaptive factor into the EKF to refine the gain matrix, thereby improving the capture of the model's dynamic properties. This multi-model approach reduces estimation errors and enhances the robustness, accuracy, and stability of SOC estimation. The main contributions of this paper are as follows:
+
+1) Battery parameter estimation: An ESC model with second-order RC characteristics is used for accurate characterization, with GA-PSO employed for parameter identification, validated through model testing.
+
+2) Multi-model AEKF algorithm for SOC estimation: A multi-model AEKF algorithm is designed, combining adaptive parameters for process noise with a multi-model approach to improve SOC estimation accuracy.
+
+3) Simulation comparative analysis: SOC estimation is analyzed using constant current discharge and automotive driving cycle scenarios, comparing the multi-model AEKF with the traditional AEKF, demonstrating enhanced convergence and accuracy..
+
+§ II. LITHIUM-ION BATTERY SOC ESTIMATION METHOD
+
+This paper presents an ESC model based on a second-order RC equivalent circuit, incorporating the hysteresis phenomenon observed during battery charging and discharging. The model captures the battery's dynamic behavior, static characteristics, and hysteresis effects, as shown in Fig. 1.
+
+ < g r a p h i c s >
+
+Fig. 1. ESC model of second-order RC
+
+To identify the unknown parameters in the ESC model, the GA-PSO algorithm, an integration of Genetic Algorithm and Particle Swarm Optimization, is utilized. The process initiates with GA generating an initial population of parameter sets, which are subsequently evaluated by comparing the model's predictions with experimental battery data. GA operations, including selection, crossover, and mutation, are employed to refine these parameters, while PSO dynamically adjusts their search direction. After several iterations, the algorithm converges on the optimal parameter set, facilitating precise SOC estimation.
+
+Building on the ESC model and parameter identification, a multi-model AEKF framework is developed to enhance SOC estimation. This framework employs multi-model fusion, integrating the estimates from several models to improve filter performance and robustness. The battery SOC is quantized into discrete sets, with $\mathrm{n}$ AEKF models constructed. The conditional probability of each SOC is calculated using Bayesian rules, and the SOC with the highest probability is selected for each time step. By using conditional probability as the switching rule, the multi-model AEKF adapts to varying operating conditions and improves SOC estimation accuracy and stability. The Bayesian rule used to compute these conditional probabilities is given by the following formula:
+
+$$
+p\left( {{s}_{i} \mid {Y}_{k}}\right) = \frac{p\left( {{y}_{k} \mid {Y}_{k - 1},{s}_{i}}\right) p\left( {{Y}_{k - 1} \mid {s}_{i}}\right) p\left( {s}_{i}\right) }{\mathop{\sum }\limits_{{i = 1}}^{N}p\left( {{y}_{k} \mid {Y}_{k - 1},{s}_{i}}\right) p\left( {{Y}_{k - 1} \mid {s}_{i}}\right) p\left( {s}_{i}\right) } \tag{1}
+$$
+
+where $p\left( {s}_{i}\right)$ denotes the prior probability, reflecting the initial estimate of the state ${s}_{i}$ in the absence of any measurement information. The entire expression delineates the posterior probability of each potential state given all previous measurements ${Y}_{k - 1}$ and the current measurement ${s}_{i}$ .
+
+MATLAB simulations were conducted to model constant current discharge and automotive driving cycle discharge scenarios. A comparative experiment was set up between the traditional AEKF and the multi-model AEKF, focusing on evaluating their convergence performance and accuracy under conditions of unstable initial parameters and complex variations in discharge current.
+
+§ III. CONCLUSION
+
+This paper focuses on the estimation performance of SOC in lithium-ion batteries. A second-order RC ESC model is considered, and the battery parameters are identified using the GA-PSO algorithm. Additionally, accurate estimation of battery SOC is achieved through the implementation of a multi-model Adaptive Kalman Filter. To validate the effectiveness of the proposed method, a series of simulation comparisons are conducted. The simulation results demonstrate that the proposed multi-model AEKF algorithm exhibits fast convergence and high estimation accuracy in predicting battery SOC, showcasing its superior performance in SOC estimation.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/ED7EDryw3i/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/ED7EDryw3i/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..5a1d09381e7c7e2fdf48a748a82dfa2602a51d3b
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/ED7EDryw3i/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,297 @@
+# Dynamic Target Pursuit by Multi-UAV Under Communication Coverage: ACO-MATD3 Approach
+
+${1}^{\text{st }}$ Zhuang Cao
+
+School of Information and Communication Engineering Hainan University
+
+Haikou, Hainan
+
+hnucz@hainanu.edu.cn
+
+${2}^{\text{nd }}$ Di Wu*
+
+School of Information and Communication Engineering
+
+Hainan University
+
+Haikou, Hainan
+
+hainuicaplab@hainanu.edu.cn
+
+Abstract-This study proposes a new approach for cooperative pursuit of dynamic targets under communication coverage involving multi-unmanned aerial vehicles (UAVs). This approach combines the ant colony optimization algorithm with the multi-agent twin delay deep deterministic policy gradient, called ACO-MATD3. The ACO-MATD3 algorithm dynamically adjusts hyper-parameters based on varying stages and requirements, greatly enhancing the stability and performance of cooperative multi-UAV pursuit tasks, especially under strong communication coverage. Experimental results demonstrate that the ACO-MATD3 algorithm significantly outperforms other algorithms in terms of mean reward and communication return.
+
+Index Terms-Multi-UAV, Pursuit, Communication coverage, Ant colony optimization algorithm, Multi-agent reinforcement learning
+
+## I. INTRODUCTION
+
+In recent years, multi-unmanned aerial vehicles (UAVs) have found extensive applications in fields like agriculture [1], environmental monitoring [2], and communication [3], [4], due to their flexibility and ease of deployment. As technology progresses, UAVs are tasked with more complex challenges such as pursuing dynamic targets, where UAVs need to consistently pursue and approach a moving target in complex environments through strategic adjustments. This pursuit involves a strategic interaction between the UAVs and the targets, where effective decision-making is vital for success and showcases the UAV's intelligence. Therefore, developing effective pursuit strategies is crucial.
+
+Significant research has been conducted on the pursuit of UAVs using traditional methods. For instance, the study in [5] developed a cooperative pursuit-evasion strategy for UAVs in a complex 3D environment, utilizing a heterogeneous system to enhance spatial perception and decision-making. However, this approach encounters challenges related to scalability, computational demands, and robustness in dynamic environments. In [6], the problem of minimizing the time for a UAV to pursuit a moving ground target by optimizing the pursuit strategy using sensor data. Additionally, a hierarchical game structure was proposed in [7] to enhance the cooperative pursuit-evasion capabilities of UAVs in dynamic environments. Despite these advancements, the high computational complexity of these methods and the necessity to predefine the UAVs' flight paths limit their applicability in unknown environments.
+
+Fortunately, advancements in deep reinforcement learning (DRL) have introduced new methods for addressing UAV pursuit problems. Techniques such as the deep deterministic policy gradient (DDPG) [8] and twin delay deep deterministic policy gradient (TD3) [9] enable simultaneous learning of value and policy functions, thereby enhancing algorithm efficiency and stability. However, in multi-agent environments, interactions between agents can lead to policy non-convergence when DRL algorithms are applied directly. To address this issue, multi-agent reinforcement learning (MARL) algorithms, including the multi-agent deep deterministic policy gradient (MADDPG) [10] and multi-agent twin delay deep deterministic policy gradient (MATD3) [11], have been developed. The MATD3 is an improvement based on MADDPG. These algorithms improve stability and collaboration among agents by employing a centralized training and decentralized execution (CTDE) mechanism [10].
+
+Based on these DRL methods mentioned above, several studies have attempted to utilize DRL to solve UAV pursuit tasks. An approach proposed for UAV pursuit-evasion games utilizes hierarchical maneuvering decision-making with soft actor-critic algorithm [12] to enhance autonomy and strategic flexibility in complex environments. However, this method needs to work on high-dimensional state spaces. Another study [13] proposed a UAV pursuit policy combining DDPG with imitation learning to improve sample exploration efficiency, resulting in better performance and faster convergence than traditional DDPG method. A multi-UAV pursuit-evasion game was also explored in [14], utilizing online motion planning and DRL to enhance UAV interactions in complex environments. However, these studies still do not address the challenge of maintaining communication among UAVs while performing their tasks.
+
+Based on the above related research, we propose an algorithm that combines MATD3 and ant colony optimization (ACO) algorithm to address the multi-UAV cooperative pursuit problem under communication coverage, called ACO-MATD3. The algorithm can adaptively select the optimal hyperpa-rameters at different stages during the training process. As a result, the multi-UAV system learns a policy that allows it to pursuit dynamic targets in the airspace without prior knowledge, while maintaining strong communication coverage from base stations (BSs). The main contributions of this paper are as follows:
+
+---
+
+This work is partly distributed under the "South China Sea Rising Star" Education Platform Foundation of Hainan Province (JYNHXX2023-17G), the Natural Science Foundation of Hainan Province (624MS036), the Postgraduate Innovation Projects in Hainan Province (Qhys2023-290).
+
+Corresponding author: Di Wu.
+
+---
+
+(1) In contrast to non-learning based approaches [5], [6], [7], the problem of multi-UAV cooperative pursuit problem under communication coverage is formulated as a Markov game. Each UAV operates as an independent agent while cooperating with others to maximize cumulative rewards and optimize their policies.
+
+(2) Differently from other DRL-based approaches [12], [13], [14], this study investigates the communication connectivity between multi-UAV and BSs during pursuit tasks, and considers the effect of noise in the environment on communication.
+
+(3) Compared with the MATD3 [11] algorithm, the ACO-MATD3 algorithm proposed in this study can dynamically optimize the hyperparameters according to the training stage, reduce the impact of hyperparameters on performance, and improve training efficiency and effectiveness.
+
+The paper is organized as follows: Section 2 provides the problem description and system modeling. Section 3 presents the ACO-MATD3 algorithm proposed in this paper. Section 4 analyses the results of the experiment. Section 5 concludes the paper.
+
+## II. Problem Description and System Modeling
+
+In this section, we describe the multi-UAV pursuit problem under communication coverage. Then the BS antenna model and the path loss model are introduced. Finally, we illustrate the communication coverage model used in this experiment.
+
+## A. Problem Description
+
+
+
+Fig. 1: Communication coverage strength map.
+
+This experiment investigates the multi-UAV pursuit problem under communication coverage, consisting of multi-UAV, obstacles and dynamic targets, as shown in Fig. 1. Their initial positions are randomly generated. The BSs support UAV communication, with the blue shading in Fig. 1 indicating the strength of the communication coverage. During the pursuit of dynamic targets, each UAV must avoid collisions with obstacles and maintain strong communication coverage.
+
+## B. Antenna Model and Path Loss Model
+
+This experiment formulates the antenna model of the BSs through the 3GPP [15] specification. Each BS has the same height ${h}_{BS}$ and divided into three sectors, each vertically placed with a uniform linear array of 8 elements.
+
+The radiation pattern of each element is determined by combining its horizontal and vertical radiation patterns, defined as
+
+follows:
+
+$$
+{AH} = - \min \left\lbrack {{12}{\left( \frac{\phi }{{\phi }_{3dB}}\right) }^{2},{A}_{m}}\right\rbrack \tag{1}
+$$
+
+$$
+{AV} = - \min \left\lbrack {{12}{\left( \frac{\theta - {90}}{{\theta }_{3dB}}\right) }^{2},{A}_{m}}\right\rbrack \tag{2}
+$$
+
+where $\phi$ is the azimuth angle indicating the angle of the antenna in the horizontal plane, and $\theta$ is the elevation angle indicating the angle of the antenna in the vertical plane. Both are in degrees, ${\phi }_{3dB}$ and ${\theta }_{3dB}$ are the half-power beamwidths, ${A}_{m}$ is the element gain threshold.
+
+The total gain of the antenna elements is expressed in dB as:
+
+$$
+{G}_{{el}{e}_{dB}} = {G}_{max} + {A}_{ele} \tag{3}
+$$
+
+$$
+= {G}_{\max } + \left\{ {-\min \left\lbrack {-\left( {{AH} + {AV}}\right) ,{A}_{m}}\right\rbrack }\right\}
+$$
+
+where ${A}_{ele}$ represents the power gain of the antenna element and ${G}_{\max }$ is the maximum directional gain of the antenna element. For ease of computation, we convert ${G}_{{el}{e}_{dB}}$ to the linear scale of ${G}_{ele}$ .
+
+The combined gain of the antenna array is expressed in dB as:
+
+$$
+G = {10} \times {\log }_{10}{\left| {F}_{\text{ele }} \times AF\right| }^{2} \tag{4}
+$$
+
+where ${F}_{ele}$ is the arithmetic square root of ${G}_{ele}$ , and ${AF}$ is the antenna array factor.
+
+This experiment determines whether the communication link between the UAV and the BS sector is a line of sight (LoS) link or an non-line of sight (NLoS) link by assessing whether buildings in the environment obscure the communication link. The path loss of the LoS link from the UAV to sector $m$ is expressed in $\mathrm{{dB}}$ as:
+
+$$
+{h}_{m}^{\mathrm{{LoS}}}\left( t\right) = {28} + {22}{\log }_{10}{d}_{m}\left( t\right) + {20}{\log }_{10}{f}_{c} \tag{5}
+$$
+
+where ${d}_{m}\left( t\right)$ represents the distance between the UAV and sector $m$ , and ${f}_{c}$ denotes the carrier frequency.
+
+The path loss of the NLoS link between the UAV and sector $m$ is given in $\mathrm{{dB}}$ as:
+
+$$
+{h}_{m}^{\mathrm{{NLoS}}}\left( t\right) = - {17.5} + \left( {{46} - 7{\log }_{10}h\left( t\right) }\right) {\log }_{10}{d}_{m}\left( t\right) \tag{6}
+$$
+
+$$
++ {20}{\log }_{10}\left( {{40\pi }{f}_{c}/3}\right)
+$$
+
+where $h\left( t\right)$ is the height of UAV at time $t$ .
+
+In addition, the channel small-scale fading is Rician fading in the case of LoS and Rayleigh fading in the case of NLoS.
+
+## C. Communication Model
+
+The baseband equivalent channel between the UAV and the communication BS sector $m$ at time $t$ is denoted by ${H}_{m}\left( t\right)$ , where $1 \leq m \leq M$ , and $M$ represents the total number of communication BS sectors linked with the UAV throughout its flight. The baseband equivalent channel ${H}_{m}\left( t\right)$ is influenced by the BS antenna array gain $G$ , the path loss $\beta$ , and the small-scale fading $h$ . The magnitudes of ${H}_{m}\left( t\right)$ and $\beta$ are related to the position $q\left( t\right)$ of the UAV at time $t$ , while $h$ is a random variable. The signal power received by the UAV from the communication BS sector $m$ at time $t$ can be expressed as:
+
+$$
+{P}_{m}\left( t\right) = \bar{P}{\left| {H}_{m}\left( t\right) \right| }^{2} = \bar{P}{G}_{m}\left( {q\left( t\right) }\right) \beta \left( {q\left( t\right) }\right) h\left( t\right) \tag{7}
+$$
+
+where $\bar{P}$ represents the transmit power of the BS sector $m$ , which is assumed to remain constant. The path loss is calculated using the following equation:
+
+$$
+\beta \left( {q\left( t\right) }\right) = \left\{ \begin{array}{l} P{L}_{LoS},\text{ if LoS link } \\ P{L}_{NLoS},\text{ if NLoS link } \end{array}\right. \tag{8}
+$$
+
+where $P{L}_{LoS}$ and $P{L}_{NLoS}$ are the linear scales of ${h}_{m}^{\mathrm{{LoS}}}\left( t\right)$ and ${h}_{m}^{\mathrm{{NLoS}}}\left( t\right)$ , respectively.
+
+In this experiment, the signal to interference plus noise ratio (SINR) is used as a crucial criterion for evaluating the communication coverage performance of UAVs. This criterion can be expressed as:
+
+$$
+{SIN}{R}_{t} = \frac{{P}_{m}\left( t\right) }{\mathop{\sum }\limits_{{n \neq m}}{P}_{n}\left( t\right) + {\sigma }^{2}} \tag{9}
+$$
+
+where $n$ represents the BSs not associated with the UAV at time $t$ . In this case, the communication of the UAV is affected not only by interference from all non-associated BS sectors but also by the environmental noise, which impacts the quality of its communication.
+
+To ensure communication coverage while the UAV is airborne, the SINR of the UAV should not drop below a minimum threshold $\alpha$ . That is, the UAV is not under the communication coverage of the BS when $\operatorname{SINR}\left( t\right) < \alpha$ . Each UAV has an independent SINR at time $t$ .
+
+## III. Multi-UAV COOPERATIVE PURSUIT USING ACO-MATD3
+
+In this subsection, we characterize the UAV's state space, action space, and reward function within a Markov game framework and detail our proposed ACO-MATD3 algorithm.
+
+## A. Markov Game with Multi-UAV
+
+This subsection explores the framework of the Markov game as applied to multi-UAV systems. It details the state and action spaces for UAVs and defines the reward function guiding their interactions in a complex environment.
+
+The state space for each UAV $i$ at time $t$ is defined as ${s}_{it} = \left( {{s}_{ut},{s}_{ot},{SIN}{R}_{t}}\right)$ , where ${s}_{ut} = \left( {{x}_{t},{y}_{t},{v}_{xt},{v}_{yt}}\right)$ is a combination of the position and the speed. Additionally, ${s}_{ot} =$ $\left( {{l}_{uu},{l}_{uo},{l}_{ut}}\right)$ represents the distance from the UAV to other UAVs, obstacles and dynamic targets, respectively. ${SIN}{R}_{t}$ denotes the SINR of the UAV at that moment.
+
+The action space for each UAV is discrete. The action of UAV $i$ is defined as ${V}_{u} = \left( {{V}_{x},{V}_{y}}\right)$ , which denotes the vector velocity on the $\mathrm{x}$ -axis and $\mathrm{y}$ -axis, respectively. It also changes its own speed when the UAV collides.
+
+The reward function for the UAVs in this experiment has three components. It encourages the UAV to quickly pursuit the dynamic target by considering the distance between them, providing a reward ${R}_{\text{goal }}$ upon successful pursuit. It also penalizes collisions to ensure safe flight and rewards higher SINR to promote flying in areas with better communication coverage. The reward function can be expressed as:
+
+$$
+r\left( {{s}_{t},{a}_{t}}\right) = {R}_{\text{dist }} + {R}_{\text{goal }} + {R}_{\text{coll }} + {R}_{{\text{SINR }}_{t}} \tag{10}
+$$
+
+## B. Fundamental of the ACO-MATD3 Approach
+
+
+
+Fig. 2: Framework of ACO-MATD3 algorithm.
+
+The ACO algorithm is an optimization algorithm that simulates the foraging behavior of ants. It directs the ant colony towards the optimal path in complex search spaces through pheromone accumulation and evaporation, combined with a probabilistic selection mechanism. This experiment combines the ACO algorithm with the MATD3 algorithm, aiming to dynamically choose the most appropriate learning rate $\alpha$ , discount factor $\gamma$ , and batch size $\mathcal{B}$ based on the current situation at different stages. This integration enhances the adaptability and robustness of the ACO-MATD3 algorithm. The framework of the algorithm is illustrated in Fig. 2.
+
+We define a search space containing three hyperparameters: $\alpha ,\gamma$ , and $\mathcal{B}$ . Each hyperparameter has multiple candidate values, and the range of values for these hyperparameters is given in detail in the next chapter. Additionally, we initialize a pheromone matrix.
+
+In the initialization phase, we establish an initial colony of 100 ants. Each ant's hyperparameter configuration is derived by calculating selection probabilities based on the current values in the pheromone matrix. These probabilities then guide the random selection of hyperparameters from the corresponding spaces. The selection probability for each hyperparameter value is calculated as follows:
+
+$$
+p\left( {v}_{i}\right) = \frac{\tau \left( {v}_{i}\right) }{\mathop{\sum }\limits_{{k = 1}}^{n}\tau \left( {v}_{k}\right) } \tag{11}
+$$
+
+where $p\left( {v}_{i}\right)$ represents the probability of selecting the $i$ -th value, $\tau \left( {v}_{i}\right)$ denotes the pheromone level associated with the $i$ -th value, and $n$ is the total number of possible values for the hyperparameter. This approach ensures that the search space is thoroughly explored, enabling the algorithm to evaluate a wide array of potential solutions right from the start.
+
+In the multi-UAV system, each UAV uses hyperparameters derived from the current ant's configuration to execute a pursuit task, and the resulting reward values are recorded. If the reward from a particular set of hyperparameters exceeds the highest reward recorded in previous iterations, that configuration is designated as the optimal set for the current phase.
+
+After each iteration, the pheromone level is adjusted according to the optimal hyperparameter configuration determined during the evaluation process. During this update process, the pheromone level for the chosen optimal configuration is increased to reinforce its selection in future iterations. Simultaneously, the pheromone levels for the other hyper-parameters are reduced in accordance with the evaporation rate to ensure diversity in the search process and prevent premature convergence. This pheromone updating method can be succinctly described as follows:
+
+$$
+\tau \left( {v}_{i}\right) \leftarrow \tau \left( {v}_{i}\right) + {\Delta \tau } \tag{12}
+$$
+
+$$
+\tau \left( {v}_{i}\right) \leftarrow \tau \left( {v}_{i}\right) \times \left( {1 - \rho }\right) \tag{13}
+$$
+
+where ${\Delta \tau }$ represents the increment added to the pheromone level upon a successful iteration, $\rho$ is the evaporation rate that moderates the decrease in pheromone levels to facilitate sustained exploration and exploitation balance. This dynamic adjustment ensures that the search algorithm not only intensifies exploration around proven successful parameters but also explores new potential areas effectively.
+
+In the ACO-MATD3 algorithm, the target Q-value for UAV $i$ is calculated as:
+
+$$
+{y}_{i} = {r}_{i} + \gamma \mathop{\min }\limits_{{j = 1,2}}{Q}_{{w}_{i, j}^{\prime }}\left( {{x}^{\prime },{a}_{1}^{\prime },\ldots ,{a}_{N}^{\prime }}\right) \tag{14}
+$$
+
+where ${r}_{i}$ is the reward received by UAV $i,\gamma$ is the discount factor, ${Q}_{{w}_{i, j}^{\prime }}$ is the $j$ -th target critic network of UAV $i, x$ is the joint next state of all UAVs, and ${a}_{i}^{\prime }$ represents the joint actions of all UAVs at the next time.
+
+The loss function for updating the critic networks is:
+
+$$
+L\left( {w}_{i}\right) = {\mathbb{E}}_{\left( {x,{a}_{i}, r,{x}^{\prime }}\right) \sim D}\left\lbrack {\left( {y}_{i} - {Q}_{{w}_{i}}\left( x,{a}_{1},\ldots ,{a}_{N}\right) \right) }^{2}\right\rbrack \tag{15}
+$$
+
+where ${w}_{i}$ represents the parameters of the critic network for UAV $i, D$ is the experience replay buffer.
+
+The policy update rule for the actor networks is given by:
+
+$$
+{\nabla }_{{\theta }_{i}}J\left( {\theta }_{i}\right) =
+$$
+
+$$
+{\mathbb{E}}_{x,{a}_{i} \sim D}\left\lbrack {\left. {\nabla }_{{\theta }_{i}}{\pi }_{{\theta }_{i}}\left( {s}_{i}\right) {\nabla }_{{a}_{i}}{Q}_{{w}_{i}}\left( x,{a}_{1},\ldots ,{a}_{N}\right) \right| }_{{a}_{i} = {\pi }_{{\theta }_{i}}\left( {s}_{i}\right) }\right\rbrack
+$$
+
+(16)where ${\theta }_{i}$ represents the parameters of the actor network for UAV $i,{s}_{i}$ is the state of $i$ th UAV, ${\pi }_{{\theta }_{i}}\left( {s}_{i}\right)$ is the policy of UAV $i$ .
+
+## IV. SIMULATION RESULTS AND DISCUSSION
+
+## A. Parameter Setting
+
+In this experiment, we build a $2\mathrm{\;{km}} \times 2\mathrm{\;{km}}$ urban area scenario with numerous buildings, each with a maximum height ${h}_{bd}$ of 90 meters. The presence of a LoS link is determined by examining the linear connection between the BSs and the UAVs, considering the distribution of buildings. There are seven BSs in this area, totaling $M = {21}$ sectors. The transmit power of each sector is $\bar{P} = {20}\mathrm{\;{dBm}}$ . The half-power beamwidth ${\phi }_{3dB}$ and ${\theta }_{3dB}$ both are ${65}^{ \circ }$ . The SINR interruption threshold is ${\gamma }_{th} = 1\mathrm{\;{dB}}$ . The noise power ${\sigma }^{2}$ of $5\mathrm{{dBm}}$ .
+
+The hyperparameter search spaces for the ACO-MATD3 algorithm are: learning rate $= \{ {0.005},{0.01},{0.015}\}$ , discount factor $= \{ {0.93},{0.95},{0.97}\}$ , batch size $= \{ {512},{1024}\}$ . The remaining algorithm parameters and the parameters for the DRL algorithms are provided in Table 1.
+
+TABLE I: DRL algorithm parameters setting
+
+| Command | Intent | Keywords | Thresholds |
| talk_about_system | For PiBot to play a series of audio files, explaining itself, its capabilities, its components and its features. | (hablame, háblame, cuéntame, cuentame, explicame, explícate), (ti), (sobre), (capacidades), (componentes) | 0.2 |
| talk_about_machine_care | For PiBot to play a series of audio files, explaining Machine Care, a strategical business partner for PiBot development. | (hablame, háblame, cuentame, cuéntame, explicame), (machin, machine, care, quer) | 0.6 |
| talk_about_event | For PiBot to play a series of audio files, explaining the ENCLELAC event which invited PiBot to attend. | (háblame, hablame, cuéntame, cuentame, explicame), (evento, clelac, conferencia, enclelac, claustro) | 0.6 |
| come_towards_me | Command for a future action intended to command PiBot to navigate towards the person closest to the sound direction. | (ven, vente, acercate, aproxima, aproximate), (aca, acá, aquí) | 0.6 |
| patrol | Command for future to start patrolling actions on PiBot, navigating autonomously in a set of predefined points. | (patrullaje, patrullar, vuelta), (empieza, comienza, ponte) | 0.2 |
| look_at_me | Command for future action intended to command PiBot to rotate to face the sound direction source. | (voltea, volteame, observame, observa, boltea, mi- rame, mira), (empieza, comienza, ponte) | 0.3 |
| stop_action | Some states are indefinite and are only stopped by this action. Additionally, the audio sequences can be stopped with this command. | (detente, alto, parate, cancela, basta, termina, inter- rumpe, suspende, aborta) | 0.2 |
| continue | Signals PiBot to continue with its last state, either continuing from the last played audio, or continuing an indefinite task. | (continúa, continua, reanuda) | 1 |
| TARTET | | |
+
+List of commands programmed in the speech recognition implementation for PiBot. The list presents the identifier, the intention of the matter FOR THIS STATE (IN THE STATE MACHINE), KEYWORDS FOR EACH COMMAND, AND THE MINIMUM SCORE THRESHOLD TO SURPASS TO BE CONSIDERED A POSSIBLE CANDIDATE.
+
+Table I lists the programmed commands, their intended actions, associated keywords, and the minimum score thresholds required for selection.
+
+## C. State Machine Node
+
+The State Machine Node manages the execution of tasks corresponding to received commands using the smach library. Each state within the state machine performs specific functions and makes decisions based on the commands received. The final state, FinishProgram, handles the concluding logic before terminating the program. Figure 3 depicts the structure of the state machine, which begins in the Initial Setup state and transitions to the Waiting Mode state once all necessary nodes are active. Upon receiving a command, the state machine transitions to the appropriate state to execute the corresponding task and then returns to Waiting Mode after completion or interruption. If the state machine is stopped, it moves to the End state.
+
+The Initial Setup state verifies that all required ROS nodes, specifically the Inference Node and Command Detection Node, are operational. This verification accounts for the approximately 20-second power-on time. Once confirmed, the state machine transitions to the Waiting Mode state, where it remains ready to receive and process commands from the "/state_topic" ROS topic.
+
+In the Waiting Mode state, the state machine continuously monitors for incoming commands. Upon receiving a command, it transitions to the corresponding state to execute the associated actions. All states responsible for providing explanations are fully operational, playing a series of audio files as intended. Users can interrupt these actions with the 'stop_action' command, which halts audio playback and records the last played audio. The 'continue' command also allows users to resume the last interrupted action. While commands such as 'patrol', 'come_towards_me', and 'look_at_me' are recognized and processed, the actions they trigger are scheduled for future development.
+
+Each state within the state machine is designed to handle specific functionalities, with clearly defined transitions ensuring a reliable and adaptable system. Audio feedback is integral to the state machine, enhancing user interaction by confirming received commands. When a command is successfully detected, the system plays a randomly selected confirmation audio from a pool of pre-recorded phrases. This approach confirms command recognition and adds variety to interactions, aiming to improve the user experience. Similarly, continuation commands trigger randomized audio feedback to inform users that the system has resumed its previous action.
+
+## IV. TESTING AND VALIDATION OF SPEECH RECOGNITION IMPLEMENTATION
+
+Evaluating the performance of the speech recognition system implemented in PiBot is essential to ensure effective interaction and accurate command execution and to highlight critical areas of opportunity for future work in this platform. This section details the testing methodology, including the audio samples, testing environments, and the metrics used to assess system reliability and accuracy. The testing was performed on a computer running ROS Melodic on Windows 11 through the Windows Subsystem for Linux with Ubuntu 18.04. The focus was on evaluating the algorithmic accuracy and reliability consistent across different platforms.
+
+
+
+Fig. 2. Comparison of Audio Waveforms Across Various Scenarios (spoken text said is "oye pibot hablame de ti"): a) Original clear audio waveform, b) Noise-reduced waveform from the original clear audio, c) Original audio waveform with added background noise, d) Noise-reduced waveform of the audio with added background noise.
+
+
+
+Fig. 3. Illustration of the State Machine structure implemented on PiBot, to perform different algorithms based on the command received.
+
+| Filename | Inferred Text | Detected Command | WER | Command Accuracy |
| comienza_patrullar.wav | oyemiotoveja patrullar | | 0.8 | 0 |
| comienza_patrullar_laboratorio.wav | oye ven pibot comienza a patrullar | patrol | 0.2 | 1 |
| comienza_patrullar_office.wav | oye pibot comienza a patrullar | patrol | 0 | 1 |
| cuentame_laboratorio.wav | oye pibot cuentame sobre ti | talk_about_system | 0 | 1 |
| cuentame_sobre_ti_biblio.wav | ove pibot cuentame sobre ti | talk_about_system | 0 | 1 |
| cuentame_ti_office.wav | oye pibot cuentame sobre ti | talk_about_system | 0 | 1 |
| hablameenclelac.wav | ove pibot hablame de | | 0.2 | 0 |
| hablameenclelac_biblio.wav | eoundedte | | 1 | 0 |
| hablameenclelac office.wav | ove pibot hablame de enclelac | talk_about_event | 0 | 1 |
| hablameevento biblio.wav | ove pibot o aca del evento | talk_about_event | 0.4 | 1 |
| hablameevento office.wav | ove pibot hablame de evento | talk_about_event | 0.2 | 1 |
| hablamedeeenclelac_laboratorio.wav | ove pibot hablame enclelac | talk_about_event | 0.2 | 1 |
| hablamedelevento laboratorio.wav | ove pibot hablame de evento | | 0.2 | 0 |
| hablamedeti biblio.wav | ove pibot hablame de ti | talk_about_system | 0 | 1 |
| hablamedeti_laboratorio.wav | ove pibot hablame de ti | talk_about_system | 0 | 1 |
| hablamedeti office.wav | ove pibot hablame de ti | talk_about_system | 0 | 1 |
| mc_biblio.wav | ove pibot hablame de aca | | 0.33 | 0 |
| mc laboratiorio.wav | ove pibot hablame de | | 0.33 | 0 |
| mc office.wav | ove pibot hablame de machin | talk about machine care | 0.33 | 1 |
| patrullar_laboratorio.wav | ove mibun patrullar | | 0.6 | 0 |
| pontepatrullar biblio.wav | ove pibot bonda patrullar | patrol | 0.4 | 1 |
| ponte_patrullar_office.wav | ove pibot aca patrullar | patrol | 0.4 | 1 |
| sabes evento biblio.wav | oye pibot que moes de le ven | | 0.67 | 0 |
| sabes evento laboratorio.wav | ove pibot que sabes de evento | talk about event | 0.17 | 1 |
| sabes evento office.wav | ove pibot que sabes de evento | talk about event | 0.17 | 1 |
| ven aca office.wav | ove pibot ven par aca | come towards me | 0.2 | 1 |
| ven biblio.wav | ove pibot clelac | talk about event | 0.6 | 0 |
| ven laboratorio.wav | ove pibot ven par aca | come towards me | 0.2 | 1 |
| voltea biblio.wav | ove pibot voltea | look at me | 0 | 1 |
| voltea laboratorio.wav | ove pibot voltea | look at me | 0 | 1 |
| voltea office.wav | oye pibot voltea | look at me | 0 | 1 |
+
+TABLE V
+
+LIST OF GENERATED AUDIO FILES WITH THEIR TRANSCRIPTION, INFERED TEXT, DETECTED COMMAND, WER, AND COMMAND ACCURACY METRICS.
+
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/Jz7lDfrb1j/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/Jz7lDfrb1j/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..bba31e3d31c83ca63a2c48848d99b0058ccb7a5a
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/Jz7lDfrb1j/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,288 @@
+§ WORK IN PROGRESS: ENHANCING HUMAN-ROBOT INTERACTION THROUGH A SPEECH AND COMMAND RECOGNITION SYSTEM FOR A SERVICE ROBOT USING ROS MELODIC
+
+*Note: Sub-titles are not captured in Xplore and should not be used
+
+Luis Emiliano Rodríguez Raygoza
+
+Tecnologico de Monterrey
+
+School of Engineering and Sciences Monterrey, México
+
+a01252086@tec.mx
+
+Jorge De-J. Lozoya-Santos
+
+Tecnologico de Monterrey School of Engineering and Sciences Monterrey, México jorge.lozoya@tec.mx
+
+Luis C. Félix-Herrán
+
+Tecnologico de Monterrey
+
+School of Engineering and Sciences
+
+Monterrey, México
+
+lcfelix@tec.mx
+
+Juan C. Tudon-Martinez
+
+Tecnologico de Monterrey
+
+School of Engineering and Sciences
+
+Monterrey, México
+
+jc.tudon@tec.mx
+
+${Abstract}$ -This paper presents the development and evaluation of a Speech and Command Recognition system integrated into PiBot, an autonomous service robot developed at Tecnológico de Monterrey. The system executes on Robot Operating System (ROS) Melodic framework running on a Jetson TX2 embedded computer to enable natural language interaction through Automated Speech Recognition (ASR). The study focuses on the challenges and opportunities of implementing speech recognition in real-world environments, particularly within constrained hardware platforms. The system achieved a 25% Word Error Rate (WER) and a 73% Command Accuracy, with performance varying across different testing environments. The system achieved a 25% Word Error Rate (WER) and a 73% Command Accuracy, with performance varying across different testing environments. Difficulties were noted in recognizing uncommon or non-Spanish words. A comparison with state-of-the-art models indicates room for improvement. Future work will focus on fine-tuning the model using datasets with ground truth transcriptions to enhance reliability in complex, noise-prone settings.
+
+Index Terms-Automated Speech Recognition (ASR), Human-Robot Interaction (HRI), Service Robots, Command Detection, Embedded Systems
+
+§ I. INTRODUCTION
+
+In recent years, robotics has made notable progress, with service robots becoming prominent solutions designed to communicate, interact, and assist customers [8]. As society moves toward greater automation, effective human-robot interaction is increasingly important. Among the key elements facilitating this interaction, speech algorithms are essential tools and widely used approaches in Human-Robot Interaction (HRI) [7]. Speech functions both as an input and an output in dialogue systems. As an input, it allows robots to recognize spoken language through Speech-to-Text (STT) or Automated Speech Recognition (ASR). As an output, speech synthesis converts textual responses into spoken language, enabling natural language interaction [1].
+
+This paper presents preliminary work focusing on the development and evaluation of these systems to identify areas for future improvement. The system is integrated into PiBot, an autonomous service robot developed at Tecnológico de Monterrey, with its design and development previously described [5]. PiBot's algorithms run within the Robot Operating System (ROS) Melodic framework on Ubuntu 18.04, utilizing the processing capabilities of a Jetson TX2 embedded computer. While this combination of software and hardware is functional, it presents limitations due to the constraints of embedded computer architecture, reliance on battery power, and limited availability of GPU-accelerated library versions. Additionally, the Jetson TX2, an older model, poses specific challenges impacting the system's performance and flexibility.
+
+This paper examines the challenges and opportunities of integrating speech recognition algorithms within a service robot, emphasizing the practical implementation and evaluation of these systems in real-world settings. Through experimentation and analysis, we aim to identify the strengths and limitations of current speech recognition technologies when deployed on constrained hardware platforms like the Jetson TX2, providing insights that may inform future enhancements in human-robot interaction in complex, noise-prone environments.
+
+The paper is organized as follows: Section II describes the system integration and technological configuration. Section III outlines the configuration and operation of the developed processing nodes, detailing their roles in the processing pipeline. Section IV discusses the testing and validation methodology used to evaluate the system. The validation results are presented in Section V, and conclusions are drawn in Section VI.
+
+§ II. PIBOT'S SPEECH RECOGNITION INTEGRATION
+
+The development of the Speech and Command Recognition System for PiBot is a significant enhancement, significantly improving its interactive capabilities. This system, detailed in this section, facilitates verbal communication between humans and PiBot, extending interactions beyond the existing terminal and web interface. It also sets the stage for future voice-activated motion tasks with their respective algorithms. The integration process, which began with the ReSpeaker Mic Array, is a crucial step in this journey. This device, connected via USB, provides raw audio data and the relative direction of sound, which will be used to activate motion tasks through speech, thereby enhancing PiBot's functionality.
+
+The implementation of the Speech and Command Recognition System for PiBot is designed to enable it to respond to vocal instructions, a common interface method for HRI. The process begins by connecting the ReSpeaker Mic Array and setting up an inference node. This node is dedicated to pre-processing the audio signal, performing inference on the audio data, and post-processing the results to obtain interpreted speech. The system then uses score criteria to determine whether a command is present in the inferred text. If a command is detected, it is forwarded to a state machine to execute the appropriate task on PiBot.
+
+§ A. TECHNOLOGICAL FRAMEWORK AND ADAPTATIONS
+
+This section describes the hardware and software architecture that integrates the Speech and Command Recognition system into PiBot. The integration is supported by a Jetson TX2 embedded computer running Ubuntu 18.04 with the JetPack 4.6.5 SDK and a ReSpeaker 2.0 Mic Array connected via USB for capturing audio input. The Jetson TX2 serves as the central processing unit, handling all computational tasks including audio inference, navigation, and sensor fusion. The JetPack SDK includes essential libraries, such as CUDA 10.2 and cuDNN 8.2.1, providing GPU acceleration to handle the demanding deep learning inference tasks required for real-time operation [4].
+
+The ROS Melodic framework provides a robust environment for developing modular nodes that handle specific tasks within the speech recognition pipeline. The Jetson TX2's GPU accelerates the inference process of the ASR model, enabling real-time speech processing. ROS topics and services are used for inter-node communication, allowing the ReSpeaker Node to publish audio data, the Inference Node to perform GPU-accelerated speech-to-text conversion, and the Command Detection Node to interpret commands. The State Machine Node orchestrates the execution of commands, leveraging ROS's actionlib for asynchronous task handling, which ensures that multiple actions can be managed concurrently.
+
+Due to the limitations of the Jetson TX2 hardware, several strategies were explored for configuring the necessary software environment. The Nvidia JetPack SDK is crucial for hardware-accelerated AI development, but due to compatibility constraints, the available machine learning frameworks are limited to older versions. Initially, a conda environment was considered for managing the library versions required for inference, but the lack of ARM-compatible versions proved to be a significant barrier. A compatible PyTorch Docker container was also investigated, offering GPU-accelerated support for speech recognition. Despite the potential, this approach faced practical challenges related to processing demands and frequent image deletions.
+
+Ultimately, we installed a specific version of PyTorch (provided by Nvidia) that works with CUDA 10.2, enabling us to perform GPU-accelerated inference for speech recognition tasks. This required transitioning from the torchaudio library to the librosa library for certain audio processing tasks, maintaining the same inference approach with some modifications. The difference in performance was significant: GPU-accelerated inference took 3-4 seconds, while CPU-based inference took approximately 55 seconds, emphasizing the necessity of GPU acceleration for achieving near real-time response. Figure X illustrates the hardware and software integration within PiBot, including the flow between ROS nodes, the Jetson TX2, and the ReSpeaker Mic Array.
+
+§ III. OPERATION OF PROCESSING NODES
+
+This section details the setup and functionality of the processing nodes developed for the speech and command recognition system, highlighting their roles within the framework. The system handles audio input, speech recognition, command detection, and command execution through four distinct nodes. Each node operates within the processing pipeline, collectively ensuring the system's functionality. The initial node was adapted from an existing ROS Melodic package [2], which facilitates communication with the ReSpeaker 2.0 Mic Array. This array captures audio input and provides directional sound data using its quad-microphone setup. The directional information is intended for future enhancements, such as activating motion tasks based on speech direction.
+
+The second node, developed specifically for this implementation, handles inference. It receives audio segments, preprocesses the data to reduce background noise, performs GPU-accelerated inference using the jonatasgrosman/wav2vec2- large-xlsr-53-spanish model [3], and post-processes the results to identify keywords. The third node processes the inferred text to determine if it contains a command from a predefined set of keywords and thresholds. Finally, the fourth node functions as a state machine, waiting for commands and executing the corresponding tasks on PiBot.
+
+Figure 1 illustrates the interconnection of these nodes, the topics they broadcast, and the data types transmitted between them. This visual aid clarifies the communication flow and the sequential processing steps from one node to the next. Each node and the algorithms employed are further explained in the subsequent subsections.
+
+ < g r a p h i c s >
+
+Fig. 1. Schematic representation of the audio processing framework, showcasing the workflow from audio capture to command execution. It begins with the ReSpeaker Node processing audio data, followed by the Inference Node for text inference and processing, and the Command Detection Node for command detection and selection. The State Machine Node completes the sequence by executing the corresponding actions.
+
+§ A. INFERENCE NODE CONFIGURATION AND OPERATION
+
+The Inference Node initiates the speech recognition process by handling audio segments, pre-processing them, performing inference, and converting the results into text. The node subscribes to the "/speech_audio" ROS topic, where audio segments are continuously published by the ReSpeaker node. Upon receiving an audio message, the data undergoes several processing steps before the inferred text is published to the "/audio_text_topic" as a String message for the next node.
+
+Audio processing begins with the initialization of necessary libraries. The 'rospy' library facilitates communication between ROS nodes, while 'librosa' and 'soundfile' are used for audio processing. Additional libraries support array manipulation, audio transformations, and machine learning tasks. The core of the inference task utilizes the wav2vec2-large-xlsr-53-spanish model, a speech recognition model already fine-tuned with the Common Voice Corpus 6.1 dataset for Spanish. This model, sources from Hugging Face [3], operates on wav files sampled at ${16},{000}\mathrm{\;{Hz}}$ . The dataset used for fine-tuning, Common Voice Corpus 6.1, provides a diverse range of transcriptions. For our implementation, we leveraged this pre-existing fine-tuned model to process our collected audio data without further modification. Global variables are set during initialization, and the GPU device is configured. A similarity threshold is established for fuzzy word matching, and a spell checker and a set of keywords are initialized to address common inference errors.
+
+Once the model is ready, which takes approximately 20 seconds, the ROS node and subscriber are activated to listen for audio data on the "/speech_audio" topic. The main processing occurs in the callback function, which is triggered upon receiving an audio message. The audio data is converted to a .wav file and loaded into a GPU-compatible tensor. Preprocessing is performed using the 'noisereduce' library to minimize background noise, as illustrated in Figure 2, which shows the effects of noise reduction on different audio signals.
+
+The filtered audio is then processed by the Wav2Vec 2.0 model, which transcribes the spoken content into text. Postprocessing involves mapping the inferred text to correct common transcription errors. This includes mapping terms like 'piot' to 'pibot', 'pivot' to 'pibot', and 'machin' to 'machine'. Additionally, the 'fuzzywuzzy' library performs word corrections based on the Levenshtein Distance, allowing for corrections with a similarity threshold of ${70}\%$ . Keywords such as 'pibot', 'pibotino', and 'patrullar' are specifically targeted for this correction process. The refined text is then published to the "/audio_text_topic" ROS topic for the Command Detection Node to process.
+
+§ B. COMMAND DETECTION NODE
+
+The Command Detection Node converts inferred text into detected commands to execute later. It subscribes to the "/audio_text_topic" ROS topic to receive text inputs from the Inference Node. Upon receiving a message, the node processes the text by tokenizing it into individual words for detailed analysis. Special attention is given to the keywords "pibot" and "pibotino," which identify the robot intended to receive the commands. Detecting these keywords ensures that only relevant commands are processed, filtering out unrelated speech.
+
+After recognizing the robot identifier, the node maps keywords associated with each potential command. These keywords are grouped by synonyms to improve the accuracy of the scoring mechanism, which determines the most likely intended command. This grouping prevents score inflation from repetitive similar words, ensuring a more accurate interpretation. The algorithm then evaluates the scores for each potential command against predefined thresholds. If a command's score exceeds its threshold and is the highest among the candidates, it is selected for execution. The selected command is published to the "/state_topic" ROS topic as a String message for the State Machine Node to execute relevant algorithms.
+
+TABLE I
+
+max width=
+
+Command Intent Keywords Thresholds
+
+1-4
+talk_about_system For PiBot to play a series of audio files, explaining itself, its capabilities, its components and its features. (hablame, háblame, cuéntame, cuentame, explicame, explícate), (ti), (sobre), (capacidades), (componentes) 0.2
+
+1-4
+talk_about_machine_care For PiBot to play a series of audio files, explaining Machine Care, a strategical business partner for PiBot development. (hablame, háblame, cuentame, cuéntame, explicame), (machin, machine, care, quer) 0.6
+
+1-4
+talk_about_event For PiBot to play a series of audio files, explaining the ENCLELAC event which invited PiBot to attend. (háblame, hablame, cuéntame, cuentame, explicame), (evento, clelac, conferencia, enclelac, claustro) 0.6
+
+1-4
+come_towards_me Command for a future action intended to command PiBot to navigate towards the person closest to the sound direction. (ven, vente, acercate, aproxima, aproximate), (aca, acá, aquí) 0.6
+
+1-4
+patrol Command for future to start patrolling actions on PiBot, navigating autonomously in a set of predefined points. (patrullaje, patrullar, vuelta), (empieza, comienza, ponte) 0.2
+
+1-4
+look_at_me Command for future action intended to command PiBot to rotate to face the sound direction source. (voltea, volteame, observame, observa, boltea, mi- rame, mira), (empieza, comienza, ponte) 0.3
+
+1-4
+stop_action Some states are indefinite and are only stopped by this action. Additionally, the audio sequences can be stopped with this command. (detente, alto, parate, cancela, basta, termina, inter- rumpe, suspende, aborta) 0.2
+
+1-4
+continue Signals PiBot to continue with its last state, either continuing from the last played audio, or continuing an indefinite task. (continúa, continua, reanuda) 1
+
+1-4
+X TARTET X X
+
+1-4
+
+List of commands programmed in the speech recognition implementation for PiBot. The list presents the identifier, the intention of the matter FOR THIS STATE (IN THE STATE MACHINE), KEYWORDS FOR EACH COMMAND, AND THE MINIMUM SCORE THRESHOLD TO SURPASS TO BE CONSIDERED A POSSIBLE CANDIDATE.
+
+Table I lists the programmed commands, their intended actions, associated keywords, and the minimum score thresholds required for selection.
+
+§ C. STATE MACHINE NODE
+
+The State Machine Node manages the execution of tasks corresponding to received commands using the smach library. Each state within the state machine performs specific functions and makes decisions based on the commands received. The final state, FinishProgram, handles the concluding logic before terminating the program. Figure 3 depicts the structure of the state machine, which begins in the Initial Setup state and transitions to the Waiting Mode state once all necessary nodes are active. Upon receiving a command, the state machine transitions to the appropriate state to execute the corresponding task and then returns to Waiting Mode after completion or interruption. If the state machine is stopped, it moves to the End state.
+
+The Initial Setup state verifies that all required ROS nodes, specifically the Inference Node and Command Detection Node, are operational. This verification accounts for the approximately 20-second power-on time. Once confirmed, the state machine transitions to the Waiting Mode state, where it remains ready to receive and process commands from the "/state_topic" ROS topic.
+
+In the Waiting Mode state, the state machine continuously monitors for incoming commands. Upon receiving a command, it transitions to the corresponding state to execute the associated actions. All states responsible for providing explanations are fully operational, playing a series of audio files as intended. Users can interrupt these actions with the 'stop_action' command, which halts audio playback and records the last played audio. The 'continue' command also allows users to resume the last interrupted action. While commands such as 'patrol', 'come_towards_me', and 'look_at_me' are recognized and processed, the actions they trigger are scheduled for future development.
+
+Each state within the state machine is designed to handle specific functionalities, with clearly defined transitions ensuring a reliable and adaptable system. Audio feedback is integral to the state machine, enhancing user interaction by confirming received commands. When a command is successfully detected, the system plays a randomly selected confirmation audio from a pool of pre-recorded phrases. This approach confirms command recognition and adds variety to interactions, aiming to improve the user experience. Similarly, continuation commands trigger randomized audio feedback to inform users that the system has resumed its previous action.
+
+§ IV. TESTING AND VALIDATION OF SPEECH RECOGNITION IMPLEMENTATION
+
+Evaluating the performance of the speech recognition system implemented in PiBot is essential to ensure effective interaction and accurate command execution and to highlight critical areas of opportunity for future work in this platform. This section details the testing methodology, including the audio samples, testing environments, and the metrics used to assess system reliability and accuracy. The testing was performed on a computer running ROS Melodic on Windows 11 through the Windows Subsystem for Linux with Ubuntu 18.04. The focus was on evaluating the algorithmic accuracy and reliability consistent across different platforms.
+
+ < g r a p h i c s >
+
+Fig. 2. Comparison of Audio Waveforms Across Various Scenarios (spoken text said is "oye pibot hablame de ti"): a) Original clear audio waveform, b) Noise-reduced waveform from the original clear audio, c) Original audio waveform with added background noise, d) Noise-reduced waveform of the audio with added background noise.
+
+ < g r a p h i c s >
+
+Fig. 3. Illustration of the State Machine structure implemented on PiBot, to perform different algorithms based on the command received.
+
+max width=
+
+Desired Command Transcription
+
+1-2
+look_at_me oye pibot voltea
+
+1-2
+talk_about_system oye pibot hablame de ti
+
+1-2
+talk_about_event oye pibot hablame del evento
+
+1-2
+talk_about_event oye pibot hablame de clelac
+
+1-2
+look_at_me oye pibot voltea
+
+1-2
+patrol oye pibot ponte a patrullar
+
+1-2
+talk_about_machine_care oye pibot hablame de machine care
+
+1-2
+come_towards_me oye pibot ven para aca
+
+1-2
+talk_about_system oye pibot cuentame sobre ti
+
+1-2
+talk_about_event oye pibot que sabes del evento
+
+1-2
+patrol oye pibot comienza a patrullar
+
+1-2
+
+TABLE II
+
+LIST OF PHRASES WITH THE INTENDED COMMAND USED ON THE SPEECH RECOGNITION TESTING.
+
+§ A. RECORDING AND PREPARATION OF AUDIO DATA
+
+Audio recordings for this validation were captured using PiBot's ReSpeaker Mic Array. A Python script defined each audio segment's start and end times based on keyboard inputs. This resulted in individual .wav files named according to the spoken phrase and the recording location; all recordings were mono-channel with a sample rate of ${16},{000}\mathrm{\;{Hz}}$ . Some recordings included English words such as "machine" and "care" to test the system's handling of multilingual inputs even when fine-tuned with a Spanish dataset. Table II lists the phrases and their corresponding intended commands. It is important to note that all commands are in Spanish, aligning with the system's target language.
+
+§ B. TESTING ENVIRONMENTS
+
+The speech recognition system was tested in three different environments to evaluate its performance under varying noise conditions:
+
+ * Office Floor: Recordings were made on the second floor of the CETEC tower at Tecnológico de Monterrey's In-novaction floor, where undergraduate students presented their final projects. This environment featured significant background noise due to multiple conversations and activities, providing a challenging setting for speech recognition.
+
+ * Library: PiBot was positioned on the first floor of Tecnológico de Monterrey's library. Electrical escalators, nearby shops, and student activity contributed to background noise, testing the system's ability to function in a moderately noisy environment.
+
+ * Laboratory: Recordings in the laboratory were conducted in a wide, open space with minimal background noise. This environment served as a control to assess the system's performance in ideal conditions.
+
+During testing, the speaker maintained a consistent speaking volume and pace. The impact of factors such as background noise, echo, and microphone distance were analyzed to understand their effects on WER and Command Accuracy.
+
+Figures 4, 5, and 6 show PiBot's placement in each of these environments.
+
+ < g r a p h i c s >
+
+Fig. 4. PiBot located at Innovaction, where the set of phrases were recorded.
+
+ < g r a p h i c s >
+
+Fig. 5. PiBot located at Tec's Library, where the set of phrases were recorded.
+
+§ C. EVALUATION METRICS
+
+Two primary metrics were used to evaluate the system's performance: Word Error Rate (WER) and Command Prediction Accuracy.
+
+1) Word Error Rate (WER): WER measures the difference between the recognized word sequence and the ground truth transcription by calculating the minimum number of substitutions, insertions, and deletions required to transform one sequence into the other. It is calculated using the following formula:
+
+$$
+\text{ Word Error Rate } = \frac{S + I + D}{N} \tag{1}
+$$
+
+ * S (Substitutions): The number of words in the recognized transcription that differ from the ground truth.
+
+ * I (Insertions): The number of additional words present in the recognized transcription that are not in the ground truth.
+
+ * D (Deletions): The number of words from the ground truth that are missing in the recognized transcription.
+
+ < g r a p h i c s >
+
+Fig. 6. PiBot located at Tec's Laboratory, where the set of phrases were recorded.
+
+ * N (Number of words): The total number of words in the ground truth transcription.
+
+WER was calculated using the jewel Python library, which compares the inferred transcription against the ground truth.
+
+2) Command Prediction Accuracy: Command Prediction Accuracy evaluates whether the system correctly identifies and executes the intended command. This metric is binary: a score of 1 is assigned if the inferred command matches the ground truth command, and a score of 0 otherwise.
+
+§ V. RESULTS
+
+The tests were designed to evaluate the performance of the speech recognition pipeline within operational scenarios, specifically examining how spoken words trigger commands in PiBot's state machine. These evaluations assess the system's reliability and identify potential areas for enhancement. The results are detailed in Table V, which includes each file's name, inferred text, selected command, Word Error Rate (WER), and command accuracy for each scenario. Additionally, Table III summarizes the metrics to provide an overview of overall averages and location-specific performance.
+
+max width=
+
+X Word Error Rate Command Accuracy
+
+1-3
+Library 42% 50%
+
+1-3
+Laboratory 19% 70%
+
+1-3
+Office 13% 100%
+
+1-3
+Average 25% 73%
+
+1-3
+3|c|TARLE III
+
+1-3
+
+AVERAGES OF WER AND COMMAND ACCURACY AVERAGES IN DIFFERENT LOCATIONS AND OVERALL.
+
+PiBot's speech recognition system achieved an overall command accuracy of ${73}\%$ and a WER of ${25}\%$ . Compared to advanced models such as OpenAI's Whisper, which achieves a WER below 9% and maintains performance in noisy environments [6], there is potential for further improvement in PiBot's system. The variation in WER and Command Accuracy across different environments suggests that factors beyond general noise levels influence system performance. In the library, the open space and echoes likely contributed to a higher WER of 42% and lower Command Accuracy of 50%. Despite high background noise in the office, the confined space may have allowed the microphone array to better capture the speaker's voice, resulting in a lower WER of 13% and high Command Accuracy. The laboratory, covered by glass walls and an open ceiling, showed intermediate results. These findings could indicate that environmental acoustics, such as echo and reverberation, and the directional characteristics of background noise, significantly impact the system's effectiveness.
+
+§ VI. CONCLUSIONS
+
+This study evaluated a speech recognition and command detection system for the PiBot platform, achieving an average Word Error Rate (WER) of 25% and a Command Accuracy of 73%. The system's performance varied across different testing environments, with the library setting exhibiting the highest WER of ${42}\%$ and the lowest Command Accuracy of ${50}\%$ . Conversely, despite its high background noise, the office environment demonstrated a WER of 13% and a Command Accuracy of ${100}\%$ . The laboratory environment, characterized by minimal background noise, showed a WER of ${19}\%$ and a Command Accuracy of 70%.
+
+These results might indicate that factors beyond the general noise level influence the system's performance. The unexpectedly high accuracy in the noisy office environment suggests that the system can perform well in high background noise scenarios, while the specific conditions remain unclear. In contrast, the library's moderate noise levels adversely affected both WER and Command Accuracy, being the only open location. Additionally, the system faced challenges in recognizing certain words, particularly those uncommon or not in Spanish, such as "Enclelac" and "Machine Care." This difficulty aligns with existing research, which indicates that proper nouns and less frequent terms are more susceptible to recognition errors in speech systems [1]. To address the difficulties in recognizing uncommon or non-Spanish words, we implemented post-processing techniques in the Command Detection Node, specifically mapping commonly unrecognized words to the correct terms. PiBot, the system's name, was common but had special difficulty due to the wide range of inferences.
+
+However, due to limitations and lack of a dataset for this initial setup, these methods had limited effectiveness, particularly in complex acoustic environments. To improve this, future work should focus on fine-tuning the model using a more comprehensive dataset, which should include a wide range set of words and phrases that the system is expected to handle. This dataset should be captured using the ReSpeaker microphone array across various environments with different noise levels. Such customization will likely enhance the model's ability to recognize specific terms and increase overall command accuracy, as the current lack of effective keyword detection negatively impacts the performance of short command phrases.
+
+Furthermore, advancing pre-processing techniques, mainly through more effective noise reduction methods, could significantly increase the system's robustness and accuracy. Some noise reduction techniques include other libraries for spectral subtraction and deep learning-based noise impression algorithms that do not affect speech recognition tasks. These improvements are essential for ensuring reliable and practical real-world applications of PiBot in various and potentially challenging acoustic environments.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NKhQ1UEQFb/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NKhQ1UEQFb/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..b2307313eb8ca3779a910bbb09fb6b754e5ebb2e
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NKhQ1UEQFb/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,601 @@
+# Safety-critical Obstacle Avoidance Control of Autonomous Surface Vehicles with Uncertainties and Disturbances
+
+${1}^{\text{st }}$ Gege Dong
+
+College of Marine Electrical Engineering
+
+Dalian Maritime University
+
+Dalian, China
+
+donggege0507@163.com
+
+${2}^{\text{nd }}$ Li-Ying Hao*
+
+College of Marine Electrical Engineering
+
+Dalian Maritime University
+
+Dalian, China
+
+haoliying_0305@163.com
+
+${Abstract}$ -This paper proposes a safety-critical obstacle avoidance control approach for autonomous surface vehicles (ASVs) with disturbances and uncertainties. The existing exponential control barrier functions (ECBF) are extended to handle unknown disturbances, leading to the development of input-to-state safe exponential control barrier functions (ISSf-ECBFs). An extended state observer is used to estimate unknown external marine disturbances and internal model uncertainties, based on which an anti-disturbance controller is designed. Based on the proposed ISSf-ECBFs, a quadratic programming problem is formulated to determine the optimal control input. It is proven that the closed-loop system is input-to-state safe and the errors of the closed-loop system are uniformly ultimately bounded. Simulations validate the effectiveness of the proposed control strategy.
+
+Index Terms-Autonomous surface vehicles (ASVs), safety-critical control, obstacles avoidance, input-to-state safe exponential control barrier functions (ISSf-ECBFs)
+
+## I. INTRODUCTION
+
+Autonomous Surface Vehicles (ASVs) are gaining attention for their ability to enhance maritime operations [1]-[3]. With advanced sensors and navigation systems, ASVs can navigate complex environments and perform diverse tasks [4]. They are increasingly utilized in search and rescue, fisheries management, hydrographic surveying, and offshore energy, making them a focal point for researchers in ASV control [5]-[7].
+
+ASVs navigating in dynamic marine environments face numerous challenges, primarily internal model uncertainties and external disturbances [8]. Internal uncertainties arise from modeling inaccuracies, parameter variations, and sensor noise. Additionally, ASVs must navigate unpredictable ocean conditions, such as waves, currents, and winds. These factors can adversely affect the performance of control strategy. To address this challenge, researchers have proposed various methods to enhance the robustness of the system, such as sliding mode control [9], adaptive control, and neural network control [10].
+
+The Extended State Observer (ESO) can estimate disturbances in real-time and dynamically adjust the control strategy. By treating internal model uncertainties and external disturbances as lumped disturbances for estimation, the reliance on the model can be reduced, thereby enhancing the robustness of the system.
+
+In complex maritime environments, ASVs face significant threats from various obstacles, including vessels, islands, and reefs [11]. To mitigate these risks, researchers have proposed several obstacle avoidance strategies, such as the artificial potential method [12], the velocity obstacle method [13], and the dynamic window approach [14]. Control barrier functions (CBFs), introduced in [15], have proven effective in ensuring real-time safety. In [16], the nominal controller was modified to formally adhere to safety constraints for successful obstacle avoidance. However, the control strategy in [16] did not account for model uncertainties or disturbances. To address this, [17] introduced input-to-state safe control barrier functions (ISSf-CBFs). Furthermore, [18] proposed a framework to ensure safety for uncertain nonlinear systems with structured parametric uncertainty. In [19], a collision avoidance strategy for ASVs was proposed using ISSf-CBFs. However, these functions have a relative degree of one, limiting their use in higher-order systems. To address this, [20] introduced exponential control barrier functions (ECBFs). [21] further explored ISSf-ECBFs under known perturbation bounds, but measuring such disturbances is challenging. Therefore, a safety-critical controller based on ISSf-ECBFs is crucial for ASVs dealing with unknown model uncertainties and external disturbances.
+
+This paper presents a safety-critical control strategy for Autonomous Surface Vehicles (ASVs) that accounts for external marine disturbances and internal model uncertainties. The key contributions are as follows:
+
+1) While the existing method [21] constructs safety constraints only under known disturbances or their upper bounds, this paper extends the results of input-to-state safe control barrier functions (ISSf-ECBFs) to develop safety constraints for unknown disturbances.
+
+---
+
+This work was funded by the National Natural Science Foundation of China (51939001, 52171292, 51979020, 61976033), Dalian Outstanding Young Talents Program (2022RJ05), the Topnotch Young Talents Program of China (36261402), and the Liaoning Revitalization Talents Program (XLYC2007188).
+
+---
+
+2) Unlike previous work [12], [22]-[24], this paper formulates a safety-critical controller based on ISSf-ECBFs by constructing a quadratic programming problem to facilitate collision avoidance with obstacles.
+
+The structure of the paper includes the following sections: The preliminaries and problem statement are covered in Section II. Section III gives the safety-critical controller design and Section IV is stability and safety analysis. Simulations are carried out in Section V. Section VI summarizes this article.
+
+## II. PRELIMINARIES AND PROBLEM STATEMENT
+
+## A. Notation
+
+In this paper, the notation $\parallel \cdot \parallel$ denotes the 2-norm of a vector, and $\mathfrak{R}$ represents the set of real numbers. The symbols ${\lambda }_{\min }\left( \cdot \right)$ and ${\lambda }_{\max }\left( \cdot \right)$ indicate the smallest and largest eigenvalues of a symmetric matrix, respectively.
+
+Let $\beta \left( r\right)$ be a scalar continuous function defined for $r \in$ $\lbrack - b, a)$ . If $a = \infty , b = 0$ , and $\beta \left( r\right) \rightarrow \infty$ as $r \rightarrow \infty$ , then $\beta \left( r\right)$ belongs to class ${\mathcal{K}}_{\infty }$ . If $a, b = \infty ,\beta \left( r\right) \rightarrow \infty$ as $r \rightarrow \infty$ , and $\beta \left( r\right) \rightarrow - \infty$ as $r \rightarrow - \infty$ , it represents an extended class ${\mathcal{K}}_{\infty }$ , denoted as ${\mathcal{K}}_{\infty , e}$ .
+
+B. Input-to-state Safe Exponential Control Barrier Functions Consider the following system
+
+$$
+\dot{x} = f\left( x\right) + g\left( x\right) u + {d}_{w} \tag{1}
+$$
+
+where $x\left( t\right) \in {\Re }^{n}$ denotes state and $u \in {\Re }^{m}$ denotes control input. The term ${d}_{w}$ denotes bounded disturbances. The function $f\left( x\right) \in {\Re }^{n}$ and $g\left( x\right) \in {\Re }^{n \times m}$ are locally Lipschitz continuous.
+
+Definition 1. [25] The set $\mathcal{C} \in {\Re }^{n}$ is described as
+
+$$
+\mathcal{C} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) \geq 0}\right\}
+$$
+
+$$
+\partial \mathcal{C} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) = 0}\right\}
+$$
+
+$$
+\operatorname{Int}\left( \mathcal{C}\right) \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) > 0}\right\} \tag{2}
+$$
+
+where $h\left( \cdot \right) \in {\Re }^{n} \mapsto \Re$ represents a continuously differentiable function, and $\mathcal{C}$ is referred to as the safe set. If for all ${x}_{0} \in \mathcal{C}$ it holds that $x\left( t\right) \in \mathcal{C}$ for every $t \in I\left( {x}_{0}\right)$ , then the set $\mathcal{C}$ is considered forward invariant. Consequently, the system described by (1) with ${d}_{w}\left( t\right) = 0$ can be deemed safe on $\mathcal{C}$ .
+
+Definition 2. The relative degree of $S\left( x\right) : {\Re }^{n} \rightarrow \Re$ with respect to the system (1) refers to the number of derivatives required along the dynamics of (1) before the control input $u$ explicitly appears.
+
+Definition 3. [17] For system (1), an extended set ${\mathcal{C}}_{d} \supset \mathcal{C}$ is expressed as follows
+
+$$
+{\mathcal{C}}_{d} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) + {\beta }_{d}\left( {\begin{Vmatrix}{d}_{w}\left( t\right) \end{Vmatrix}}_{\infty }\right) \geq 0}\right\}
+$$
+
+$$
+\partial {\mathcal{C}}_{d} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) + {\beta }_{d}\left( {\begin{Vmatrix}{d}_{w}\left( t\right) \end{Vmatrix}}_{\infty }\right) = 0}\right\}
+$$
+
+$$
+\operatorname{Int}\left( {\mathcal{C}}_{d}\right) \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) + {\beta }_{d}\left( {\begin{Vmatrix}{d}_{w}\left( t\right) \end{Vmatrix}}_{\infty }\right) > 0}\right\} \tag{3}
+$$
+
+where ${\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty } \leq {\bar{d}}_{w}$ , a positive constant, and $S\left( x\right)$ is a continuous function, with ${\beta }_{d}\left( \cdot \right) \in {\mathcal{K}}_{\infty , e}$ .
+
+Definition 4. (ISSf [17]) If the control input $u$ and the function ${\beta }_{d}$ ensure the forward invariance of the set ${\mathcal{C}}_{d}$ , then the system (1) with disturbances is ISSf on $\mathcal{C}$ .
+
+Definition 5. (ISSf-ECBF [17]) Considering the sets ${\mathcal{C}}_{d}$ defined by (3), $S\left( x\right)$ , which has a relative degree $\rho > 1$ , qualifies as an ISSf-ECBF for the system described in (1). This holds true if, for all $x \in {\Re }^{n}$ , there exist a bound ${\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty } \leq {\bar{\tau }}_{w}$ and a function $\gamma \left( \cdot \right) \in {\mathcal{K}}_{\infty , e}$ that satisfies
+
+$$
+\mathop{\sup }\limits_{{u \in \mathcal{U}}}\left\lbrack {{\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w}}\right.
+$$
+
+$$
+\left. {+{\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}}\right\rbrack \geq - \gamma \left( {\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }\right) \tag{4}
+$$
+
+The terms ${\mathcal{L}}_{f}^{\rho }$ and ${\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}$ represent the Lie derivatives of the function $S\left( x\right) .{\mathcal{T}}_{s} = {\left\lbrack \begin{array}{llll} {p}_{0} & {p}_{1} & \ldots & {p}_{\iota } \end{array}\right\rbrack }^{T}$ where ${p}_{i}$ is positive constant. ${\mathcal{H}}_{s} = {\left\lbrack \begin{array}{llll} S\left( x\right) & {\mathcal{L}}_{f}S\left( x\right) & \ldots & {\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) \end{array}\right\rbrack }^{T}$ .
+
+Lemma 1. If $S\left( x\right)$ functions as an ISSf-ECBF for the system (1) in the set $\mathcal{C}$ , then any controller $u \in \mathcal{U}$ that is Lipschitz continuous and valid for all $x \in {\Re }^{n}$ must satisfy
+
+$$
+\mathcal{U}\left( x\right) = \left\{ {u \in {\Re }^{m} : {\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u}\right.
+$$
+
+$$
+\left. {+{\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w} + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s} \geq - \gamma \left( {\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }\right) }\right\} . \tag{5}
+$$
+
+This implies that the set ${\mathcal{C}}_{d}$ is forward invariant. In other words, the system (1) is ISSf on the set $\mathcal{C}$ .
+
+### C.ASV Model
+
+The kinematics and kinetics of ASV can be described as:
+
+$$
+\dot{\eta }\left( t\right) = R\left( \psi \right) \nu \left( t\right) \tag{6}
+$$
+
+$$
+M\dot{\nu }\left( t\right) = f\left( \nu \right) + {d}_{w}\left( t\right) + \tau \left( t\right)
+$$
+
+where $\eta \left( t\right) = {\left\lbrack \begin{array}{ll} \bar{p}\left( t\right) & \psi \left( t\right) \end{array}\right\rbrack }^{T} \in {\Re }^{3}$ represents the position and heading of ASV. $R\left( \psi \right) = \operatorname{diag}\left\{ {{R}_{2}\left( \psi \right) ,1}\right\}$ is a rotate matrix
+
+with
+
+$$
+{R}_{2}\left( \psi \right) = \left\lbrack \begin{matrix} \cos \left( \psi \right) & - \sin \left( \psi \right) \\ \sin \left( \psi \right) & \cos \left( \psi \right) \end{matrix}\right\rbrack . \tag{7}
+$$
+
+The vector $\nu \left( t\right) = {\left\lbrack \begin{array}{lll} u\left( t\right) & v\left( t\right) & r\left( t\right) \end{array}\right\rbrack }^{T} \in {\Re }^{3}$ represents the surge velocity, sway velocity, and yaw velocity, respectively. The matrix $M$ denotes the inertial matrix. $f\left( \nu \right)$ represents the Coriolis and centripetal matrix, damping matrix, and unmodeled hydrodynamics. The vector $\tau \left( t\right)$ signifies the forces produced by the actuators. The external disturbances, caused by wind, waves, and ocean currents, are represented by ${d}_{w}\left( t\right) = {\left\lbrack \begin{array}{lll} {d}_{w1}\left( t\right) & {d}_{w2}\left( t\right) & {d}_{w3}\left( t\right) \end{array}\right\rbrack }^{T} \in {\Re }^{3}.$
+
+Letting $q\left( t\right) = R\left( \psi \right) \nu \left( t\right)$ ,(6) can be rewritten as
+
+$$
+\dot{p} = q \tag{8}
+$$
+
+$$
+\dot{q} = \xi + R{M}^{-1}\tau
+$$
+
+where $\xi = R{M}^{-1}\left( {{d}_{w} + f\left( \nu \right) }\right) + \dot{R}\nu$ .
+
+The desired parameterized path is set as ${p}_{0}\left( \theta \right) =$ ${\left\lbrack {x}_{0}\left( \theta \right) ,{y}_{0}\left( \theta \right) ,{\psi }_{0}\left( \theta \right) \right\rbrack }^{\mathrm{T}},{\psi }_{0}\left( \theta \right) = \arctan \left( {{y}_{0}^{\theta }\left( \theta \right) /{x}_{0}^{\theta }\left( \theta \right) }\right)$ where $\theta$ represents path variable. ${y}_{0}^{\theta }\left( \theta \right)$ and ${x}_{0}^{\theta }\left( \theta \right)$ is the partial derivative of ${y}_{0}\left( \theta \right)$ and ${x}_{0}\left( \theta \right)$ , respectively. In addition, it is assumed that ${p}_{0}^{\theta }\left( \theta \right)$ is bounded.
+
+## D. Problem Formulation
+
+The safety-critical obstacle avoidance controller of ASV is required to achieve the following tasks:
+
+(1) Geometric task: Ensure that the ASV follows the desired path, meaning that
+
+$$
+\mathop{\lim }\limits_{{t \rightarrow \infty }}\begin{Vmatrix}{p\left( t\right) - {p}_{0}\left( \theta \right) }\end{Vmatrix} < {l}_{1} \tag{9}
+$$
+
+where ${l}_{1} \in \mathfrak{R}$ denotes a small positive constant.
+
+(2) Dynamic task: The derivative of the path variable $\theta$ converge to the desired speed
+
+$$
+\mathop{\lim }\limits_{{t \rightarrow \infty }}\begin{Vmatrix}{\dot{\theta }\left( t\right) - {u}_{d}\left( t\right) }\end{Vmatrix} < {l}_{2} \tag{10}
+$$
+
+where ${u}_{d}\left( t\right)$ represents desired speed and ${l}_{2}$ is a small positive constant.
+
+(3) Obstacle avoidance task: To prevent collisions between the ASV and obstacles, the following condition must be met
+
+$$
+\begin{Vmatrix}{\bar{p}\left( t\right) - {\bar{p}}_{k}\left( t\right) }\end{Vmatrix} > {r}_{k} + {d}_{k} \tag{11}
+$$
+
+where ${\bar{p}}_{k}\left( t\right) ,{r}_{k}$ , and ${d}_{k}$ represent the position, the radius, and the minimum obstacle avoidance distance of the $k$ th obstacle.
+
+## III. Main Results
+
+## A. ISSf-ECBF with Unknown Disturbancces
+
+While the previous studies have made substantial progress, they were mainly directed at scenarios with known disturbances or predefined upper bounds. To alleviate this limitation, the following theorem is presented to account for unknown disturbances.
+
+Theorem 1. Given the ISSf-ECBF $S\left( x\right)$ as defined in Definition 5 for the system (1) on the set $\mathcal{C}$ , if there exists a bound ${\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty } \leq {\bar{d}}_{w}$ such that for every $x \in {\Re }^{n}$ , the following inequality holds
+
+$$
+\mathop{\sup }\limits_{{u \in \mathcal{U}}}\left\lbrack {{\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}}\right.
+$$
+
+$$
+\left. {-{\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }\right\rbrack \geq 0 \tag{12}
+$$
+
+and the admissible control set satisfies as
+
+$$
+\mathcal{U}\left( x\right) = \left\{ {u \in {\Re }^{m} : {\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}}\right.
+$$
+
+$$
+- {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) \geq 0\} . \tag{13}
+$$
+
+Then, we can obtain that the system (1) is ISSf on $\mathcal{C}$ .
+
+Proof. For $u \in \mathcal{U}\left( x\right)$ , one has
+
+$$
+{\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w} + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}
+$$
+
+$$
+\geq {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) + {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w}
+$$
+
+$$
+\geq {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right)
+$$
+
+$$
+- \parallel \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\parallel {\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }. \tag{14}
+$$
+
+Adding and subtracting $\frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }^{2}}{4}$ yields
+
+$$
+\dot{h} \geq {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x} - \frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }}{2}\right) }^{2} - \frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }^{2}}{4}
+$$
+
+$$
+\geq - \frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }^{2}}{4} \tag{15}
+$$
+
+which is of the form (4).
+
+Remark 1. Compared with [21], the proposed ISSf-ECBF can deal with unknown perturbations. Although asymptotically stable ESO is used in reference 21, in practice, the disturbance estimation error is difficult to be 0 . Thus, it is essential to develop ISSf-ECBFs that ensure safety in the presence of unknown disturbances.
+
+## B. Anti-disturbance Controller Design
+
+In this section, we will focus on designing a safety-critical controller. The control architecture for the proposed strategy is illustrated in Figure 1.
+
+
+
+Fig. 1. Control architecture of the safety-critical controller for the ASV.
+
+Firstly, we utilize the ESO to obtain the estimations of the model uncertainties, external disturbances in this part. In addition, the ESO relies on the following assumption.
+
+Assumption 1. $\dot{\xi }\left( t\right)$ is a bounded function meeting
+
+$$
+\parallel \dot{\xi }\left( t\right) \parallel \leq {\xi }^{ * } \tag{16}
+$$
+
+where ${\xi }^{ * }$ is a positive constant.
+
+Then, the ESO is devised to estimate model uncertainties, external disturbances.
+
+$$
+\left\{ \begin{array}{l} \dot{\widehat{q}}\left( t\right) = - {K}_{1}\widetilde{q}\left( t\right) + \widehat{\xi }\left( t\right) + R{M}^{-1}\tau \\ \dot{\widehat{\xi }}\left( t\right) = - {K}_{2}\widetilde{q}\left( t\right) \end{array}\right. \tag{17}
+$$
+
+where $\widehat{q}\left( t\right)$ and $\widehat{\xi }\left( t\right)$ represent the estimates of $q\left( t\right)$ and $\xi \left( t\right)$ . The observer matrices ${\left\lbrack \begin{array}{ll} {K}_{1} & {K}_{2} \end{array}\right\rbrack }^{T} = {\left\lbrack \begin{array}{ll} {2w}{I}_{3} & {w}^{2}{I}_{3} \end{array}\right\rbrack }^{T}$ where $w$ is the observer bandwidth.
+
+Defining $\widetilde{q}\left( t\right) = \widehat{q}\left( t\right) - q\left( t\right)$ , and $\widetilde{\xi }\left( t\right) = \widehat{\xi }\left( t\right) - \xi \left( t\right)$ are the estimates. The dynamics of $\widetilde{q}\left( t\right)$ and $\widetilde{\xi }\left( t\right)$ can be written as
+
+$$
+\left\{ \begin{array}{l} \dot{\widetilde{q}}\left( t\right) = - {K}_{1}\widetilde{q}\left( t\right) + \widetilde{\xi }\left( t\right) \\ \dot{\widetilde{\xi }}\left( t\right) = - {K}_{2}\widetilde{q}\left( t\right) - \dot{\xi }\left( t\right) . \end{array}\right. \tag{18}
+$$
+
+Next, (18) can be rewritten as
+
+$$
+{\dot{E}}_{o}\left( t\right) = T{E}_{o}\left( t\right) - D\dot{\xi }\left( t\right) \tag{19}
+$$
+
+where ${E}_{o}\left( t\right) = {\left\lbrack \begin{array}{ll} {\widetilde{q}}^{\mathrm{T}}\left( t\right) & {\widetilde{\xi }}^{\mathrm{T}}\left( t\right) \end{array}\right\rbrack }^{\mathrm{T}} \in {\Re }^{6}$ and
+
+$$
+T = \left\lbrack \begin{array}{ll} - {K}_{1} & {I}_{3} \\ - {K}_{2} & {0}_{3} \end{array}\right\rbrack , D = \left\lbrack \begin{array}{l} {0}_{3} \\ {I}_{3} \end{array}\right\rbrack .
+$$
+
+Then, the following tracking error is defined as ${e}_{1} = p -$ ${p}_{0}\left( \theta \right)$ . By taking the derivative of ${e}_{1}$ and using (6), we can
+
+get
+
+$$
+{\dot{e}}_{1} = q - {p}_{0}^{\theta }\left( \theta \right) \dot{\theta }. \tag{20}
+$$
+
+Let ${u}_{d} - \vartheta \left( t\right) = \dot{\theta }\left( t\right)$ , one can obtain
+
+$$
+{\dot{e}}_{1} = q - {p}_{0}^{\theta }\left( \theta \right) \left( {{u}_{d} - \vartheta }\right) . \tag{21}
+$$
+
+The kinematic guidance law ${q}_{d}$ is designed as follows to stabilize ${e}_{1}$ :
+
+$$
+{q}_{d} = - {k}_{1}{e}_{1} + {p}_{0}^{\theta }\left( \theta \right) {u}_{d} \tag{22}
+$$
+
+and
+
+$$
+\dot{\vartheta } = - \ell \left( {\vartheta + \mu {p}_{0}^{\theta }{\left( \theta \right) }^{\mathrm{T}}{e}_{1}}\right) \tag{23}
+$$
+
+where ${k}_{1} = \operatorname{diag}\left\{ {{k}_{11},{k}_{12},{k}_{13}}\right\} ,\ell$ and $\mu$ are positive constants.
+
+To proceed, defining ${e}_{2} = q - {\widehat{q}}_{d}$ , where ${\widehat{q}}_{d}$ is the estimate of ${q}_{d}$ . ${\widehat{q}}_{d}$ can be obtained by using the following filtering scheme:
+
+$$
+{t}_{d}{\dot{\widehat{q}}}_{d} + {\widehat{q}}_{d} = {q}_{d},\;{\widehat{q}}_{d}\left( 0\right) = {q}_{d}\left( 0\right) \tag{24}
+$$
+
+where ${t}_{d}$ is a positive constant. Let
+
+$$
+{e}_{d} = {\widehat{q}}_{d} - {q}_{d}. \tag{25}
+$$
+
+And ${\dot{q}}_{d} \triangleq a = {\left\lbrack \begin{array}{lll} {a}_{1} & {a}_{2} & {a}_{3} \end{array}\right\rbrack }^{T} \cdot {a}_{j}$ is bounded by $\left| {a}_{j}\right| \leq {a}_{j}^{ * }$ , $j = 1,2,3$ , where ${a}_{j}^{ * }$ is a positive constant. For details, please refer to [26].
+
+Then, the time derivative of ${e}_{2}$ yields
+
+$$
+{\dot{e}}_{2} = \xi + R{M}^{-1}\tau \left( t\right) + \frac{{e}_{d}}{{t}_{d}}. \tag{26}
+$$
+
+To stabilize ${e}_{2}$ , the anti-disturbance control law is developed as follows:
+
+$$
+{\tau }_{c}\left( t\right) = M{R}^{T}\left( {-\widehat{\xi } - {e}_{1} - \frac{{e}_{d}}{{t}_{d}} - {k}_{2}{e}_{2}}\right) \tag{27}
+$$
+
+where ${k}_{2} = \operatorname{diag}\left\{ {{k}_{21},{k}_{22},{k}_{23}}\right\}$ . Denote $\tau = {\tau }_{c} + {\tau }_{e}$ .
+
+## C. Safety-critical Obstacle Avoidance Controller
+
+In this part, considering the collision with obstacles and ASV to design the optimal surge and sway force of safety conditions. From (8), we can get
+
+$$
+\dot{\bar{p}} = \bar{q} \tag{28}
+$$
+
+$$
+\dot{\bar{q}} = {\widehat{\xi }}_{2} + {\tau }_{2} - {\widetilde{\xi }}_{2}
+$$
+
+where $\bar{p}$ denotes ${\left\lbrack x\left( t\right) , y\left( t\right) \right\rbrack }^{T},\bar{q}$ denotes ${R}_{2}\left( \psi \right) {\left\lbrack u, v\right\rbrack }^{T}.{\widehat{\xi }}_{2},{\widetilde{\xi }}_{2}$ and ${\tau }_{2}$ is the first two dimensions of $\widehat{\xi },\widetilde{\xi }$ and $\tau$ , respectively. ${\bar{p}}_{k} = {\left\lbrack {x}_{k},{y}_{k}\right\rbrack }^{T}$ is position of $k$ th obstacle.
+
+We choose the following candidate ISSf-ECBF
+
+$$
+{S}_{k}\left( s\right) = {\begin{Vmatrix}{\bar{p}}_{ek}\end{Vmatrix}}^{2} - {\left( {r}_{k} + {d}_{k}\right) }^{2} \tag{29}
+$$
+
+where ${\bar{p}}_{ek} = \bar{p} - {\bar{p}}_{k}, s = {\left\lbrack {\bar{p}}^{T},{\bar{q}}^{T}\right\rbrack }^{T}$ . To achieve the objective of obstacle avoidance, the set $\mathcal{C}$ can be obtained
+
+$$
+\mathcal{C} = \left\{ {\bar{p} \in {\mathbb{R}}^{2} : {S}_{k}\left( s\right) = {\begin{Vmatrix}{\bar{p}}_{ek}\end{Vmatrix}}^{2} - {\left( {r}_{k} + {d}_{k}\right) }^{2} \geq 0}\right\} \tag{30}
+$$
+
+For ease of notation, it is denoted by ${S}_{k}$ in the sequel. The safety constraint with ${S}_{k}\left( s\right)$ is described as
+
+$$
+\mathcal{U} = \left\{ {{\tau }_{2} : {\mathcal{L}}_{f}^{2}{S}_{k} + {\mathcal{L}}_{g}{\mathcal{L}}_{f}{S}_{k}{\tau }_{2} - {\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) }\right.
+$$
+
+$$
+\left. {+{\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s} \geq 0}\right\} \tag{31}
+$$
+
+where ${\mathcal{L}}_{f}^{2}{S}_{k} = 2{\bar{q}}^{T}\bar{q} + 2{\bar{p}}_{ek}^{T}{\widehat{\xi }}_{2},{\mathcal{L}}_{g}{\mathcal{L}}_{f}{S}_{k} = 2{\bar{p}}_{ek}^{T},{\mathcal{T}}_{s} =$ ${\left\lbrack {\beta }^{2},2\beta \right\rbrack }^{T}$ . For the ASV, ensuring safety takes precedence over geometric objectives. Based on the safety constraint (31), the following quadratic programming problem is constructed.
+
+$$
+{\tau }^{ * } = \mathop{\operatorname{argmin}}\limits_{{\tau \in {\Re }^{m}}}J\left( \tau \right) = {\begin{Vmatrix}\tau - {\tau }_{c}\end{Vmatrix}}^{2}
+$$
+
+$$
+\text{s.t.} - {\mathcal{L}}_{g}{\mathcal{L}}_{f}{S}_{k}\tau \leq \phi \tag{32}
+$$
+
+where $\phi = 2{\bar{q}}^{T}\bar{q} - - {\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) + 2{\bar{p}}_{ek}^{T}{\widehat{\xi }}_{2} + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}$ . The ${\tau }^{ * }$ is obtained by solving the above quadratic programming problem.
+
+Remark 2. The proposed safety-critical controller can avoid obstacles while ensuring minimal impact on the given tracking task.
+
+## IV. STABILITY AND SAFETY ANALYSIS
+
+In this section, we will conduct stability and safety analysis of the closed-loop system.
+
+## A. Stability Analysis
+
+Lemma 2. The observer error subsystem in (19) is ISS, and the error signals being $\widetilde{q}$ and $\widetilde{f}$ are bounded by
+
+$$
+\begin{Vmatrix}{{E}_{o}\left( t\right) }\end{Vmatrix} \leq \sqrt{\frac{{\lambda }_{\max }\left( N\right) }{{\lambda }_{\min }\left( N\right) }}\max \left\{ {\begin{Vmatrix}{{E}_{o}\left( {t}_{0}\right) }\end{Vmatrix}{e}^{-{\gamma }_{1}\left( {t - {t}_{0}}\right) /2},}\right.
+$$
+
+$$
+\frac{2\parallel {ND}\parallel {\xi }^{ * }}{\varsigma {}_{1}\kappa }\} ,\forall t \geq {t}_{0} \tag{33}
+$$
+
+where ${\gamma }_{1} = \left( {\left\lbrack {{\varsigma }_{1}\left( {1 - \kappa }\right) }\right\rbrack /\left\lbrack {{\lambda }_{\max }\left( N\right) }\right\rbrack }\right)$ and $0 < \kappa < 1$ provided that
+
+$$
+{T}^{T}N + {NT} \leq - {\varsigma }_{1}I \tag{34}
+$$
+
+where ${\varsigma }_{1} \in \mathfrak{R}$ is a positive constant.
+
+Proof. Choose the following Lyapunov function
+
+$$
+{V}_{1} = \left( {1/2}\right) {E}_{o}^{\mathrm{T}}\left( t\right) N{E}_{o}\left( t\right) . \tag{35}
+$$
+
+Taking (34) into account, one has ${\dot{V}}_{1} = {E}_{o}{\left( t\right) }^{\mathrm{T}}N\left( {T{E}_{o}\left( t\right) - }\right.$ $\left. {D\dot{\xi }\left( t\right) }\right) \leq - \frac{{\varsigma }_{1}}{2}{\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2} + \begin{Vmatrix}{{E}_{o}\left( t\right) }\end{Vmatrix}\parallel {ND}\parallel \parallel \dot{\xi }\left( t\right) \parallel$ Since $\begin{Vmatrix}{{E}_{o}\left( t\right) }\end{Vmatrix} \geq \left\lbrack \left( {2\parallel {ND}\parallel \parallel \dot{\xi }\left( t\right) \parallel /{\varsigma }_{1}\kappa }\right) \right\rbrack$ , we have
+
+$$
+{\dot{V}}_{1} \leq - \frac{{\varsigma }_{1}}{2}\left( {1 - \kappa }\right) {\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2}. \tag{36}
+$$
+
+It follows that the observer error subsystem described by (19) is ISS. It is important to note that ${V}_{1}$ is bounded and satisfies the inequality $\left( {\left\lbrack {{\lambda }_{\min }\left( N\right) }\right\rbrack /2}\right) {\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2} \leq {V}_{1} \leq$ $\left( {\left\lbrack {{\lambda }_{\max }\left( N\right) }\right\rbrack /2}\right) {\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2}$ . From this, we can derive (33).
+
+Next, we will outline the stability analysis of the closed-loop system.
+
+Lemma 3. Taking into account the error dynamics represented by (21) and (26), the error signals ${e}_{1},{e}_{2},{e}_{r},\vartheta$ , and $\widetilde{\gamma }$ are uniformly ultimately bounded by
+
+$$
+\parallel E\parallel \leq \sqrt{\frac{{\lambda }_{\max }\left( Q\right) }{{\lambda }_{\min }\left( Q\right) }}\max \left\{ {\begin{Vmatrix}{E\left( {t}_{0}\right) }\end{Vmatrix}{e}^{-{\gamma }_{2}\left( {t - {t}_{0}}\right) /2},}\right.
+$$
+
+$$
+\frac{{E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }{\epsilon {\varsigma }_{2}},\forall t \geq {t}_{0} \tag{37}
+$$
+
+where $Q = \operatorname{diag}\{ 1,1/\ell \mu \} ,{\gamma }_{2} = 2{\varsigma }_{2}\left( {1 - \epsilon }\right) /{\lambda }_{\max }\left( Q\right)$ .
+
+Proof. The constructed Lyapunov function is
+
+$$
+{V}_{2} = \frac{1}{2}\left( {{e}_{1}^{\mathrm{T}}{e}_{1} + {e}_{2}^{\mathrm{T}}{e}_{2} + {e}_{d}^{\mathrm{T}}{e}_{d}}\right) + \frac{{\vartheta }^{2}}{2\ell \mu }.
+$$
+
+According to (20),(23)-(27) , the time derivative of ${V}_{2}$ is
+
+$$
+{\dot{V}}_{2} = {e}_{1}^{\mathrm{T}}\left( {{e}_{2} + {e}_{d}}\right) - {e}_{1}^{\mathrm{T}}{k}_{1}{e}_{1} + {e}_{2}^{\mathrm{T}}f + {e}_{2}^{\mathrm{T}}\frac{{e}_{d}}{{t}_{d}} + {e}_{d}^{\mathrm{T}}\left( {-\frac{{e}_{d}}{{t}_{d}} - a}\right)
+$$
+
+$$
++ {e}_{2}^{\mathrm{T}}\left( {-\widehat{f} - {e}_{1} - \frac{{e}_{d}}{{t}_{d}} - {k}_{2}{e}_{2}}\right) + {e}_{2}^{\mathrm{T}}R{M}^{-1}{\tau }_{e} - \frac{{\vartheta }^{2}}{\mu }. \tag{38}
+$$
+
+Finally, we can obtain
+
+$$
+{\dot{V}}_{2} \leq - \left( {{\lambda }_{\min }\left( {k}_{1}\right) - \frac{1}{2}}\right) {\begin{Vmatrix}{e}_{1}\end{Vmatrix}}^{2} - \left( {\frac{1}{{t}_{d}} - \frac{1}{2}}\right) {\begin{Vmatrix}{e}_{d}\end{Vmatrix}}^{2} + \begin{Vmatrix}{e}_{d}\end{Vmatrix}\begin{Vmatrix}{a}^{ * }\end{Vmatrix}
+$$
+
+$$
+- {\lambda }_{\min }\left( {k}_{2}\right) {\begin{Vmatrix}{e}_{2}\end{Vmatrix}}^{2} + \begin{Vmatrix}{e}_{2}\end{Vmatrix}\parallel \widetilde{f}\parallel + \begin{Vmatrix}{e}_{2}^{\mathrm{T}}\end{Vmatrix}\begin{Vmatrix}{M{\tau }_{e}}\end{Vmatrix} - \frac{{\vartheta }^{2}}{\mu }. \tag{39}
+$$
+
+Choose ${\lambda }_{\min }\left( {k}_{1}\right) - \frac{1}{2} > 0,\frac{1}{{t}_{d}} - \frac{1}{2} > 0$ . Then, define ${\varsigma }_{2} = \min \left( {{\lambda }_{\min }\left( {k}_{1}\right) - \frac{1}{2},{\lambda }_{\min }\left( {k}_{2}\right) ,\frac{1}{{t}_{d}} - \frac{1}{2},\frac{1}{\mu }}\right) ,\varpi =$ $\begin{Vmatrix}{M{\tau }_{e}}\end{Vmatrix}, E = {\left\lbrack \begin{array}{llll} {e}_{1}^{\mathrm{T}} & {e}_{2}^{\mathrm{T}} & {e}_{d}^{\mathrm{T}} & \vartheta \end{array}\right\rbrack }^{\mathrm{T}}$ . Hence,(39) becomes ${\dot{V}}_{2} \leq$ $- {\varsigma }_{2}\parallel \mid E{\parallel }^{2} + \parallel E\parallel \left( {{E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }\right)$ and ${\dot{V}}_{2} \leq - {\varsigma }_{2}\left( {1 - \epsilon }\right) \parallel E{\parallel }^{2} +$ $\parallel E\parallel \left( {-\epsilon {\varsigma }_{2}\parallel E\parallel + {E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }\right)$ where $\epsilon \leq 1$ .
+
+Note that
+
+$$
+\parallel E\parallel \geq \frac{{E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }{\epsilon {\varsigma }_{2}}
+$$
+
+renders
+
+$$
+{\dot{V}}_{2} \leq - {\varsigma }_{2}\left( {1 - \epsilon }\right) \parallel E{\parallel }^{2}. \tag{40}
+$$
+
+It can be established that the error subsystem related to obstacle avoidance control is ISS. Additionally, the errors of the closed-loop system satisfies (37). B. Safe Analysis
+
+The subsequent lemma presents the safety analysis of the ASV.
+
+Lemma 4. Given the dynamics of the ASV as outlined in (6), if $\bar{p}\left( {t}_{0}\right) \in \mathcal{U}$ and ${\bar{\tau }}^{ * } \in \mathcal{U}$ are satisfied for all $t > {t}_{0}$ , the closed-loop system will be ISSf.
+
+Proof. According to Lemma 1, if ${\bar{\tau }}^{ * } \in \mathcal{U}$ holds, then $\mathcal{C}$ is forward invariant, meaning that the set $\mathcal{C}$ is ISSf. In other words, as long as $\bar{p}\left( {t}_{0}\right)$ is within $\mathcal{C}$ , the position $\bar{p}\left( t\right)$ will remain in $\mathcal{C}$ indefinitely. Therefore, the closed-loop system is ISSf.
+
+Theorem 2. The closed-loop system is shown to achieve ISSf, indicating that collision avoidance is feasible. Furthermore, all error signals in the closed-loop system are uniformly ultimately bounded.
+
+Proof. According to Lemma 4, the ASV will meet the safety constraint, meaning that the safety objective (11) is fulfilled. We employ Lemmas 2, 3, and [27, Lemma 1], which enable us to deduce that the closed-loop system is ISSf. The norm $\parallel E\parallel$ is uniformly ultimately bounded by
+
+$$
+\parallel E\parallel \leq \sqrt{\frac{{\lambda }_{\max }\left( Q\right) }{{\lambda }_{\min }\left( Q\right) }}\left( {\sqrt{\frac{{\lambda }_{\max }\left( N\right) }{{\lambda }_{\min }\left( N\right) }}\frac{2\parallel {ND}\parallel {\xi }^{ * }}{{\varsigma }_{1}{\kappa \epsilon }{\varsigma }_{2}}}\right.
+$$
+
+$$
+\left. {+\frac{\begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }{\epsilon {\varsigma }_{2}}}\right) \text{.} \tag{41}
+$$
+
+Given that $E$ is bounded, we can deduce that ${e}_{1}$ and $\vartheta$ are also bounded. As a result, it follows that $\begin{Vmatrix}{p\left( t\right) - {p}_{0}\left( \theta \right) }\end{Vmatrix} =$ $\begin{Vmatrix}{e}_{1}\end{Vmatrix}$ and $\begin{Vmatrix}{\dot{\theta }\left( t\right) - {u}_{d}}\end{Vmatrix}$ remain bounded, that is,(9) and (10) hold.
+
+## Remark 3.
+
+## V. Simulation Results
+
+To validate the effectiveness of the proposed control strategy, this paper conducts simulations using Cybership II in [28]. The simulation parameters are set as follows: $w = {40}$ , $\Omega = {0.1}, l = 1,\mu = {0.1},\beta = {30},{d}_{k} = {0.5},{r}_{k} = 1$ , ${k}_{1} = \operatorname{diag}\{ 3,2,8\} ,{k}_{2} = \operatorname{diag}\{ {16},{22},{28}\} ,{t}_{d} = {0.1},{d}_{w} =$ ${\left\lbrack \begin{array}{lll} 3\cos \left( t\right) \sin \left( {0.5t}\right) & 4\sin \left( {0.5t}\right) \cos \left( {0.5t}\right) & {0.2}\sin \left( t\right) \end{array}\right\rbrack }^{\mathrm{T}}$ . The desired parameterized path is ${x}_{d}\left( {\vartheta }_{0}\right) = {y}_{d}\left( {\vartheta }_{0}\right) = {0.06}{\vartheta }_{0} + {0.5}$ , ${\psi }_{d} = \pi /4$ . The position of static obstacle is ${\bar{p}}_{1} = {\left\lbrack \begin{array}{ll} 3 & {2.5} \end{array}\right\rbrack }^{\mathrm{T}}$ , ${\bar{p}}_{2} = {\left\lbrack \begin{array}{ll} 6 & 7 \end{array}\right\rbrack }^{\mathrm{T}}.$
+
+
+
+Fig. 2. Control performance of the proposed control strategy for ASV.
+
+Fig. 2 illustrates the effectiveness of the safety-critical controller proposed for the autonomous vessel. The upper section demonstrates that the vessel prioritizes obstacle avoidance to ensure safety. Once the obstacle avoidance operation is complete, it can proceed with the tracking task. Fig. 3 illustrates the observation effect of the extended state observer. The velocity and aggregated disturbances of the ASV can be accurately estimated. Fig. 4 and Fig. 5 depict the comparisons of the tracking errors and velocities, respectively.
+
+
+
+Fig. 3. The estimates of lumped disturbances.
+
+
+
+Fig. 4. Tracking errors of the ASV.
+
+
+
+Fig. 5. Velocity comparisons of the ASV.
+
+## VI. CONCLUSION
+
+This paper introduces a safety-critical control strategy for ASVs that considers external marine disturbances and internal model uncertainties. Initially, an anti-disturbance controller is devised based on the estimation of lumped disturbances using an ESO. Following this, a quadratic optimization problem is established by incorporating ISSf-ECBFs to enforce safety constraints on the control inputs. By solving this problem, a safety-critical controller is derived, significantly improving the safety and robustness of the system. The closed-loop system is demonstrated to be ISSf, with error signals shown to be uniformly ultimately bounded. The effectiveness of the proposed control strategy is verified through simulation results.
+
+## REFERENCES
+
+[1] T. Kang, N. Gu, D. Wang, L. Liu, Q. Hu, and Z. Peng, "Neurodynamics-based attack-defense guidance of autonomous surface vehicles against multiple attackers for domain protection," IEEE Transactions on Industrial Electronics, vol. 71, no. 10, pp. 12655-12663, 2024.
+
+[2] Z.-Q. Liu, X. Ge, Q.-L. Han, Y.-L. Wang, and X.-M. Zhang, "Secure cooperative path following of autonomous surface vehicles under cyber and physical attacks," IEEE Transactions on Intelligent Vehicles, vol. 8, no. 6, pp. 3680-3691, 2023.
+
+[3] C. Tang, H.-T. Zhang, H. Cao, and J. Wang, "Time-varying formation control of autonomous surface vehicles based on affine observer," IEEE Transactions on Industrial Electronics, 2024.
+
+[4] G. Wu, D. Li, H. Ding, D. Shi, and B. Han, "An overview of developments and challenges for unmanned surface vehicle autonomous berthing," Complex & Intelligent Systems, vol. 10, no. 1, pp. 981-1003, 2024.
+
+[5] H. Bao, Y. Wang, H. Zhu, and D. Wang, "Area complete coverage path planning for offshore seabed organisms fishing autonomous underwater vehicle based on improved whale optimization algorithm," IEEE Sensors Journal, vol. 24, no. 8, pp. 12887-12903, 2024.
+
+[6] D. Madeo, A. Pozzebon, C. Mocenni, and D. Bertoni, "A low-cost unmanned surface vehicle for pervasive water quality monitoring," IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 4, pp. 1433-1444, 2020.
+
+[7] T. Yang, Z. Jiang, R. Sun, N. Cheng, and H. Feng, "Maritime search and rescue based on group mobile computing for unmanned aerial vehicles and unmanned surface vehicles," IEEE Transactions on Industrial Informatics, vol. 16, no. 12, pp. 7700-7708, 2020.
+
+[8] Z. Peng, J. Wang, D. Wang, and Q.-L. Han, "An overview of recent advances in coordinated control of multiple autonomous surface vehicles," IEEE Transactions on Industrial Informatics, vol. 17, no. 2, pp. 732-745, 2020.
+
+[9] R. Cui, X. Zhang, and D. Cui, "Adaptive sliding-mode attitude control for autonomous underwater vehicles with input nonlinearities," Ocean Engineering, vol. 123, pp. 45-54, 2016.
+
+[10] Z. Peng, N. Gu, Y. Zhang, Y. Liu, D. Wang, and L. Liu, "Path-guided time-varying formation control with collision avoidance and connectivity preservation of under-actuated autonomous surface vehicles subject to unknown input gains," Ocean Engineering, vol. 191, p. 106501, 2019.
+
+[11] Z. Xiao, X. Lu, J. Ning, and D. Liu, "Colregs-compliant unmanned surface vehicles collision avoidance based on improved differential evolution algorithm," Expert Systems with Applications, vol. 237, p. 121499, 2024.
+
+[12] Z. Peng, D. Wang, T. Li, and M. Han, "Output-feedback cooperative formation maneuvering of autonomous surface vehicles with connectivity preservation and collision avoidance," IEEE Transactions on Cybernetics, vol. 50, no. 6, pp. 2527-2535, 2019.
+
+[13] Y. Wang, B. Jiang, Z.-G. Wu, S. Xie, and Y. Peng, "Adaptive sliding mode fault-tolerant fuzzy tracking control with application to unmanned marine vehicles," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 51, no. 11, pp. 6691-6700, 2020.
+
+[14] W. Guan and K. Wang, "Autonomous collision avoidance of unmanned surface vehicles based on improved a-star and dynamic window approach algorithms," IEEE Intelligent Transportation Systems Magazine, vol. 15, no. 3, pp. 36-50, 2023.
+
+[15] P. Wieland and F. Allgöwer, "Constructive safety using control barrier
+
+functions," IFAC Proceedings Volumes, vol. 40, no. 12, pp. 462-467, 2007.
+
+[16] L. Wang, A. D. Ames, and M. Egerstedt, "Safety barrier certificates for collisions-free multirobot systems," IEEE Transactions on Robotics, vol. 33, no. 3, pp. 661-674, 2017.
+
+[17] S. Kolathaya and A. D. Ames, "Input-to-state safety with control barrier functions," IEEE Control Systems Letters, vol. 3, no. 1, pp. 108-113, 2018.
+
+[18] B. T. Lopez, J.-J. E. Slotine, and J. P. How, "Robust adaptive control barrier functions: An adaptive and data-driven approach to safety," IEEE Control Systems Letters, vol. 5, no. 3, pp. 1031-1036, 2020.
+
+[19] N. Gu, D. Wang, Z. Peng, and J. Wang, "Safety-critical containment maneuvering of underactuated autonomous surface vehicles based on neurodynamic optimization with control barrier functions," IEEE Transactions on Neural Networks and Learning Systems, vol. 34, no. 6, pp. 2882-2895, 2021.
+
+[20] W. Xiao and C. Belta, "High-order control barrier functions," IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3655-3662, 2021.
+
+[21] M. Li and Z. Sun, "A new perspective on projection-to-state safety and its application to robotic arms," in Proceeding of the American Control Conference. San Diego, CA, 2023, pp. 2430-2435.
+
+[22] L.-Y. Hao, G. Dong, T. Li, and Z. Peng, "Path-following control with obstacle avoidance of autonomous surface vehicles subject to actuator faults," IEEE/CAA Journal of Automatica Sinica, vol. 11, no. 4, pp. 956-964, 2024.
+
+[23] S. He, M. Wang, S.-L. Dai, and F. Luo, "Leader-follower formation control of usvs with prescribed performance and collision avoidance," IEEE Transactions on Industrial Informatics, vol. 15, no. 1, pp. 572- 581, 2018.
+
+[24] Z. Peng, Y. Jiang, L. Liu, and Y. Shi, "Path-guided model-free flocking control of unmanned surface vehicles based on concurrent learning extended state observers," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 53, no. 8, pp. 4729-4739, 2023.
+
+[25] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, "Control barrier function based quadratic programs for safety critical systems," IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861-3876, 2016.
+
+[26] D. Wang and J. Huang, "Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form," IEEE Transactions on Neural Networks, vol. 16, no. 1, pp. 195- 202, 2005.
+
+[27] Z. Peng, D. Wang, and J. Wang, "Cooperative dynamic positioning of multiple marine offshore vessels: A modular design," IEEE/ASME Transactions on Mechatronics, vol. 21, no. 3, pp. 1210-1221, 2015.
+
+[28] R. Skjetne, T. I. Fossen, and P. V. Kokotović, "Adaptive maneuvering, with experiments, for a model ship in a marine control laboratory," Automatica, vol. 41, no. 2, pp. 289-298, 2005.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NKhQ1UEQFb/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NKhQ1UEQFb/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3d8866536331fd3d338e1be5ce0023cc8bfb2e64
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NKhQ1UEQFb/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,537 @@
+§ SAFETY-CRITICAL OBSTACLE AVOIDANCE CONTROL OF AUTONOMOUS SURFACE VEHICLES WITH UNCERTAINTIES AND DISTURBANCES
+
+${1}^{\text{ st }}$ Gege Dong
+
+College of Marine Electrical Engineering
+
+Dalian Maritime University
+
+Dalian, China
+
+donggege0507@163.com
+
+${2}^{\text{ nd }}$ Li-Ying Hao*
+
+College of Marine Electrical Engineering
+
+Dalian Maritime University
+
+Dalian, China
+
+haoliying_0305@163.com
+
+${Abstract}$ -This paper proposes a safety-critical obstacle avoidance control approach for autonomous surface vehicles (ASVs) with disturbances and uncertainties. The existing exponential control barrier functions (ECBF) are extended to handle unknown disturbances, leading to the development of input-to-state safe exponential control barrier functions (ISSf-ECBFs). An extended state observer is used to estimate unknown external marine disturbances and internal model uncertainties, based on which an anti-disturbance controller is designed. Based on the proposed ISSf-ECBFs, a quadratic programming problem is formulated to determine the optimal control input. It is proven that the closed-loop system is input-to-state safe and the errors of the closed-loop system are uniformly ultimately bounded. Simulations validate the effectiveness of the proposed control strategy.
+
+Index Terms-Autonomous surface vehicles (ASVs), safety-critical control, obstacles avoidance, input-to-state safe exponential control barrier functions (ISSf-ECBFs)
+
+§ I. INTRODUCTION
+
+Autonomous Surface Vehicles (ASVs) are gaining attention for their ability to enhance maritime operations [1]-[3]. With advanced sensors and navigation systems, ASVs can navigate complex environments and perform diverse tasks [4]. They are increasingly utilized in search and rescue, fisheries management, hydrographic surveying, and offshore energy, making them a focal point for researchers in ASV control [5]-[7].
+
+ASVs navigating in dynamic marine environments face numerous challenges, primarily internal model uncertainties and external disturbances [8]. Internal uncertainties arise from modeling inaccuracies, parameter variations, and sensor noise. Additionally, ASVs must navigate unpredictable ocean conditions, such as waves, currents, and winds. These factors can adversely affect the performance of control strategy. To address this challenge, researchers have proposed various methods to enhance the robustness of the system, such as sliding mode control [9], adaptive control, and neural network control [10].
+
+The Extended State Observer (ESO) can estimate disturbances in real-time and dynamically adjust the control strategy. By treating internal model uncertainties and external disturbances as lumped disturbances for estimation, the reliance on the model can be reduced, thereby enhancing the robustness of the system.
+
+In complex maritime environments, ASVs face significant threats from various obstacles, including vessels, islands, and reefs [11]. To mitigate these risks, researchers have proposed several obstacle avoidance strategies, such as the artificial potential method [12], the velocity obstacle method [13], and the dynamic window approach [14]. Control barrier functions (CBFs), introduced in [15], have proven effective in ensuring real-time safety. In [16], the nominal controller was modified to formally adhere to safety constraints for successful obstacle avoidance. However, the control strategy in [16] did not account for model uncertainties or disturbances. To address this, [17] introduced input-to-state safe control barrier functions (ISSf-CBFs). Furthermore, [18] proposed a framework to ensure safety for uncertain nonlinear systems with structured parametric uncertainty. In [19], a collision avoidance strategy for ASVs was proposed using ISSf-CBFs. However, these functions have a relative degree of one, limiting their use in higher-order systems. To address this, [20] introduced exponential control barrier functions (ECBFs). [21] further explored ISSf-ECBFs under known perturbation bounds, but measuring such disturbances is challenging. Therefore, a safety-critical controller based on ISSf-ECBFs is crucial for ASVs dealing with unknown model uncertainties and external disturbances.
+
+This paper presents a safety-critical control strategy for Autonomous Surface Vehicles (ASVs) that accounts for external marine disturbances and internal model uncertainties. The key contributions are as follows:
+
+1) While the existing method [21] constructs safety constraints only under known disturbances or their upper bounds, this paper extends the results of input-to-state safe control barrier functions (ISSf-ECBFs) to develop safety constraints for unknown disturbances.
+
+This work was funded by the National Natural Science Foundation of China (51939001, 52171292, 51979020, 61976033), Dalian Outstanding Young Talents Program (2022RJ05), the Topnotch Young Talents Program of China (36261402), and the Liaoning Revitalization Talents Program (XLYC2007188).
+
+2) Unlike previous work [12], [22]-[24], this paper formulates a safety-critical controller based on ISSf-ECBFs by constructing a quadratic programming problem to facilitate collision avoidance with obstacles.
+
+The structure of the paper includes the following sections: The preliminaries and problem statement are covered in Section II. Section III gives the safety-critical controller design and Section IV is stability and safety analysis. Simulations are carried out in Section V. Section VI summarizes this article.
+
+§ II. PRELIMINARIES AND PROBLEM STATEMENT
+
+§ A. NOTATION
+
+In this paper, the notation $\parallel \cdot \parallel$ denotes the 2-norm of a vector, and $\mathfrak{R}$ represents the set of real numbers. The symbols ${\lambda }_{\min }\left( \cdot \right)$ and ${\lambda }_{\max }\left( \cdot \right)$ indicate the smallest and largest eigenvalues of a symmetric matrix, respectively.
+
+Let $\beta \left( r\right)$ be a scalar continuous function defined for $r \in$ $\lbrack - b,a)$ . If $a = \infty ,b = 0$ , and $\beta \left( r\right) \rightarrow \infty$ as $r \rightarrow \infty$ , then $\beta \left( r\right)$ belongs to class ${\mathcal{K}}_{\infty }$ . If $a,b = \infty ,\beta \left( r\right) \rightarrow \infty$ as $r \rightarrow \infty$ , and $\beta \left( r\right) \rightarrow - \infty$ as $r \rightarrow - \infty$ , it represents an extended class ${\mathcal{K}}_{\infty }$ , denoted as ${\mathcal{K}}_{\infty ,e}$ .
+
+B. Input-to-state Safe Exponential Control Barrier Functions Consider the following system
+
+$$
+\dot{x} = f\left( x\right) + g\left( x\right) u + {d}_{w} \tag{1}
+$$
+
+where $x\left( t\right) \in {\Re }^{n}$ denotes state and $u \in {\Re }^{m}$ denotes control input. The term ${d}_{w}$ denotes bounded disturbances. The function $f\left( x\right) \in {\Re }^{n}$ and $g\left( x\right) \in {\Re }^{n \times m}$ are locally Lipschitz continuous.
+
+Definition 1. [25] The set $\mathcal{C} \in {\Re }^{n}$ is described as
+
+$$
+\mathcal{C} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) \geq 0}\right\}
+$$
+
+$$
+\partial \mathcal{C} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) = 0}\right\}
+$$
+
+$$
+\operatorname{Int}\left( \mathcal{C}\right) \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) > 0}\right\} \tag{2}
+$$
+
+where $h\left( \cdot \right) \in {\Re }^{n} \mapsto \Re$ represents a continuously differentiable function, and $\mathcal{C}$ is referred to as the safe set. If for all ${x}_{0} \in \mathcal{C}$ it holds that $x\left( t\right) \in \mathcal{C}$ for every $t \in I\left( {x}_{0}\right)$ , then the set $\mathcal{C}$ is considered forward invariant. Consequently, the system described by (1) with ${d}_{w}\left( t\right) = 0$ can be deemed safe on $\mathcal{C}$ .
+
+Definition 2. The relative degree of $S\left( x\right) : {\Re }^{n} \rightarrow \Re$ with respect to the system (1) refers to the number of derivatives required along the dynamics of (1) before the control input $u$ explicitly appears.
+
+Definition 3. [17] For system (1), an extended set ${\mathcal{C}}_{d} \supset \mathcal{C}$ is expressed as follows
+
+$$
+{\mathcal{C}}_{d} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) + {\beta }_{d}\left( {\begin{Vmatrix}{d}_{w}\left( t\right) \end{Vmatrix}}_{\infty }\right) \geq 0}\right\}
+$$
+
+$$
+\partial {\mathcal{C}}_{d} \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) + {\beta }_{d}\left( {\begin{Vmatrix}{d}_{w}\left( t\right) \end{Vmatrix}}_{\infty }\right) = 0}\right\}
+$$
+
+$$
+\operatorname{Int}\left( {\mathcal{C}}_{d}\right) \triangleq \left\{ {x \in {\Re }^{n} \mid S\left( x\right) + {\beta }_{d}\left( {\begin{Vmatrix}{d}_{w}\left( t\right) \end{Vmatrix}}_{\infty }\right) > 0}\right\} \tag{3}
+$$
+
+where ${\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty } \leq {\bar{d}}_{w}$ , a positive constant, and $S\left( x\right)$ is a continuous function, with ${\beta }_{d}\left( \cdot \right) \in {\mathcal{K}}_{\infty ,e}$ .
+
+Definition 4. (ISSf [17]) If the control input $u$ and the function ${\beta }_{d}$ ensure the forward invariance of the set ${\mathcal{C}}_{d}$ , then the system (1) with disturbances is ISSf on $\mathcal{C}$ .
+
+Definition 5. (ISSf-ECBF [17]) Considering the sets ${\mathcal{C}}_{d}$ defined by (3), $S\left( x\right)$ , which has a relative degree $\rho > 1$ , qualifies as an ISSf-ECBF for the system described in (1). This holds true if, for all $x \in {\Re }^{n}$ , there exist a bound ${\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty } \leq {\bar{\tau }}_{w}$ and a function $\gamma \left( \cdot \right) \in {\mathcal{K}}_{\infty ,e}$ that satisfies
+
+$$
+\mathop{\sup }\limits_{{u \in \mathcal{U}}}\left\lbrack {{\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w}}\right.
+$$
+
+$$
+\left. {+{\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}}\right\rbrack \geq - \gamma \left( {\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }\right) \tag{4}
+$$
+
+The terms ${\mathcal{L}}_{f}^{\rho }$ and ${\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}$ represent the Lie derivatives of the function $S\left( x\right) .{\mathcal{T}}_{s} = {\left\lbrack \begin{array}{llll} {p}_{0} & {p}_{1} & \ldots & {p}_{\iota } \end{array}\right\rbrack }^{T}$ where ${p}_{i}$ is positive constant. ${\mathcal{H}}_{s} = {\left\lbrack \begin{array}{llll} S\left( x\right) & {\mathcal{L}}_{f}S\left( x\right) & \ldots & {\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) \end{array}\right\rbrack }^{T}$ .
+
+Lemma 1. If $S\left( x\right)$ functions as an ISSf-ECBF for the system (1) in the set $\mathcal{C}$ , then any controller $u \in \mathcal{U}$ that is Lipschitz continuous and valid for all $x \in {\Re }^{n}$ must satisfy
+
+$$
+\mathcal{U}\left( x\right) = \left\{ {u \in {\Re }^{m} : {\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u}\right.
+$$
+
+$$
+\left. {+{\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w} + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s} \geq - \gamma \left( {\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }\right) }\right\} . \tag{5}
+$$
+
+This implies that the set ${\mathcal{C}}_{d}$ is forward invariant. In other words, the system (1) is ISSf on the set $\mathcal{C}$ .
+
+§ C.ASV MODEL
+
+The kinematics and kinetics of ASV can be described as:
+
+$$
+\dot{\eta }\left( t\right) = R\left( \psi \right) \nu \left( t\right) \tag{6}
+$$
+
+$$
+M\dot{\nu }\left( t\right) = f\left( \nu \right) + {d}_{w}\left( t\right) + \tau \left( t\right)
+$$
+
+where $\eta \left( t\right) = {\left\lbrack \begin{array}{ll} \bar{p}\left( t\right) & \psi \left( t\right) \end{array}\right\rbrack }^{T} \in {\Re }^{3}$ represents the position and heading of ASV. $R\left( \psi \right) = \operatorname{diag}\left\{ {{R}_{2}\left( \psi \right) ,1}\right\}$ is a rotate matrix
+
+with
+
+$$
+{R}_{2}\left( \psi \right) = \left\lbrack \begin{matrix} \cos \left( \psi \right) & - \sin \left( \psi \right) \\ \sin \left( \psi \right) & \cos \left( \psi \right) \end{matrix}\right\rbrack . \tag{7}
+$$
+
+The vector $\nu \left( t\right) = {\left\lbrack \begin{array}{lll} u\left( t\right) & v\left( t\right) & r\left( t\right) \end{array}\right\rbrack }^{T} \in {\Re }^{3}$ represents the surge velocity, sway velocity, and yaw velocity, respectively. The matrix $M$ denotes the inertial matrix. $f\left( \nu \right)$ represents the Coriolis and centripetal matrix, damping matrix, and unmodeled hydrodynamics. The vector $\tau \left( t\right)$ signifies the forces produced by the actuators. The external disturbances, caused by wind, waves, and ocean currents, are represented by ${d}_{w}\left( t\right) = {\left\lbrack \begin{array}{lll} {d}_{w1}\left( t\right) & {d}_{w2}\left( t\right) & {d}_{w3}\left( t\right) \end{array}\right\rbrack }^{T} \in {\Re }^{3}.$
+
+Letting $q\left( t\right) = R\left( \psi \right) \nu \left( t\right)$ ,(6) can be rewritten as
+
+$$
+\dot{p} = q \tag{8}
+$$
+
+$$
+\dot{q} = \xi + R{M}^{-1}\tau
+$$
+
+where $\xi = R{M}^{-1}\left( {{d}_{w} + f\left( \nu \right) }\right) + \dot{R}\nu$ .
+
+The desired parameterized path is set as ${p}_{0}\left( \theta \right) =$ ${\left\lbrack {x}_{0}\left( \theta \right) ,{y}_{0}\left( \theta \right) ,{\psi }_{0}\left( \theta \right) \right\rbrack }^{\mathrm{T}},{\psi }_{0}\left( \theta \right) = \arctan \left( {{y}_{0}^{\theta }\left( \theta \right) /{x}_{0}^{\theta }\left( \theta \right) }\right)$ where $\theta$ represents path variable. ${y}_{0}^{\theta }\left( \theta \right)$ and ${x}_{0}^{\theta }\left( \theta \right)$ is the partial derivative of ${y}_{0}\left( \theta \right)$ and ${x}_{0}\left( \theta \right)$ , respectively. In addition, it is assumed that ${p}_{0}^{\theta }\left( \theta \right)$ is bounded.
+
+§ D. PROBLEM FORMULATION
+
+The safety-critical obstacle avoidance controller of ASV is required to achieve the following tasks:
+
+(1) Geometric task: Ensure that the ASV follows the desired path, meaning that
+
+$$
+\mathop{\lim }\limits_{{t \rightarrow \infty }}\begin{Vmatrix}{p\left( t\right) - {p}_{0}\left( \theta \right) }\end{Vmatrix} < {l}_{1} \tag{9}
+$$
+
+where ${l}_{1} \in \mathfrak{R}$ denotes a small positive constant.
+
+(2) Dynamic task: The derivative of the path variable $\theta$ converge to the desired speed
+
+$$
+\mathop{\lim }\limits_{{t \rightarrow \infty }}\begin{Vmatrix}{\dot{\theta }\left( t\right) - {u}_{d}\left( t\right) }\end{Vmatrix} < {l}_{2} \tag{10}
+$$
+
+where ${u}_{d}\left( t\right)$ represents desired speed and ${l}_{2}$ is a small positive constant.
+
+(3) Obstacle avoidance task: To prevent collisions between the ASV and obstacles, the following condition must be met
+
+$$
+\begin{Vmatrix}{\bar{p}\left( t\right) - {\bar{p}}_{k}\left( t\right) }\end{Vmatrix} > {r}_{k} + {d}_{k} \tag{11}
+$$
+
+where ${\bar{p}}_{k}\left( t\right) ,{r}_{k}$ , and ${d}_{k}$ represent the position, the radius, and the minimum obstacle avoidance distance of the $k$ th obstacle.
+
+§ III. MAIN RESULTS
+
+§ A. ISSF-ECBF WITH UNKNOWN DISTURBANCCES
+
+While the previous studies have made substantial progress, they were mainly directed at scenarios with known disturbances or predefined upper bounds. To alleviate this limitation, the following theorem is presented to account for unknown disturbances.
+
+Theorem 1. Given the ISSf-ECBF $S\left( x\right)$ as defined in Definition 5 for the system (1) on the set $\mathcal{C}$ , if there exists a bound ${\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty } \leq {\bar{d}}_{w}$ such that for every $x \in {\Re }^{n}$ , the following inequality holds
+
+$$
+\mathop{\sup }\limits_{{u \in \mathcal{U}}}\left\lbrack {{\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}}\right.
+$$
+
+$$
+\left. {-{\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }\right\rbrack \geq 0 \tag{12}
+$$
+
+and the admissible control set satisfies as
+
+$$
+\mathcal{U}\left( x\right) = \left\{ {u \in {\Re }^{m} : {\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}}\right.
+$$
+
+$$
+- {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) \geq 0\} . \tag{13}
+$$
+
+Then, we can obtain that the system (1) is ISSf on $\mathcal{C}$ .
+
+Proof. For $u \in \mathcal{U}\left( x\right)$ , one has
+
+$$
+{\mathcal{L}}_{f}^{\rho }S\left( x\right) + {\mathcal{L}}_{g}{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) u + {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w} + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}
+$$
+
+$$
+\geq {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) + {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}{d}_{w}
+$$
+
+$$
+\geq {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\right)
+$$
+
+$$
+- \parallel \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x}\parallel {\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }. \tag{14}
+$$
+
+Adding and subtracting $\frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }^{2}}{4}$ yields
+
+$$
+\dot{h} \geq {\left( \frac{\partial \left( {{\mathcal{L}}_{f}^{\rho - 1}S\left( x\right) }\right) }{\partial x} - \frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }}{2}\right) }^{2} - \frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }^{2}}{4}
+$$
+
+$$
+\geq - \frac{{\begin{Vmatrix}{d}_{w}\end{Vmatrix}}_{\infty }^{2}}{4} \tag{15}
+$$
+
+which is of the form (4).
+
+Remark 1. Compared with [21], the proposed ISSf-ECBF can deal with unknown perturbations. Although asymptotically stable ESO is used in reference 21, in practice, the disturbance estimation error is difficult to be 0 . Thus, it is essential to develop ISSf-ECBFs that ensure safety in the presence of unknown disturbances.
+
+§ B. ANTI-DISTURBANCE CONTROLLER DESIGN
+
+In this section, we will focus on designing a safety-critical controller. The control architecture for the proposed strategy is illustrated in Figure 1.
+
+ < g r a p h i c s >
+
+Fig. 1. Control architecture of the safety-critical controller for the ASV.
+
+Firstly, we utilize the ESO to obtain the estimations of the model uncertainties, external disturbances in this part. In addition, the ESO relies on the following assumption.
+
+Assumption 1. $\dot{\xi }\left( t\right)$ is a bounded function meeting
+
+$$
+\parallel \dot{\xi }\left( t\right) \parallel \leq {\xi }^{ * } \tag{16}
+$$
+
+where ${\xi }^{ * }$ is a positive constant.
+
+Then, the ESO is devised to estimate model uncertainties, external disturbances.
+
+$$
+\left\{ \begin{array}{l} \dot{\widehat{q}}\left( t\right) = - {K}_{1}\widetilde{q}\left( t\right) + \widehat{\xi }\left( t\right) + R{M}^{-1}\tau \\ \dot{\widehat{\xi }}\left( t\right) = - {K}_{2}\widetilde{q}\left( t\right) \end{array}\right. \tag{17}
+$$
+
+where $\widehat{q}\left( t\right)$ and $\widehat{\xi }\left( t\right)$ represent the estimates of $q\left( t\right)$ and $\xi \left( t\right)$ . The observer matrices ${\left\lbrack \begin{array}{ll} {K}_{1} & {K}_{2} \end{array}\right\rbrack }^{T} = {\left\lbrack \begin{array}{ll} {2w}{I}_{3} & {w}^{2}{I}_{3} \end{array}\right\rbrack }^{T}$ where $w$ is the observer bandwidth.
+
+Defining $\widetilde{q}\left( t\right) = \widehat{q}\left( t\right) - q\left( t\right)$ , and $\widetilde{\xi }\left( t\right) = \widehat{\xi }\left( t\right) - \xi \left( t\right)$ are the estimates. The dynamics of $\widetilde{q}\left( t\right)$ and $\widetilde{\xi }\left( t\right)$ can be written as
+
+$$
+\left\{ \begin{array}{l} \dot{\widetilde{q}}\left( t\right) = - {K}_{1}\widetilde{q}\left( t\right) + \widetilde{\xi }\left( t\right) \\ \dot{\widetilde{\xi }}\left( t\right) = - {K}_{2}\widetilde{q}\left( t\right) - \dot{\xi }\left( t\right) . \end{array}\right. \tag{18}
+$$
+
+Next, (18) can be rewritten as
+
+$$
+{\dot{E}}_{o}\left( t\right) = T{E}_{o}\left( t\right) - D\dot{\xi }\left( t\right) \tag{19}
+$$
+
+where ${E}_{o}\left( t\right) = {\left\lbrack \begin{array}{ll} {\widetilde{q}}^{\mathrm{T}}\left( t\right) & {\widetilde{\xi }}^{\mathrm{T}}\left( t\right) \end{array}\right\rbrack }^{\mathrm{T}} \in {\Re }^{6}$ and
+
+$$
+T = \left\lbrack \begin{array}{ll} - {K}_{1} & {I}_{3} \\ - {K}_{2} & {0}_{3} \end{array}\right\rbrack ,D = \left\lbrack \begin{array}{l} {0}_{3} \\ {I}_{3} \end{array}\right\rbrack .
+$$
+
+Then, the following tracking error is defined as ${e}_{1} = p -$ ${p}_{0}\left( \theta \right)$ . By taking the derivative of ${e}_{1}$ and using (6), we can
+
+get
+
+$$
+{\dot{e}}_{1} = q - {p}_{0}^{\theta }\left( \theta \right) \dot{\theta }. \tag{20}
+$$
+
+Let ${u}_{d} - \vartheta \left( t\right) = \dot{\theta }\left( t\right)$ , one can obtain
+
+$$
+{\dot{e}}_{1} = q - {p}_{0}^{\theta }\left( \theta \right) \left( {{u}_{d} - \vartheta }\right) . \tag{21}
+$$
+
+The kinematic guidance law ${q}_{d}$ is designed as follows to stabilize ${e}_{1}$ :
+
+$$
+{q}_{d} = - {k}_{1}{e}_{1} + {p}_{0}^{\theta }\left( \theta \right) {u}_{d} \tag{22}
+$$
+
+and
+
+$$
+\dot{\vartheta } = - \ell \left( {\vartheta + \mu {p}_{0}^{\theta }{\left( \theta \right) }^{\mathrm{T}}{e}_{1}}\right) \tag{23}
+$$
+
+where ${k}_{1} = \operatorname{diag}\left\{ {{k}_{11},{k}_{12},{k}_{13}}\right\} ,\ell$ and $\mu$ are positive constants.
+
+To proceed, defining ${e}_{2} = q - {\widehat{q}}_{d}$ , where ${\widehat{q}}_{d}$ is the estimate of ${q}_{d}$ . ${\widehat{q}}_{d}$ can be obtained by using the following filtering scheme:
+
+$$
+{t}_{d}{\dot{\widehat{q}}}_{d} + {\widehat{q}}_{d} = {q}_{d},\;{\widehat{q}}_{d}\left( 0\right) = {q}_{d}\left( 0\right) \tag{24}
+$$
+
+where ${t}_{d}$ is a positive constant. Let
+
+$$
+{e}_{d} = {\widehat{q}}_{d} - {q}_{d}. \tag{25}
+$$
+
+And ${\dot{q}}_{d} \triangleq a = {\left\lbrack \begin{array}{lll} {a}_{1} & {a}_{2} & {a}_{3} \end{array}\right\rbrack }^{T} \cdot {a}_{j}$ is bounded by $\left| {a}_{j}\right| \leq {a}_{j}^{ * }$ , $j = 1,2,3$ , where ${a}_{j}^{ * }$ is a positive constant. For details, please refer to [26].
+
+Then, the time derivative of ${e}_{2}$ yields
+
+$$
+{\dot{e}}_{2} = \xi + R{M}^{-1}\tau \left( t\right) + \frac{{e}_{d}}{{t}_{d}}. \tag{26}
+$$
+
+To stabilize ${e}_{2}$ , the anti-disturbance control law is developed as follows:
+
+$$
+{\tau }_{c}\left( t\right) = M{R}^{T}\left( {-\widehat{\xi } - {e}_{1} - \frac{{e}_{d}}{{t}_{d}} - {k}_{2}{e}_{2}}\right) \tag{27}
+$$
+
+where ${k}_{2} = \operatorname{diag}\left\{ {{k}_{21},{k}_{22},{k}_{23}}\right\}$ . Denote $\tau = {\tau }_{c} + {\tau }_{e}$ .
+
+§ C. SAFETY-CRITICAL OBSTACLE AVOIDANCE CONTROLLER
+
+In this part, considering the collision with obstacles and ASV to design the optimal surge and sway force of safety conditions. From (8), we can get
+
+$$
+\dot{\bar{p}} = \bar{q} \tag{28}
+$$
+
+$$
+\dot{\bar{q}} = {\widehat{\xi }}_{2} + {\tau }_{2} - {\widetilde{\xi }}_{2}
+$$
+
+where $\bar{p}$ denotes ${\left\lbrack x\left( t\right) ,y\left( t\right) \right\rbrack }^{T},\bar{q}$ denotes ${R}_{2}\left( \psi \right) {\left\lbrack u,v\right\rbrack }^{T}.{\widehat{\xi }}_{2},{\widetilde{\xi }}_{2}$ and ${\tau }_{2}$ is the first two dimensions of $\widehat{\xi },\widetilde{\xi }$ and $\tau$ , respectively. ${\bar{p}}_{k} = {\left\lbrack {x}_{k},{y}_{k}\right\rbrack }^{T}$ is position of $k$ th obstacle.
+
+We choose the following candidate ISSf-ECBF
+
+$$
+{S}_{k}\left( s\right) = {\begin{Vmatrix}{\bar{p}}_{ek}\end{Vmatrix}}^{2} - {\left( {r}_{k} + {d}_{k}\right) }^{2} \tag{29}
+$$
+
+where ${\bar{p}}_{ek} = \bar{p} - {\bar{p}}_{k},s = {\left\lbrack {\bar{p}}^{T},{\bar{q}}^{T}\right\rbrack }^{T}$ . To achieve the objective of obstacle avoidance, the set $\mathcal{C}$ can be obtained
+
+$$
+\mathcal{C} = \left\{ {\bar{p} \in {\mathbb{R}}^{2} : {S}_{k}\left( s\right) = {\begin{Vmatrix}{\bar{p}}_{ek}\end{Vmatrix}}^{2} - {\left( {r}_{k} + {d}_{k}\right) }^{2} \geq 0}\right\} \tag{30}
+$$
+
+For ease of notation, it is denoted by ${S}_{k}$ in the sequel. The safety constraint with ${S}_{k}\left( s\right)$ is described as
+
+$$
+\mathcal{U} = \left\{ {{\tau }_{2} : {\mathcal{L}}_{f}^{2}{S}_{k} + {\mathcal{L}}_{g}{\mathcal{L}}_{f}{S}_{k}{\tau }_{2} - {\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) }\right.
+$$
+
+$$
+\left. {+{\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s} \geq 0}\right\} \tag{31}
+$$
+
+where ${\mathcal{L}}_{f}^{2}{S}_{k} = 2{\bar{q}}^{T}\bar{q} + 2{\bar{p}}_{ek}^{T}{\widehat{\xi }}_{2},{\mathcal{L}}_{g}{\mathcal{L}}_{f}{S}_{k} = 2{\bar{p}}_{ek}^{T},{\mathcal{T}}_{s} =$ ${\left\lbrack {\beta }^{2},2\beta \right\rbrack }^{T}$ . For the ASV, ensuring safety takes precedence over geometric objectives. Based on the safety constraint (31), the following quadratic programming problem is constructed.
+
+$$
+{\tau }^{ * } = \mathop{\operatorname{argmin}}\limits_{{\tau \in {\Re }^{m}}}J\left( \tau \right) = {\begin{Vmatrix}\tau - {\tau }_{c}\end{Vmatrix}}^{2}
+$$
+
+$$
+\text{ s.t. } - {\mathcal{L}}_{g}{\mathcal{L}}_{f}{S}_{k}\tau \leq \phi \tag{32}
+$$
+
+where $\phi = 2{\bar{q}}^{T}\bar{q} - - {\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) }^{T}\left( \frac{\partial \left( {{\mathcal{L}}_{f}{S}_{k}}\right) }{\partial x}\right) + 2{\bar{p}}_{ek}^{T}{\widehat{\xi }}_{2} + {\mathcal{T}}_{s}^{T}{\mathcal{H}}_{s}$ . The ${\tau }^{ * }$ is obtained by solving the above quadratic programming problem.
+
+Remark 2. The proposed safety-critical controller can avoid obstacles while ensuring minimal impact on the given tracking task.
+
+§ IV. STABILITY AND SAFETY ANALYSIS
+
+In this section, we will conduct stability and safety analysis of the closed-loop system.
+
+§ A. STABILITY ANALYSIS
+
+Lemma 2. The observer error subsystem in (19) is ISS, and the error signals being $\widetilde{q}$ and $\widetilde{f}$ are bounded by
+
+$$
+\begin{Vmatrix}{{E}_{o}\left( t\right) }\end{Vmatrix} \leq \sqrt{\frac{{\lambda }_{\max }\left( N\right) }{{\lambda }_{\min }\left( N\right) }}\max \left\{ {\begin{Vmatrix}{{E}_{o}\left( {t}_{0}\right) }\end{Vmatrix}{e}^{-{\gamma }_{1}\left( {t - {t}_{0}}\right) /2},}\right.
+$$
+
+$$
+\frac{2\parallel {ND}\parallel {\xi }^{ * }}{\varsigma {}_{1}\kappa }\} ,\forall t \geq {t}_{0} \tag{33}
+$$
+
+where ${\gamma }_{1} = \left( {\left\lbrack {{\varsigma }_{1}\left( {1 - \kappa }\right) }\right\rbrack /\left\lbrack {{\lambda }_{\max }\left( N\right) }\right\rbrack }\right)$ and $0 < \kappa < 1$ provided that
+
+$$
+{T}^{T}N + {NT} \leq - {\varsigma }_{1}I \tag{34}
+$$
+
+where ${\varsigma }_{1} \in \mathfrak{R}$ is a positive constant.
+
+Proof. Choose the following Lyapunov function
+
+$$
+{V}_{1} = \left( {1/2}\right) {E}_{o}^{\mathrm{T}}\left( t\right) N{E}_{o}\left( t\right) . \tag{35}
+$$
+
+Taking (34) into account, one has ${\dot{V}}_{1} = {E}_{o}{\left( t\right) }^{\mathrm{T}}N\left( {T{E}_{o}\left( t\right) - }\right.$ $\left. {D\dot{\xi }\left( t\right) }\right) \leq - \frac{{\varsigma }_{1}}{2}{\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2} + \begin{Vmatrix}{{E}_{o}\left( t\right) }\end{Vmatrix}\parallel {ND}\parallel \parallel \dot{\xi }\left( t\right) \parallel$ Since $\begin{Vmatrix}{{E}_{o}\left( t\right) }\end{Vmatrix} \geq \left\lbrack \left( {2\parallel {ND}\parallel \parallel \dot{\xi }\left( t\right) \parallel /{\varsigma }_{1}\kappa }\right) \right\rbrack$ , we have
+
+$$
+{\dot{V}}_{1} \leq - \frac{{\varsigma }_{1}}{2}\left( {1 - \kappa }\right) {\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2}. \tag{36}
+$$
+
+It follows that the observer error subsystem described by (19) is ISS. It is important to note that ${V}_{1}$ is bounded and satisfies the inequality $\left( {\left\lbrack {{\lambda }_{\min }\left( N\right) }\right\rbrack /2}\right) {\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2} \leq {V}_{1} \leq$ $\left( {\left\lbrack {{\lambda }_{\max }\left( N\right) }\right\rbrack /2}\right) {\begin{Vmatrix}{E}_{o}\left( t\right) \end{Vmatrix}}^{2}$ . From this, we can derive (33).
+
+Next, we will outline the stability analysis of the closed-loop system.
+
+Lemma 3. Taking into account the error dynamics represented by (21) and (26), the error signals ${e}_{1},{e}_{2},{e}_{r},\vartheta$ , and $\widetilde{\gamma }$ are uniformly ultimately bounded by
+
+$$
+\parallel E\parallel \leq \sqrt{\frac{{\lambda }_{\max }\left( Q\right) }{{\lambda }_{\min }\left( Q\right) }}\max \left\{ {\begin{Vmatrix}{E\left( {t}_{0}\right) }\end{Vmatrix}{e}^{-{\gamma }_{2}\left( {t - {t}_{0}}\right) /2},}\right.
+$$
+
+$$
+\frac{{E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }{\epsilon {\varsigma }_{2}},\forall t \geq {t}_{0} \tag{37}
+$$
+
+where $Q = \operatorname{diag}\{ 1,1/\ell \mu \} ,{\gamma }_{2} = 2{\varsigma }_{2}\left( {1 - \epsilon }\right) /{\lambda }_{\max }\left( Q\right)$ .
+
+Proof. The constructed Lyapunov function is
+
+$$
+{V}_{2} = \frac{1}{2}\left( {{e}_{1}^{\mathrm{T}}{e}_{1} + {e}_{2}^{\mathrm{T}}{e}_{2} + {e}_{d}^{\mathrm{T}}{e}_{d}}\right) + \frac{{\vartheta }^{2}}{2\ell \mu }.
+$$
+
+According to (20),(23)-(27), the time derivative of ${V}_{2}$ is
+
+$$
+{\dot{V}}_{2} = {e}_{1}^{\mathrm{T}}\left( {{e}_{2} + {e}_{d}}\right) - {e}_{1}^{\mathrm{T}}{k}_{1}{e}_{1} + {e}_{2}^{\mathrm{T}}f + {e}_{2}^{\mathrm{T}}\frac{{e}_{d}}{{t}_{d}} + {e}_{d}^{\mathrm{T}}\left( {-\frac{{e}_{d}}{{t}_{d}} - a}\right)
+$$
+
+$$
++ {e}_{2}^{\mathrm{T}}\left( {-\widehat{f} - {e}_{1} - \frac{{e}_{d}}{{t}_{d}} - {k}_{2}{e}_{2}}\right) + {e}_{2}^{\mathrm{T}}R{M}^{-1}{\tau }_{e} - \frac{{\vartheta }^{2}}{\mu }. \tag{38}
+$$
+
+Finally, we can obtain
+
+$$
+{\dot{V}}_{2} \leq - \left( {{\lambda }_{\min }\left( {k}_{1}\right) - \frac{1}{2}}\right) {\begin{Vmatrix}{e}_{1}\end{Vmatrix}}^{2} - \left( {\frac{1}{{t}_{d}} - \frac{1}{2}}\right) {\begin{Vmatrix}{e}_{d}\end{Vmatrix}}^{2} + \begin{Vmatrix}{e}_{d}\end{Vmatrix}\begin{Vmatrix}{a}^{ * }\end{Vmatrix}
+$$
+
+$$
+- {\lambda }_{\min }\left( {k}_{2}\right) {\begin{Vmatrix}{e}_{2}\end{Vmatrix}}^{2} + \begin{Vmatrix}{e}_{2}\end{Vmatrix}\parallel \widetilde{f}\parallel + \begin{Vmatrix}{e}_{2}^{\mathrm{T}}\end{Vmatrix}\begin{Vmatrix}{M{\tau }_{e}}\end{Vmatrix} - \frac{{\vartheta }^{2}}{\mu }. \tag{39}
+$$
+
+Choose ${\lambda }_{\min }\left( {k}_{1}\right) - \frac{1}{2} > 0,\frac{1}{{t}_{d}} - \frac{1}{2} > 0$ . Then, define ${\varsigma }_{2} = \min \left( {{\lambda }_{\min }\left( {k}_{1}\right) - \frac{1}{2},{\lambda }_{\min }\left( {k}_{2}\right) ,\frac{1}{{t}_{d}} - \frac{1}{2},\frac{1}{\mu }}\right) ,\varpi =$ $\begin{Vmatrix}{M{\tau }_{e}}\end{Vmatrix},E = {\left\lbrack \begin{array}{llll} {e}_{1}^{\mathrm{T}} & {e}_{2}^{\mathrm{T}} & {e}_{d}^{\mathrm{T}} & \vartheta \end{array}\right\rbrack }^{\mathrm{T}}$ . Hence,(39) becomes ${\dot{V}}_{2} \leq$ $- {\varsigma }_{2}\parallel \mid E{\parallel }^{2} + \parallel E\parallel \left( {{E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }\right)$ and ${\dot{V}}_{2} \leq - {\varsigma }_{2}\left( {1 - \epsilon }\right) \parallel E{\parallel }^{2} +$ $\parallel E\parallel \left( {-\epsilon {\varsigma }_{2}\parallel E\parallel + {E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }\right)$ where $\epsilon \leq 1$ .
+
+Note that
+
+$$
+\parallel E\parallel \geq \frac{{E}_{o} + \begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }{\epsilon {\varsigma }_{2}}
+$$
+
+renders
+
+$$
+{\dot{V}}_{2} \leq - {\varsigma }_{2}\left( {1 - \epsilon }\right) \parallel E{\parallel }^{2}. \tag{40}
+$$
+
+It can be established that the error subsystem related to obstacle avoidance control is ISS. Additionally, the errors of the closed-loop system satisfies (37). B. Safe Analysis
+
+The subsequent lemma presents the safety analysis of the ASV.
+
+Lemma 4. Given the dynamics of the ASV as outlined in (6), if $\bar{p}\left( {t}_{0}\right) \in \mathcal{U}$ and ${\bar{\tau }}^{ * } \in \mathcal{U}$ are satisfied for all $t > {t}_{0}$ , the closed-loop system will be ISSf.
+
+Proof. According to Lemma 1, if ${\bar{\tau }}^{ * } \in \mathcal{U}$ holds, then $\mathcal{C}$ is forward invariant, meaning that the set $\mathcal{C}$ is ISSf. In other words, as long as $\bar{p}\left( {t}_{0}\right)$ is within $\mathcal{C}$ , the position $\bar{p}\left( t\right)$ will remain in $\mathcal{C}$ indefinitely. Therefore, the closed-loop system is ISSf.
+
+Theorem 2. The closed-loop system is shown to achieve ISSf, indicating that collision avoidance is feasible. Furthermore, all error signals in the closed-loop system are uniformly ultimately bounded.
+
+Proof. According to Lemma 4, the ASV will meet the safety constraint, meaning that the safety objective (11) is fulfilled. We employ Lemmas 2, 3, and [27, Lemma 1], which enable us to deduce that the closed-loop system is ISSf. The norm $\parallel E\parallel$ is uniformly ultimately bounded by
+
+$$
+\parallel E\parallel \leq \sqrt{\frac{{\lambda }_{\max }\left( Q\right) }{{\lambda }_{\min }\left( Q\right) }}\left( {\sqrt{\frac{{\lambda }_{\max }\left( N\right) }{{\lambda }_{\min }\left( N\right) }}\frac{2\parallel {ND}\parallel {\xi }^{ * }}{{\varsigma }_{1}{\kappa \epsilon }{\varsigma }_{2}}}\right.
+$$
+
+$$
+\left. {+\frac{\begin{Vmatrix}{a}^{ * }\end{Vmatrix} + \varpi }{\epsilon {\varsigma }_{2}}}\right) \text{ . } \tag{41}
+$$
+
+Given that $E$ is bounded, we can deduce that ${e}_{1}$ and $\vartheta$ are also bounded. As a result, it follows that $\begin{Vmatrix}{p\left( t\right) - {p}_{0}\left( \theta \right) }\end{Vmatrix} =$ $\begin{Vmatrix}{e}_{1}\end{Vmatrix}$ and $\begin{Vmatrix}{\dot{\theta }\left( t\right) - {u}_{d}}\end{Vmatrix}$ remain bounded, that is,(9) and (10) hold.
+
+§ REMARK 3.
+
+§ V. SIMULATION RESULTS
+
+To validate the effectiveness of the proposed control strategy, this paper conducts simulations using Cybership II in [28]. The simulation parameters are set as follows: $w = {40}$ , $\Omega = {0.1},l = 1,\mu = {0.1},\beta = {30},{d}_{k} = {0.5},{r}_{k} = 1$ , ${k}_{1} = \operatorname{diag}\{ 3,2,8\} ,{k}_{2} = \operatorname{diag}\{ {16},{22},{28}\} ,{t}_{d} = {0.1},{d}_{w} =$ ${\left\lbrack \begin{array}{lll} 3\cos \left( t\right) \sin \left( {0.5t}\right) & 4\sin \left( {0.5t}\right) \cos \left( {0.5t}\right) & {0.2}\sin \left( t\right) \end{array}\right\rbrack }^{\mathrm{T}}$ . The desired parameterized path is ${x}_{d}\left( {\vartheta }_{0}\right) = {y}_{d}\left( {\vartheta }_{0}\right) = {0.06}{\vartheta }_{0} + {0.5}$ , ${\psi }_{d} = \pi /4$ . The position of static obstacle is ${\bar{p}}_{1} = {\left\lbrack \begin{array}{ll} 3 & {2.5} \end{array}\right\rbrack }^{\mathrm{T}}$ , ${\bar{p}}_{2} = {\left\lbrack \begin{array}{ll} 6 & 7 \end{array}\right\rbrack }^{\mathrm{T}}.$
+
+ < g r a p h i c s >
+
+Fig. 2. Control performance of the proposed control strategy for ASV.
+
+Fig. 2 illustrates the effectiveness of the safety-critical controller proposed for the autonomous vessel. The upper section demonstrates that the vessel prioritizes obstacle avoidance to ensure safety. Once the obstacle avoidance operation is complete, it can proceed with the tracking task. Fig. 3 illustrates the observation effect of the extended state observer. The velocity and aggregated disturbances of the ASV can be accurately estimated. Fig. 4 and Fig. 5 depict the comparisons of the tracking errors and velocities, respectively.
+
+ < g r a p h i c s >
+
+Fig. 3. The estimates of lumped disturbances.
+
+ < g r a p h i c s >
+
+Fig. 4. Tracking errors of the ASV.
+
+ < g r a p h i c s >
+
+Fig. 5. Velocity comparisons of the ASV.
+
+§ VI. CONCLUSION
+
+This paper introduces a safety-critical control strategy for ASVs that considers external marine disturbances and internal model uncertainties. Initially, an anti-disturbance controller is devised based on the estimation of lumped disturbances using an ESO. Following this, a quadratic optimization problem is established by incorporating ISSf-ECBFs to enforce safety constraints on the control inputs. By solving this problem, a safety-critical controller is derived, significantly improving the safety and robustness of the system. The closed-loop system is demonstrated to be ISSf, with error signals shown to be uniformly ultimately bounded. The effectiveness of the proposed control strategy is verified through simulation results.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NLNBi9lbov/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NLNBi9lbov/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..87e893ff56ac54ef5ddb559fa73017f2081d48d8
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/NLNBi9lbov/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,462 @@
+§ REMOTE SENSING OBJECT DETECTION BASED ON FUSION OF SPATIAL AND CHANNEL ATTENTION
+
+Wenyun Sun
+
+School of Artificial Intelligence
+
+Nanjing University of
+
+Information Science and
+
+Technology
+
+Nanjing, China
+
+wenyunsun@nuist.edu.cn
+
+Long Ji
+
+School of Computer Science
+
+Nanjing University of
+
+Information Science and
+
+Technology
+
+Nanjing, China
+
+202212490374@nuist.edu.cn
+
+Abstract-Remote sensing object detection faces unique challenges due to objects' varied scales and orientations. To address these challenges, we propose the Spatial Channel Attention Fusion Module (SCAF-Module), designed to enhance detection accuracy by integrating multi-scale convolutions, adaptive rotated convolutions, and parallel spatial channel attention mechanisms. The experiments, conducted using the DOTA-v1.0 and HRSC2016 datasets, demonstrate the efficacy of the SCAF-Module. We achieved mean Average Precision (mAP) scores of 80.94% and 98.23% on these datasets, respectively. Comparative experiments reveal that the SCAF-Module surpasses several advanced models, including the baseline Oriented R-CNN. Additionally, ablation studies highlight the significance of the spatial and channel attention mechanisms and the impact of rotated convolutions on detection performance. The SCAF-Module presents a robust and adaptable framework for remote sensing object detection, offering significant improvements over existing methods. This work paves the way for further optimization and application of the module in other challenging remote sensing tasks.
+
+Keywords-Remote Sensing, Object Detection, Attention Mechanism, Neural Networks
+
+§ I. INTRODUCTION
+
+Remote sensing imagery plays a pivotal role in a wide array of applications, including environmental monitoring, urban planning, disaster management, and agricultural assessment. These applications demand accurate and efficient object detection methods to extract meaningful information from complex and large-scale images. However, detecting objects in remote sensing images presents unique challenges compared to natural image datasets. The diverse scales, orientations, and dense distribution of objects within cluttered backgrounds significantly hinder the performance of traditional detection algorithms.
+
+Conventional object detection models, such as YOLO [1] and Faster R-CNN [2], have achieved significant success in natural images. However, these models often encounter limitations when applied to remote sensing imagery. Specifically, traditional methods struggle to manage objects with varying scales, accurately detect objects with arbitrary orientations, and adapt to the complex background clutter typical in remote sensing environments. Additionally, most conventional models rely on axis-aligned bounding boxes, which are not well-suited for the detection of oriented objects, leading to reduced accuracy. Several approaches have been proposed to address these issues, such as incorporating rotated bounding boxes and employing multi-scale feature extraction techniques. However, these methods often involve trade-offs between computational efficiency and detection accuracy, and may still fall short in environments with highly varied object scales and orientations. Moreover, these models' integration of attention mechanisms has been relatively limited, often focusing on spatial or channel attention separately, rather than leveraging their combined potential.
+
+To overcome these challenges, we propose the Spatial Channel Attention Fusion Module (SCAF-Module). This module integrates multi-scale convolutions, adaptive rotated convolutions, and parallel spatial channel attention mechanisms to enhance detection accuracy. The multi-scale convolutions include a $3 \times 3$ rotated convolution, a $3 \times 3$ dilated rotated convolution, and a $5 \times 5$ dilated convolution, each contributing to the detection of objects at various scales and orientations. The spatial and channel attention mechanisms further refine these features, allowing the model to selectively focus on important regions and adapt to different object characteristics.
+
+We define multi-scale convolution as the application of convolutional layers with different receptive fields, designed to capture features at various scales [3]. Adaptive rotated convolution [4] is defined as a type of convolution that adapts to the orientation of objects, enhancing the model's sensitivity to rotated objects. The Spatial Channel Attention Fusion is a technique that enhances feature representation by focusing on significant spatial regions and feature channels [5].
+
+Our contributions are as follows:
+
+ * Multi-scale and Adaptive Rotated Convolutions: By incorporating convolutions with different receptive fields and adaptive rotated convolutions, the SCAF-Module effectively captures objects of varying scales and orientations.
+
+ * Spatial and Channel Attention Mechanisms: These mechanisms enhance the model's ability to focus on significant regions and channels, improving detection performance.
+
+ * Comprehensive Evaluation: Extensive experiments on the DOTA-v1.0 and HRSC2016 datasets demonstrate the SCAF-Module's effectiveness, achieving mAP scores of 80.94% and 98.23%, respectively.
+
+This work was supported by the National Natural Science Foundation of China under Grant No. 61702340.
+
+The remainder of this paper is organized as follows: Section II reviews related work in remote sensing object detection. Section III details the proposed SCAF-Module and its integration into the backbone network. Section IV presents the experiment setup, results, and ablation studies. Finally, Section TABLE VI. concludes the paper and discusses future work.
+
+§ II. RELATED WORK
+
+§ A. REMOTE SENSING OBJECT DETECTION FRAMEWORKS
+
+Object detection in remote sensing images has garnered significant attention due to its critical applications in areas such as urban planning, disaster management, and environmental monitoring. Traditional object detection frameworks designed for natural images, such as YOLO and Faster R-CNN, have been adapted to remote sensing scenarios. However, these frameworks face challenges unique to remote sensing images, such as the need to detect objects at various orientations and scales. Consequently, specialized frameworks have been developed to address these challenges, focusing on rotated object detection, multi-scale feature extraction, and robust performance in diverse and cluttered environments.
+
+To mitigate the abundance of rotated anchors and to minimize the disparity between the feature representations and the actual objects, Ding et al. [6] have introduced the RoI transformer. This technique, which extracts rotated RoIs from the horizontal ones yielded by the RPN, substantially enhances the precision of detecting objects with orientation. Nonetheless, the incorporation of fully-connected layers and the RoI alignment process during the learning phase adds a layer of complexity and computational demands to the network.
+
+To tackle the detection of small-scale, densely packed, and rotated objects, Yang et al. [7] have crafted an oriented object detection approach that integrates with the established Faster R-CNN framework. Additionally, a novel representation for oriented objects, known as gliding vertexes [8], has been put forward. This method refines the detection process by acquiring four vertex gliding offsets from the regression component of the Faster R-CNN architecture.
+
+Despite these advancements, the reliance on horizontal RoIs for classification and oriented bounding box regression in these methods leads to significant misalignment issues between the objects and their corresponding features. Furthermore, various studies have delved into one-stage or anchor-free oriented object detection frameworks, which forgo the need for region proposal generation and RoI alignment, directly outputting object classes and oriented bounding boxes. For instance, a refined one-stage oriented object detector [9] has been proposed, featuring two pivotal enhancements: feature refinement and progressive regression, addressing the misalignment of features. A new label assignment strategy for one-stage oriented object detection, inspired by RetinaNet, dynamically assigns anchors as either positive or negative through an innovative matching approach. A single-shot alignment network (S ${}^{2}$ ANet) [10] has been introduced for oriented object detection, focusing on harmonizing the classification score with location precision through deep feature alignment. Lastly, a dynamic refinement network (DRN) [11]has been conceptualized for oriented object detection, leveraging the anchor-free detection approach of CenterNet [12].
+
+§ B. ADDRESSING METRIC AND LOSS INCONSISTENCY
+
+The issue of inconsistency between metrics and loss functions is prevalent in horizontal bounding box object detection and becomes even more pronounced in remote sensing object detection due to the introduction of angle parameters. In horizontal bounding box detection, new IoU (Intersection over Union) calculation methods like DIoU (Distance Intersection over Union) [13] and GIoU (Generalized Intersection over Union) [14] have been proposed to alleviate inconsistency problems. However, these methods are non-differentiable and thus not directly applicable to remote sensing object detection.
+
+Existing solutions to inconsistency in remote sensing object detection are limited, primarily focusing on designing new loss functions. These can be categorized into bounding box-based, pixel-based, and Gaussian distribution-based loss functions. Most current detection methods calculate the IoU of two inclined bounding boxes, often using smooth L1 as the regression loss function. However, for near-square targets, high IoU can still result in a significant loss. To address this, Yang et al. [7] proposed the IoU-smooth L1 loss, which combines IoU and smooth L1 to mitigate the problem. The overlapping forms of two inclined bounding boxes vary greatly. Zheng et al. [15] addressed this by proposing a rotation-robust IoU (RIoU) calculation method for $3\mathrm{D}$ object detection, which can also be applied to 2D rotated object detection. This method defines a pair of projected rectangles to calculate the overlap area, allowing for the regression of bounding boxes at any angle. For anchor-free detection methods, Guo et al. [16] proposed using a convex hull formed by a set of irregular points to represent each rotated target, then optimizing the detector using a convex hull-based CIoU (Complete Intersection over Union) loss. Additionally, the smooth L1 loss is insensitive to large aspect ratio targets. To address this, Chen et al. [17] proposed PIoU (Pixels Intersection over Union) loss, which determines whether pixels are within the rotated box and calculates the rotated IoU by accumulating these pixels. This loss function can be applied to both anchor-based and anchor-free frameworks, though its accuracy needs improvement.
+
+§ C. GAUSSIAN-BASED LOSS
+
+Recently, methods based on 2D Gaussian distributions have garnered significant attention. Yang et al. [18] analyzed the impact of angle differences, center point deviations, and different aspect ratios between rotated candidate boxes and ground truth on loss function changes. They designed a new loss function based on Gaussian Wasserstein distance. The approach involves converting rotated bounding boxes to 2D Gaussian distributions, calculating the Gaussian Wasserstein distance between the distributions of the ground truth and predicted boxes to derive the new loss function. However, this method lacks scale invariance, and optimizing only the rotation center can lead to positional deviations in the detection results. To address the scale variation issue brought by Gaussian Wasserstein distance, KL divergence [19] has been used as a substitute for loss calculation. This method, similar to Gaussian Wasserstein distance, derives a theoretical explanation for selecting distribution distance metrics to maintain detection accuracy and scale invariance after transforming parameters into 2D Gaussian distributions. Both loss functions introduce additional hyperparameters, but the key to maintaining consistency between evaluation and loss is ensuring their trends remain consistent. Inspired by Kalman filtering, the KFIoU (Kalman Filter Intersection over Union) loss [20] was proposed. The basic steps involve modeling the ground truth and predicted bounding boxes as Gaussian distributions, aligning the center points of the two distributions, obtaining the Gaussian distribution of the overlapping area through Kalman filtering, and converting it back to a rotated bounding box. This approach approximates the rotated IoU.
+
+§ D. BACKBONE NETWORK DESIGN
+
+Designing an effective backbone network for remote sensing images is crucial due to the varying scales and orientations of objects. Li et al. [3] introduced a strategy incorporating large kernel convolutions with different receptive fields into the backbone network. This approach dynamically adjusts the receptive fields to capture features at various scales. However, large kernel convolutions may lead to information loss for small objects, as their large receptive fields might cover multiple objects or noise regions. Pu et al. [4] employed adaptive rotated convolutions, rotating the convolutional kernels to achieve rotational sampling. While effective, rotating large kernels significantly increases computational complexity without a corresponding improvement in accuracy. This trade-off highlights the need for a balanced approach that can capture features at different scales and orientations without excessive computational cost.
+
+§ E. ATTENTION MECHANISMS
+
+Attention mechanisms serve as a straightforward yet potent means of augmenting neural network representations across a multitude of applications. The channel-wise attention mechanism, exemplified by the SE block [5], leverages the insights from global averaging to recalibrate the importance of feature channels. Concurrently, spatial attention schemes such as those found in GENet [21], GCNet [22], and SGE [23], fortify the network's capacity to incorporate contextual cues through spatial filtering techniques. The CBAM [24] and BAM [25] architectures amalgamate channel and spatial attention, capitalizing on the strengths of both to refine feature representation.
+
+Our proposed method focuses on the integration of backbone network design and attention mechanisms. Li et al. [3] incorporated prior knowledge from remote sensing images to develop large kernel selective convolutions; however, these large kernel convolutions can reduce the network's sensitivity to small objects, which are prevalent in remote sensing imagery. Pu et al. [4] designed rotated convolutions that are sensitive to object angles, yet this approach introduces significant computational overhead. In contrast, our work combines both strategies by implementing spatial attention, utilizing smaller rotated convolutions alongside larger standard convolutions. This approach effectively balances the detection of small and large objects without substantially increasing computational costs. Additionally, channel attention mechanisms [5] are employed to suppress irrelevant features while enhancing the importance of relevant ones, thereby improving overall detection performance.
+
+§ III. PROPOSED METHOD
+
+In this section, we introduce the architecture and components of the Spatial Channel Attention Fusion Module (SCAF-Module), designed to enhance remote sensing object detection by integrating multi-scale convolutions, adaptive rotated convolutions, and parallel spatial channel attention mechanisms. We also detail the overall backbone structure, which incorporates the SCAF-Module into a hierarchical framework to effectively capture and represent features at multiple levels. The SCAF-Module is built upon the following assumptions: (1) Objects in remote sensing images vary greatly in scale and orientation [20], necessitating a detection method that can handle these variations effectively. (2) Multi-scale [3] and adaptive rotated convolutions [4] are effective in capturing detailed features across different scales and orientations. (3) Spatial and channel attention mechanisms [5] can further enhance the detection performance by emphasizing important features.
+
+§ A. SPATIAL CHANNEL ATTENTION FUSION MODULE
+
+The Spatial Channel Attention Fusion Module (SCAF-Module) is designed to enhance feature representation by integrating spatial [3]and channel attention [5] mechanisms with multi-scale and rotated convolutions [4]. This module aims to address the challenges posed by the diverse scales and orientations of objects in remote sensing images. The overall structure of the SCAF-Module is depicted in Fig. 1.
+
+§ 1) MULTI-SCALE CONVOLUTIONS
+
+The SCAF-Module begins by processing the input feature map $X$ through three convolutional layers, each with a different receptive field to capture features at various scales. The three convolutional layers include:
+
+ * 3 - 3 Rotated Convolution: Captures fine-grained details and small-scale features with enhanced sensitivity to object orientations, improving detection accuracy for rotated objects.
+
+ * $3 \times 3$ Rotated Dilated Convolution (dilation rate = 2): Utilizes a dilation rate of 2 to expand the receptive field without increasing the number of parameters, capturing medium-scale features while maintaining orientation adaptability.
+
+ * $5 \times 5$ Standard Dilated Convolution (dilation rate = 2): Further expands the receptive field to capture larger-scale features, ensuring comprehensive feature extraction across various object scales.
+
+These layers generate three feature maps ${F}_{1},{F}_{2}$ , and ${F}_{3}$ respectively. The use of different receptive fields allows the module to balance sensitivity to both large and small objects, which is crucial for the diverse object sizes found in remote sensing images.
+
+§ 2) ROTATED CONVOLUTION
+
+Rotated convolution is designed to address the unique challenges posed by remote sensing images, particularly the diverse orientations of objects. Traditional convolutional layers are limited by their fixed orientation, which can hinder the model's ability to accurately capture features of rotated objects. The rotated convolution mechanism introduces a way to dynamically adjust the orientation of the convolutional filters, enabling better alignment with the objects in the input image. The primary advantage of rotated convolution is its ability to rotate the convolutional kernels to match the orientation of the target objects. This is particularly beneficial for remote sensing applications where objects such as buildings, vehicles, and agricultural fields can appear at various angles. By aligning the convolutional filters with the orientation of these objects, the rotated convolution can more effectively capture the relevant features, leading to improved detection accuracy. The process of rotated convolution involves the following steps:
+
+ < g r a p h i c s >
+
+Fig. 1. The overall structure of the SCAF-Module.
+
+ * Kernel Rotation: The convolutional kernel is treated as a set of sampling points in the kernel space. These sampling points are then rotated by an angle $\theta$ , which is dynamically determined based on the input feature map. This rotation allows the kernel to align with the orientation of the objects in the image.
+
+ * Bilinear Interpolation: After rotating the sampling points, bilinear interpolation is used to map the original convolution parameters to the new rotated positions. This ensures that the rotated kernel retains the characteristics of the original kernel while adapting to the new orientation.
+
+ * Dynamic Angle Generation: The rotation angle $\theta$ is not fixed but is generated dynamically by a routing function based on the input features. This allows the model to adapt to different orientations present in the input image, providing a flexible and robust solution for capturing rotated objects.
+
+Rotated convolution addresses the limitations of traditional convolutional layers by introducing orientation adaptability. This innovation is crucial for improving the accuracy of object detection in remote sensing images, where the orientation of objects is often varied and unpredictable. By aligning the convolutional filters with the objects' orientations, the SCAF-Module can more effectively capture and represent the relevant features, leading to superior detection performance.
+
+§ 3) SPATIAL ATTENTION MECHANISM
+
+The spatial attention mechanism is designed to focus on important regions within the feature maps, addressing the variability in the shapes and scales of objects. As shown in Fig. 2, the spatial attention mechanism operates as follows:
+
+ * Concatenation: The feature maps ${F}_{1},{F}_{2}$ , and ${F}_{3}$ are concatenated along the channel dimension to form a combined feature map $F$ .
+
+ * Pooling: The combined feature map $F$ undergoes average pooling and max pooling along the channel dimension, producing the average feature map ${AF}$ and maximum feature map ${MF}$ .
+
+ * Fusion: The pooled feature maps ${AF}$ and ${MF}$ are concatenated along the channel dimension to form the fused feature map ${AMF}$ .
+
+ * Convolution and Activation The fused feature map ${AMF}$ is passed through a convolutional layer and a sigmoid activation function to produce the spatial attention map ${SF}$ :
+
+$$
+{SF} = \sigma \left( {\operatorname{Conv}\left( \left\lbrack {\operatorname{AvgPool}\left( F\right) ,\operatorname{MaxPool}\left( F\right) }\right\rbrack \right) }\right) , \tag{1}
+$$
+
+where $\sigma$ denotes the sigmoid activation function, which maps the output to a range of $\left\lbrack {0,1}\right\rbrack$ , serving as the spatial attention weights. This mechanism enables the model to selectively focus on significant regions in the feature maps, enhancing the detection of objects with varying shapes and scales.
+
+§ 4) CHANNEL ATTENTION MECHANISM
+
+The channel attention mechanism dynamically adjusts the weights of different channels, emphasizing channels that are more informative for the task at hand. This mechanism is crucial for optimizing the feature representation by selectively enhancing the most relevant channels based on the global context of the feature map. As shown in Fig. 3, the channel attention mechanism operates as follows:
+
+ < g r a p h i c s >
+
+Fig. 2. Spatial attention mechanism.
+
+ * Global Average Pooling: Each feature map ${F}_{i}$ undergoes global average pooling to capture the global context of the feature map, resulting in a descriptor vector $z$ . The global average pooling operation aggregates the spatial information across the entire feature map, producing a channel-wise descriptor that summarizes the global context.
+
+ * Squeeze and Excitation: The descriptor vector $z$ is passed through two convolutional layers. The first convolutional layer reduces the number of channels by a ratio (typically 16), and the second convolutional layer restores the original number of channels. This sequence of operations is designed to learn the importance of each channel dynamically:
+
+$$
+{s}_{c} = \sigma \left( {{\operatorname{Conv}}_{2}\left( {\operatorname{ReLU}\left( {{\operatorname{Conv}}_{1}\left( z\right) }\right) }\right) }\right) , \tag{2}
+$$
+
+where ${\mathrm{{Conv}}}_{1}$ and ${\mathrm{{Conv}}}_{2}$ are the convolutional layers used for compression and excitation, respectively, and $\sigma$ denotes the sigmoid activation function.
+
+ * Reweighting: The learned channel weights ${s}_{c}$ are applied to the corresponding feature map ${F}_{i}$ , producing the channel-weighted feature map $C{F}_{i}$ :
+
+$$
+C{F}_{i} = {F}_{i} \odot {s}_{c}, \tag{3}
+$$
+
+where $\odot$ denotes the element-wise multiplication.
+
+The core idea behind this mechanism is to use global information to recalibrate the feature map in a channel-wise manner, enhancing the model's ability to focus on the most informative channels and improving the overall feature representation. By integrating this SE-based channel attention mechanism, the SCAF-Module can effectively capture and utilize the global context, leading to improved performance in remote sensing object detection tasks.
+
+§ 5) FEATURE FUSION
+
+The outputs from the spatial and channel attention mechanisms are element-wise multiplied and summed to produce a new feature map ${FF}$ :
+
+$$
+{FF} = \mathop{\sum }\limits_{{i = 1}}^{3}S{F}_{i} \odot C{F}_{i}. \tag{4}
+$$
+
+where $\odot$ denotes the element-wise multiplication. This fusion step integrates spatial and channel attention, enhancing the feature representation by focusing on both important regions and informative channels. Finally, the fused feature map ${FF}$ is element-wise multiplied with the input feature map $X$ to produce the final output $Y$ of the module:
+
+$$
+Y = X \odot {FF}\text{ . } \tag{5}
+$$
+
+This final step ensures that the enhanced features are integrated with the original input, maintaining the integrity of the input information while incorporating the attention mechanisms' enhancements. The SCAF-Module, through its combination of multi-scale convolutions, spatial attention, and channel attention, effectively addresses the challenges of detecting objects with varying scales and orientations in remote sensing images.
+
+§ B. BACKBONE STRUCTURE
+
+The backbone structure of the SCAF-Module is meticulously designed to effectively capture and process the diverse features present in remote sensing images. This section provides an in-depth look at the backbone architecture, consisting of multiple stages, each composed of several blocks that integrate the SCAF-Module to enhance feature representation.
+
+§ 1) BLOCK STRUCTURE
+
+Each block within the backbone is constructed to maintain the shape and channel dimensions of the input while enhancing the feature representation through a series of operations. The block structure is as follows:
+
+ * Normalization 1: The input feature map undergoes a normalization process to stabilize and accelerate the training process.
+
+ * Fully Connected Layer: A fully connected (FC) layer is applied to the normalized features, transforming them into a different feature space.
+
+ < g r a p h i c s >
+
+Fig. 3. Channel attention mechanism.
+
+ * GELU Activation: The output from the fully connected layer is passed through a GELU (Gaussian Error Linear Unit) activation function to introduce non-linearity.
+
+ * SCAF-Module: The activated features are then processed by the SCAF-Module, which applies multi-scale convolutions, spatial attention, and channel attention to enhance the feature representation.
+
+ * Fully Connected Layer: Another fully connected layer is applied to the features output by the SCAF-Module.
+
+ * Normalization 2: A second normalization layer is used to further stabilize the feature representations.
+
+ * MLP: Finally, the features pass through a multi-layer perceptron (MLP) for additional transformation and refinement.
+
+The block incorporates two residual connections to preserve the original input features and prevent the degradation problem commonly encountered in deep networks. The first residual connection adds the input to the feature map before the second normalization step, while the second residual connection adds the input to the final output of the block, ensuring that the input-output shape and channel dimensions remain unchanged. The block structure is illustrated in Fig. 4.
+
+§ 2) STAGE STRUCTURE
+
+The backbone is organized into multiple stages, each consisting of several blocks to progressively extract and refine features at different scales and resolutions. Each stage operates as follows:
+
+ * Shape and Channel Adjustment: At the beginning of each stage, a convolutional layer adjusts the shape and channel dimensions of the input feature map to prepare it for further processing.
+
+ * Repeated Blocks: The adjusted feature map is then passed through a series of blocks. Each block applies the operations described above, progressively enhancing the feature representation.
+
+ * Normalization: The output of the final block in each stage undergoes normalization to ensure stable feature distribution before passing to the next stage.
+
+The multi-stage structure allows the backbone to capture features at varying levels of abstraction, from low-level edges and textures to high-level semantic information.
+
+3) Integration with Oriented R-CNN
+
+The backbone is integrated into the Oriented R-CNN framework, replacing the original ResNet backbone. The Oriented R-CNN is specifically designed for object detection in remote sensing images, where objects often appear in arbitrary orientations. By incorporating the SCAF-Module-based backbone, the Oriented R-CNN benefits from improved feature extraction capabilities, particularly in handling the diverse scales and orientations of objects in remote sensing imagery.
+
+The backbone structure leveraging the SCAF-Module significantly enhances the capability of the Oriented R-CNN to accurately detect and classify objects in remote sensing images. The multi-scale convolutions, combined with spatial and channel attention mechanisms, ensure that the model captures a comprehensive set of features, leading to superior detection performance.
+
+§ IV. EXPERIMENTS
+
+In this section, we present the experiments conducted to evaluate the performance of the proposed SCAF-Module. We detail the datasets used, the evaluation metrics, and the experiment setup. Finally, we present and analyze the results, demonstrating the effectiveness of our approach.
+
+§ A. DATASETS AND EVALUATION
+
+We evaluate the SCAF-Module on two widely used remote sensing image datasets: DOTA-v1.0 and HRSC2016. These datasets are chosen for their diversity in object scales, orientations, and complexity of scenes, which pose significant challenges for object detection models.
+
+DOTA-v1.0 [26]: The following fifteen object classes are covered in this dataset: Plane (PL), Baseball diamond (BD), Bridge (BR), Ground track field (GTF), Small vehicle (SV), Large vehicle (LV), Ship (SH), Tennis court (TC), Basketball court (BC), Storage tank (ST), Soccer-ball field (SBF), Roundabout (RA), Harbor (HA), Swimming pool (SP), and Helicopter (HC). The dataset contains a wide variety of object scales and orientations, making it a suitable benchmark for evaluating our model's ability to handle diverse object characteristics. Due to the large size of images, offline data augmentation is typically used. For single-scale training and testing, images are cropped to ${1024} \times {1024}$ patches with 200 pixels overlap. For multi-scale training and testing, images are first resized to 0.5, 1.0, and 1.5 times their original size, and then cropped to ${1024} \times {1024}$ patches with 500 pixels overlap.
+
+HRSC2016 [27]: This dataset focuses on ship detection and includes images captured from various angles and distances, with ships annotated using oriented bounding boxes. The variability in ship sizes and orientations makes this dataset an excellent testbed for our model's robustness.
+
+ < g r a p h i c s >
+
+Fig. 4. Block structure.
+
+The primary evaluation metric used in The experiments is the mean Average Precision (mAP), which measures the precision-recall performance across different object categories. We report the mAP scores to provide a comprehensive assessment of our model's detection capabilities.
+
+§ B. EXPERIMENT SETUP
+
+The experiments are conducted using the MMRotate framework on an NVIDIA GeForce RTX 3090 GPU with a batch size of 2 for training and evaluation. The optimizer used is AdamW with a learning rate of $5 \times {10}^{-5},{\beta }_{1} = {0.9},{\beta }_{2} =$ 0.999, and a weight decay of 0.05 . The learning rate follows a step policy with an initial linear warmup for 500 iterations starting at a third of the base learning rate, then decaying at epochs 8 and 11. Image normalization is applied with mean values [123.675, 116.28, 103.53] and standard deviations $\left\lbrack {{58.395},{57.12},{57.375}}\right\rbrack$ . The training pipeline includes resizing, random flipping (horizontal, vertical, diagonal), random rotation, normalization, padding, and data collection. The test pipeline involves multi-scale augmentation and normalization.
+
+The SCAF-Module is integrated into the Oriented R-CNN framework, replacing the original ResNet backbone, to evaluate its performance in detecting objects with arbitrary orientations in remote sensing images. For the ablation experiments, the backbone is not pre-trained on ImageNet to enhance experimental efficiency. In contrast, for the comparative experiments, the backbone undergoes pre-training on ImageNet for 300 epochs before being fine-tuned on the DOTA-v1.0 and HRSC2016 datasets to achieve higher performance.
+
+§ C. COMPARATIVE EXPERIMENTS
+
+In this section, we evaluate the performance of the SCAF-Module against six advanced models, including the baseline Oriented R-CNN, using two widely adopted remote sensing image datasets: DOTA-v1.0 and HRSC2016. The comparison focuses on mean Average Precision (mAP) as the primary metric.
+
+1) DOTA-v1.0 Dataset
+
+The DOTA-v1.0 dataset is a standard benchmark for remote sensing object detection. We conducted experiments using both single-scale and multi-scale training and testing protocols to assess the robustness of our model.
+
+For the single-scale evaluation, large images from the DOTA-v1.0 dataset were divided into ${1024} \times {1024}$ patches with a 200-pixel overlap. The results, summarized in Table I, show that the SCAF-Module achieved a significant mAP of 78.96%, outperforming all compared models. This demonstrates the module's ability to effectively capture fine-grained details and accurately detect objects at a fixed scale. In the multi-scale evaluation, images were rescaled to0.5,1.0, and 1.5 times their original sizes, with a 500-pixel overlap during patching. As shown in Table II, the SCAF-Module achieved an mAP of 80.94%, again surpassing the other models. This result highlights the module's robustness in adapting to various object scales, a critical requirement in remote sensing tasks.
+
+ < g r a p h i c s >
+
+Fig. 5. Detection Results on DOTA-V1.0 Dataset.
+
+These evaluations on the DOTA-v1.0 dataset confirm the superior performance of the SCAF-Module, particularly in its ability to enhance detection accuracy across both single and multi-scale scenarios. To visually represent the effectiveness of our model on the DOTA-v1.0 dataset, we present a series of detection result images in Fig. 5.
+
+§ 2) HRSC2016 DATASET
+
+The HRSC2016 dataset focuses on ship detection, providing a rigorous test of model precision and robustness. We evaluated our model under the PASCAL VOC 2007 and VOC 2012 metrics to ensure a thorough assessment. As detailed in Table III, the SCAF-Module achieved mAP scores of 90.61% under the VOC 2007 metric and 98.23% under the VOC 2012 metric, marking a notable improvement over the other models. This performance can be attributed to the module's advanced attention mechanisms and multi-scale convolutional structure, which enhance its ability to detect ships with varying orientations- a frequent challenge in remote sensing imagery.
+
+Overall, the results from the HRSC2016 dataset further validate the effectiveness of the SCAF-Module in detecting objects with diverse scales and orientations, reinforcing its value as a robust tool for remote sensing object detection tasks.
+
+§ D. ABLATION STUDY
+
+To understand the contribution of each component in the SCAF-Module, we conduct ablation studies on the DOTA-v1.0 dataset. The goal is to analyze the effects of Spatial Attention, Channel Attention, their order, and the use of Rotated Convolutions on the overall performance of the model.
+
+§ CONTRIBUTION OF INDIVIDUAL COMPONENTS
+
+We investigate the contributions of various components to the overall performance of our model. The baseline configuration employs multi-scale convolutional layers without any additional mechanisms. We then incrementally add the following components: rotated convolutions, channel attention, and spatial attention. All experiments are conducted using single-scale training and testing on the DOTA-v1.0 dataset.
+
+The baseline is multi-scale convolutional layers without any attention mechanisms. This setup serves as the foundational model, capturing features at different scales. We first incorporate spatial attention into the baseline. The spatial attention mechanism enables the model to focus on important spatial regions, enhancing the detection of objects of varying shapes and scales within the image. Next, we add the channel attention mechanism to the model with rotated convolutions. This mechanism allows the model to emphasize important channels, improving the representation of semantic information critical for accurate object detection. Finally, we integrate rotated convolutions into the model. Rotated convolutions enhance the model's ability to capture features at various orientations, crucial for remote sensing images where objects often appear in arbitrary orientations.
+
+The results of these experiments are summarized in Table IV. Each row in the table shows the mAP achieved by the model with the addition of the respective component. The performance improvements with each added component demonstrate the effectiveness of both the rotated convolutions and the attention mechanisms in enhancing the detection capabilities of the model.
+
+§ 1) ORDER OF SPATIAL AND CHANNEL ATTENTION
+
+We also tested different sequences of applying Spatial and Channel Attention. The results of these experiments are presented in Table V.
+
+When comparing the sequences of applying Spatial and Channel Attention, it is observed that the parallel application of both attentions yields the best results. Sequential applications result in slightly lower mAP scores, indicating that the simultaneous focus on both spatial and channel aspects is more beneficial.
+
+§ 2) EFFECTS OF ROTATED CONVOLUTIONS
+
+Finally, we tested the impact of using Rotated Convolutions in different configurations. The results of these experiments are presented in Table VI.
+
+TABLE I. AP FOR EACH CATEGORY AND OVERALL MAP ON DOTA-V1.0 (SINGLE-SCALE).
+
+max width=
+
+Model $\mathbf{{PL}}$ BD $\mathbf{{BR}}$ $\mathbf{{GTF}}$ SV LV SH $\mathbf{{TC}}$ $\mathbf{{BC}}$ ST SBF $\mathbf{{RA}}$ HA SP $\mathbf{{HC}}$ mAP
+
+1-17
+GWD [18] 88.92 77.08 45.91 69.30 72.52 64.05 76.33 90.87 79.18 80.45 57.67 64.36 63.60 64.75 48.24 69.55
+
+1-17
+${\mathrm{R}}^{3}$ Det [9] 89.30 73.36 45.10 71.21 76.51 74.01 81.03 90.89 79.01 83.54 59.37 63.47 63.04 65.93 37.02 70.19
+
+1-17
+${\mathrm{S}}^{2}\mathrm{\;A}$ - $\mathrm{{Net}}$ [10] 88.70 81.41 54.28 69.75 78.04 78.23 80.54 90.69 84.75 86.22 65.03 65.81 76.16 73.37 58.86 76.11
+
+1-17
+ReDet [28] 88.79 82.64 53.97 74.00 78.13 84.06 88.04 90.89 87.78 85.75 61.76 60.39 75.96 68.07 63.59 76.25
+
+1-17
+Oriented R- CNN [29] 88.86 83.48 55.27 76.92 74.27 82.10 87.52 90.90 85.56 85.33 65.51 66.82 74.36 70.15 57.28 76.28
+
+1-17
+LSKNet [3] 89.78 81.24 54.09 75.96 79.31 85.13 88.49 90.90 87.41 84.87 64.12 64.31 77.03 78.22 67.02 77.86
+
+1-17
+SCAF(ours) 89.72 85.25 55.38 76.10 79.55 84.85 88.43 90.85 87.46 85.71 66.99 68.54 76.85 79.79 68.87 78.96
+
+1-17
+
+TABLE II. MAP ON DOTA-V1.0 (MULTI-SCALE).
+
+max width=
+
+Model mAP
+
+1-2
+${\mathrm{R}}^{3}$ Det 76.47
+
+1-2
+${\mathrm{S}}^{2}\mathrm{\;A}$ - $\mathrm{{Net}}$ 79.42
+
+1-2
+ReDet 79.87
+
+1-2
+GWD 80.23
+
+1-2
+LSKNet 80.32
+
+1-2
+O-RCNN 80.62
+
+1-2
+SCAF(ours) 80.94
+
+1-2
+
+TABLE III. MAP ON HRSC2016 (VOC 2007 AND VOC 2012).
+
+max width=
+
+Model mAP(07) mAP(12)
+
+1-3
+${\mathrm{S}}^{2}\mathrm{\;A}$ - $\mathrm{{Net}}$ 90.17 95.01
+
+1-3
+R3Det 89.26 96.01
+
+1-3
+GWD 89.85 97.37
+
+1-3
+O-RCNN 90.50 97.60
+
+1-3
+ReDet 90.46 97.63
+
+1-3
+LSKNet 90.27 97.80
+
+1-3
+
+max width=
+
+SCAF(ours) 90.61 98.23
+
+1-3
+
+TABLE IV. CONTRIBUTION OF INDIVIDUAL COMPONENTS.
+
+max width=
+
+Configuration mAP
+
+1-2
+Baseline 67.62
+
+1-2
++ Spatial Attention 68.45
+
+1-2
++ Channel Attention 69.32
+
+1-2
++ Rotated Convolution 69.79
+
+1-2
+
+TABLE V. ORDER OF SPATIAL AND CHANNEL ATTENTION.
+
+max width=
+
+Configuration mAP
+
+1-2
+Spatial then Channel Attention 67.77
+
+1-2
+Channel then Spatial Attention 68.44
+
+1-2
+Parallel Spatial & Channel Attention 69.32
+
+1-2
+
+TABLE VI. EFFECTS OF ROTATED CONVOLUTIONS.
+
+max width=
+
+Configuration mAP
+
+1-2
+All ordinary convolutions 69.32
+
+1-2
+The first convolution rotated 69.59
+
+1-2
+The first and second convolutions rotated 69.79
+
+1-2
+All convolutions rotated 69.70
+
+1-2
+
+In the experiments focusing on the Rotated Convolutions, replacing the first and second convolutions with rotated ones gives the best performance. Using only ordinary convolutions or replacing all convolutions with rotated ones results in lower mAP scores. This suggests that a balanced combination of ordinary and rotated convolutions is most effective for capturing diverse object orientations in remote sensing images.
+
+The ablation study demonstrates the effectiveness of each component within the SCAF-Module. Spatial Attention enhances the model's ability to focus on important regions within the image, Channel Attention dynamically adjusts the significance of different feature channels, and Rotated Convolutions improve the detection of objects with varying orientations. The combination of these components results in a significant performance boost, validating the design choices made in the development of the SCAF-Module.
+
+§ V. CONCLUSION
+
+In this paper, we introduced the Spatial Channel Attention Fusion Module (SCAF-Module), a novel approach designed to enhance remote sensing object detection. By integrating multi-scale convolutions, adaptive rotated convolutions, and parallel spatial channel attention mechanisms, our model effectively addresses the challenges posed by the diverse scales and orientations of objects in remote sensing images.
+
+The experiment results on the DOTA-v1.0 and HRSC2016 datasets demonstrate the effectiveness of the SCAF-Module. Specifically, the module achieved impressive mean Average Precision (mAP) scores of 80.94% and 98.23% on the DOTA-v1.0 and HRSC2016 datasets, respectively. These results underscore the adaptability and robustness of our approach in handling various object detection scenarios in remote sensing imagery. Furthermore, the ablation studies validate the individual contributions of the spatial and channel attention mechanisms, as well as the impact of rotated convolutions on improving detection accuracy. The comparative experiments show that the SCAF-Module outperforms several advanced models, including the baseline Oriented R-CNN, highlighting its superior performance.
+
+Overall, the SCAF-Module offers a significant advancement in remote sensing object detection by providing a more comprehensive and adaptable framework. Future work will focus on further optimizing the module and exploring its application in other challenging remote sensing tasks.
+
+§ ACKNOWLEDGMENT
+
+This work was supported by the National Natural Science Foundation of China under Grant No. 61702340.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/aBxc4ADTyx/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/aBxc4ADTyx/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..8991e10965d9263b66dcb9395be244045214c140
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/aBxc4ADTyx/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,205 @@
+# Design and Implementation of Telemedicine System Using Light Fidelity and PIC16F877A Microcontroller
+
+M.R.Ezilarasan
+
+Department of Electronics and
+
+Communication Engineering
+
+Vel Tech Rangarajan Dr. Sagunthala
+
+$R\& D$ Institute of Science and Technology
+
+Chennai, India
+
+drezilarasan@veltech.edu.in
+
+Man-Fai Leung
+
+School of computing and Information
+
+Science, Faculty of science and
+
+Engineering
+
+Anglia Ruskin University, Cambridge,
+
+United kingdom
+
+Man-fai.leung@aru.ac.uk
+
+Xiangguang Dai*
+
+Chongqing Engineering Research
+
+Center of Internet of Things and
+
+Intelligent Control Technology,
+
+Chongqing Three Gorges University,
+
+Chongqing, China,
+
+daiziangguang@163.com
+
+${Abstract}$ — Medical body area networks currently use wireless communication technology to give patients and carers more freedom and convenience and radio frequency (RF) is the existing medium in healthcare applications. In this the possibility of electromagnetic waves interfering with precision medical equipment still exists, though. This study uses the newly developed wireless visible light communication (VLC) technology to provide a novel design and implementation of a medical healthcare information system. VLC is also referred as light fidelity (Li-Fi). Visible light-emitting diodes (LEDs), which are expected to overtake traditional incandescent and fluorescent lights as the dominant lighting source in the near future owing to their energy efficiency, can be used with VLC. In this research the patient's health is monitored using sensors, and an LCD will display the results. An LED lightbulb will communicate the data in the form of light. The photo-detector at the receiving end gathers the transmitted data, shows the output on an LCD, and informs the carers by beeping every 15 seconds for respiration and 30 seconds for heartbeat. These lighting fixtures can function as wireless data transmission devices in addition to sources of illumination by utilising the fast switching power of LEDs. Hospital regions with restricted radio frequencies can benefit from data services and monitoring provided by the prototype VLC-based medical healthcare system.
+
+Keywords- VLC, Li-Fi, Healthcare, Optical wireless communication, LED, Photodetector.
+
+## I. INTRODUCTION
+
+Over the last ten years, optical wireless communication (OWC) has drawn a great deal of attention. OWC is viewed as an advantageous and complementary communication method to traditional radio frequency (RF) technology [1], which operates within a licensed and regulated electromagnetic spectrum band between ${30}\mathrm{{kHz}}$ and 300 GHz. Electromagnetic spectrum range is shown in below figure 1. Electromagnetic spectrum has different frequency ranges and for applications. This spectrum, commonly used for various wireless communication services such as mobile phones, Wi-Fi, and Bluetooth, based on their frequency ranges and it is experiencing increasing challenges due to the exponential growth in wireless data traffic and the proliferation of sophisticated applications. The surge in demand for wireless data has led to spectral congestion, where the available bandwidth is insufficient to support the volume of data being transmitted. This congestion results in slower data rates, increased latency, and degraded overall performance of wireless networks. Traditional RF technologies, while effective, are struggling to keep up with the demand, leading to a so-called "spectrum crisis." This crisis is characterized by the saturation of the RF spectrum, making it difficult to achieve high maximum data rates and maintain reliable communication.
+
+
+
+In contrast, OWC utilizes the optical spectrum, which is much broader than the RF spectrum and less congested. This allows OWC to offer higher data rates, lower latency, and improved performance in environments where RF communication may be less effective. For instance, OWC can be particularly advantageous in scenarios where RF interference is a problem or where high-density data transmission is required, such as in urban areas, office buildings, and industrial settings. Moreover, OWC can operate in unlicensed bands of the optical spectrum, providing greater flexibility and cost-effectiveness for deployment. It can be used in various applications, including indoor communication using visible light (often referred to as Li-Fi), outdoor communication for point-to-point links, and underwater communication where RF signals are highly attenuated.
+
+Today, there is significance focus on researching new wireless communication alternatives that can provide massive connectivity, diverse data rates, low latency, high capacity, efficiency, and enhanced security [2]. In this context, VLC has emerged as promising wireless communication technique that could address key challenges within the wireless communications infrastructure [3]. VLC uses light that is visible as the transmission medium, leveraging LED's to transmit data. This technique offers several advantages over traditional RF-based systems. For instance, VLC has many advantages like high spped transmission, high data rate, security [4].
+
+One of the most notable advantages of VLC is its potential to provide an alternative to the heavily congested RF spectrum, offering up to 10,000 times more capacity. Additionally, the VLC spectrum is unregulated and unlicensed, presenting a vast and untapped resource for data transmission. This makes VLC is a user friendly bandwidth solution that can help with the problem RF spectrum shortage. Moreover, VLC can be used in conjunction with other essential communication systems and devices without causing interference with the electromagnetic fields generated by RF devices. This characteristic is particularly valuable in sensitive environments such as airplanes and hospitals, where RF interference can pose significant risks. There are ongoing studies exploring the transmission of medical data, such as photo plethysmography, electrocardiography, and body temperature using VLC applications $\left\lbrack {5,6}\right\rbrack$ .
+
+Another key advantage of VLC is its inherent security featureDespite RF signals, visible light is suitable for highly secure connections where wireless data transfer is meant to stay within range of an access point because it cannot pass through walls and does not expand uncontrollably. This characteristic makes VLC an ideal solution for environments where data security is paramount. For example, in corporate offices and government facilities, where sensitive information is frequently transmitted, VLC can ensure that data remains confined within the physical boundaries of a room or building. This physical limitation drastically reduces the risk of eavesdropping and unauthorized access compared to traditional RF-based systems, where signals can be intercepted from a distance. In residential settings, VLC can offer secure communication for smart home devices, preventing potential hackers from accessing personal data transmitted over wireless networks.
+
+Furthermore, VLC can be seamlessly integrated into existing lighting infrastructure, offering dual functionality in the same hardware. This dual-purpose capability means that buildings can have efficient lighting and secure data transmission without the need for additional installations, reducing costs and complexity. For instance, LED lights in a smart home could provide both illumination and secure network connectivity to various IoT devices, such as security cameras, smart locks, and home automation systems.
+
+For all these reasons, VLC can be applied to most IoT-based smart systems [7]. In the context of smart cities, VLC can be used for vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication, enhancing traffic management and safety. Intelligent transportation systems can leverage VLC to provide real-time data exchange between traffic lights and autonomous vehicles, reducing the risk of accidents and improving traffic flow. Additionally, VLC can be employed in smart grids for secure and efficient communication between various components of the electrical grid, facilitating better energy management and reducing the risk of cyberattacks on critical infrastructure. This research paper is organized as follows: Section 2 reviews the existing literature and previous studies relevant to our research, providing a comprehensive background and context while highlighting gaps and opportunities that our study aims to address. Section 3 delves into the concept of VLC, explaining its fundamental principles, advantages, applications, current state of development, and technical challenges. Section 4 presents our proposed methodology and approach, detailing the experimental setup, data collection methods, and analysis procedures, and outlining the innovative aspects of our work. Finally, Section 5 summarizes the key findings, discusses the implications and contributions to the field, and suggests potential areas.
+
+## II. RELATED WORKS
+
+Few more research in recent years, a revolutionary approach to wireless communication has emerged, known as the Internet of LED. This paradigm shift integrates the Internet of Things (IoT) with VLC using LED technology, opening up a realm of possibilities across various industries and applications. One of the most intriguing applications of this technology is seen in indoor navigation and art gallery monitoring is given in [8].In this a prestigious museum adorned with an array of LED lights embedded in the ceiling. These LEDs serve a dual purpose providing ambient illumination and acting as data transmitters. Through sophisticated algorithms and protocols, these LED arrays communicate with users' mobile devices, offering precise positioning information about products, exhibits, or artworks. This not only enhances the overall experience for visitors but also streamlines operations and enhances security in such environments.
+
+The automotive sector has also embraced the Internet of LED with enthusiasm. Modern vehicles are equipped with advanced LED headlights and taillights, which are not just efficient in lighting up the road but also serve as integral components in vehicular VLC systems [9]. These systems utilize the rapid flickering capabilities of LEDs to transmit data, enabling real-time communication for collision prevention systems and enhancing overall road safety. While traditional wireless technologies heavily rely on RF systems, such as Bluetooth, Zig bee, WLAN, and WPAN, there's a growing realization of the limitations and potential risks associated with RF in certain environments, especially healthcare settings.
+
+Concerns about electromagnetic interference and its impact on sensitive medical equipment and patient safety have spurred interest in VLC using energy-efficient LEDs as a viable alternative $\left\lbrack {{10},{11}}\right\rbrack$ . LEDs offer a multitude of advantages over conventional light sources. Their long operating lifetimes, minimal power consumption, and exceptional reliability make them ideal candidates for applications requiring continuous and efficient data transmission [12].
+
+Researchers are actively exploring how VLC using LEDs can not only address concerns about RF-related interference but also revolutionize wireless communication in critical environments like hospitals and clinics. The ongoing research and development in the Internet of LED are poised to reshape the wireless communication landscape. From enhancing indoor navigation experiences to improving road safety and revolutionizing healthcare communication, the fusion of IoT with VLC using LED technology promises efficient, secure, and reliable connectivity across diverse sectors. As this technology continues to evolve, we can expect even more innovative applications and transformative impacts on how we communicate and interact in the digital age.
+
+## III. VISIBLE LIGHT COMMUNICATION
+
+Based on a numerical estimate of optical transmission needs, [13] University in Japan was the first to propose VLC, which uses white LEDs to transport data. Evolution of light is shown in figure 2
+
+
+
+Figure 2 Evolution of Light [14]
+
+Since then, a great deal of research has been done on the use of commercial white LEDs for short-range indoor communications. Owing to the characteristic of short-range communication restricted by the light beam's range, [15] suggested that VLC is more suited for location-based service applications, including giving consumers access to location-specific information. Additionally, new developments for VLC applications are always being made.
+
+In [16] suggested a system of information that would allow medical facilities to use the hospital illumination network to offer private or public data services. Nevertheless, the study's primary focus was on the information system's architectural concepts; the actual design and implementation of medical data transfer via VLC has not yet been documented. In this work, we provide a safe and readily available substitute for RF wireless technology: a wireless VLC-based medical healthcare information system. VLC is a great option for wireless access in the medical and healthcare sectors where electromagnetic interference (EMI) and RF pollution are major concerns.
+
+The greatest frequency utilized in RF technology is 10,000 times lower than the wide bandwidth of VLC [17]. Unlike the RF band, which is crowded and has issues with frequency allocation, the VLC band is currently unlicensed. Since communication takes place inside the visible light spectrum, the Federal Communications Commission (FCC) has not established any regulations. Due to the preinstallation of LED devices indoors and their compatibility with inexpensive electronic drive circuits, VLC-based systems also offer cost-effective installation.
+
+Numerous problems with wireless RF networking methods were examined in [18]. Studies were conducted with an eye towards resolving these problems with VLC systems. In addition, a talk about applications, ways to solve current VLC problems, and upcoming advancements was given. In [19] Author has designed and implemented heterogeneous systems via wireless RF and VLC. A hybrid WiFi-VLC network makes up one system, while the other is created by aggregating Wifi and VLC in parallel utilising the Linux operating system's bonding approach. The downlink for the hybrid network is a VLC channel, which is only intended to be utilised in one way.
+
+## A. VISIBLE LIGHT COMMUNICATION IN TELEMEDICINE.
+
+Telemedicine refers to the use of telecommunications to transfer medical information and provide healthcare to patients remotely. The goal is to provide evidence-based medical treatment to anybody, anywhere, at any time. Research investigations and real-world deployments are increasingly using wireless technologies for telemedicine. It enhances the quality of care by facilitating the flexible collection of medical records and by giving users convenience and mobility. Numerous primary uses, including vital sign monitoring, electronic medical record maintenance, orders to carers, and nonmedical services like entertainment, can be realised with the use of wireless technologies.
+
+### 3.1 Proposed system design
+
+Fig. 3 shows the entire system architecture for transferring multiple medical data. The transmitter module, the receiver module, the processing module, and the monitoring system are the four components that make up this system.
+
+
+
+Figure 3. Proposed block diagram.
+
+First, patients' biomedical signals-such as their temperature and heartbeat-are converted and sent via LEDs. In this case, it is assumed that every transmitter and receiver are perfectly synchronised. At the optical receiver, the data from the optical channel are transformed into a voltage signal after passing via a photodiode. The voltage signal was then detected in the monitoring system after demodulating to a digital signal in the processing module. Each step's specifics are listed below.
+
+## B. Transmitter Section:
+
+The transmitter section consists of several key components designed to ensure proper voltage regulation and sensor interfacing. Initially, a step-down transformer is used to convert the standard ${230}\mathrm{\;V}\mathrm{{AC}}$ supply to a lower voltage of $5\mathrm{\;V}$ DC. This step-down transformer is crucial for safely powering the subsequent components. Following the transformer, a bridge rectifier is employed. The bridge rectifier's role is to convert the alternating current (AC) from the transformer into direct current (DC), which is necessary for the operation of electronic components. To further refine the DC output and ensure a stable voltage, a voltage regulator, specifically the LM7805, is used.
+
+The LM7805 maintains a consistent output of $5\mathrm{\;V}$ DC, protecting the circuitry from voltage fluctuations that could potentially cause damage or unreliable operation. Additionally, a filter capacitor with a capacitance of 1000 microfarads $\left( {\mu \mathrm{F}}\right)$ is incorporated. This capacitor smooths out any remaining ripples in the DC output, providing a clean and steady voltage for the transmitter section. Various sensors, including heartbeat, temperature, and sound sensors, are connected to the PIC16F877A microcontroller. The PIC16F877A is a low-power, high-performance microcontroller that features $8\mathrm{{KB}}$ of in-system programmable memory, allowing for flexibility in programming and updating. One of the standout features of the PIC16F877A microcontroller is its inbuilt Universal Asynchronous Transmitter/Receiver (UART). This UART module facilitates serial communication, enabling efficient data transmission between the microcontroller and other components or systems. Figure 4 illustrates the proposed block diagram for the transmitter section of the Healthcare Monitoring System (HMS) utilizing VLC This diagram provides a visual representation of how the components are interconnected and function together to monitor various health parameters.
+
+
+
+Figure 4. Block diagram of transmitter
+
+Each sensor in the system generates its own analog or digital values, depending on the type of sensor. These values are then fed into the PIC16F877A microcontroller. This microcontroller plays a crucial role in analyzing the data received from the different sensors, processing it accordingly, and displaying the processed information on an LCD screen for easy monitoring and interpretation. In addition to handling sensor data, the PIC16F877A microcontroller is interfaced with a Li-Fi transmitter, which in this case is an LED bulb. Once the data from the sensors is processed by the microcontroller, the PIC establishes serial communication with the Li-Fi transmitter. This enables the transmission of data through VLC. The Li-Fi transmitter uses the LED bulb to transmit the data by modulating the light at very high speeds. The switching frequency of the LED must be high enough to avoid any flickering, which is crucial for ensuring the safety and comfort of human eyes. The rapid modulation allows the LED to transmit data without perceptible changes in light intensity, maintaining a consistent illumination while still enabling data communication. Figure 5.illustrates the working model of the transmitter for the HMS utilizing VLC. This figure provides a detailed visual representation of how the system components interact, showcasing the integration of sensors, the PIC microcontroller, and the Li-Fi transmitter in the overall design of the transmitter section. C. Receiver section:
+
+
+
+Figure 5. Working model of transmitter for HMS using VLC
+
+This section is also known as the monitoring section because the patient's results are monitored continuously through an LCD display. The monitoring section consists of several key components, including a Li-Fi receiver (photo detector), an Arduino UNO328P, an LCD, and a buzzer. A photodiode is used as the Li-Fi receiver. The photodiode functions as a light-to-electricity converter, detecting the light signals transmitted by the Li-Fi transmitter and converting them into corresponding electrical signals. However, the initial electrical signal generated by the photodiode tends to be weak and noisy. to address these issues, the signal undergoes processing through several stages. First, it passes through signal processing and amplification units to strengthen the signal and reduce noise. Following amplification, an envelope detector is used to demodulate the signal. The envelope detector extracts the original data signal from the modulated carrier wave. Next, a low-pass filter is employed to remove high-frequency noise from the signal, ensuring that the resulting signal is clean and suitable for further processing. Once the signal has been filtered, it is fed into a voltage comparator. The voltage comparator transforms the analog signal into a digital format, making it compatible with digital processing systems. The digital signal is then passed to the Arduino UNO328P for further processing. The Arduino microcontroller analyzes the incoming data and displays the relevant information on the LCD screen, providing real-time monitoring of the patient's health parameters. Additionally, the system can trigger a buzzer to alert healthcare personnel in case of critical readings or emergencies. Figure 6 illustrates the proposed block diagram for the receiver section of the HMS utilizing VLC. This diagram visually represents the flow of data from the Li-Fi receiver to the Arduino, highlighting the key components involved in signal processing and data presentation.
+
+
+
+Figure 6 block diagram of receiver
+
+For the system to function effectively, both the transmitter and receiver must be in a line of sight (LOS) position. This requirement ensures that the light signals can be transmitted and received without any obstructions. The information received by the receiver can be depicted in digital form, allowing for detailed analysis of the patient's health by displaying the data on an LCD screen. Additionally, the buzzer is connected to provide alertness to caregivers about the patient's health condition, with alerts being issued every 15 seconds for respiration values and every 30 seconds for heartbeat values.
+
+Figure 7 shows the working model of the receiver section, detailing the flow and processing of data from the Li-Fi receiver to the LCD display and the buzzer.
+
+
+
+Figure 7 working model of receiver
+
+Figure 8 shows the Transmitter and receiver section which contains LED as transmitting medium and photo detector as receiving medium. The sensors connected with the transmitting provide the details of health condition and those are transmitted through the light. This can be applied all medical fields which can be used without any interference.
+
+
+
+Figure 8 Hardware setup of HMS using VLC
+
+TABLE 1. COMPARISON BETWEEN EXISTING RADIO FREQUENCY AND VISIBLE LIGHT COMMUNICATION
+
+| Indexes | Values | Indexes | Values |
| ${X}_{\dot{u}}$ | $- {0.72} \times {10}^{6}$ | ${X}_{\dot{u}}$ | ${5.0242} \times {10}^{4}$ |
| ${Y}_{v}$ | $- {3.6921} \times {10}^{6}$ | ${Y}_{v}$ | ${2.7229} \times {10}^{6}$ |
| ${Y}_{\dot{r}}$ | $- {1.0234} \times {10}^{6}$ | ${Y}_{r}$ | $- {4.3933} \times {10}^{6}$ |
| ${I}_{z} - {N}_{\dot{r}}$ | ${3.7454} \times {10}^{9}$ | ${Y}_{\left| v\right| v}$ | ${1.7860} \times {10}^{4}$ |
| ${X}_{\left| u\right| u}$ | ${1.0179} \times {10}^{3}$ | ${Y}_{\left| v\right| r}$ | $- {3.0068} \times {10}^{5}$ |
| ${N}_{v}$ | $- {4.3821} \times {10}^{6}$ | ${N}_{r}$ | ${4.1894} \times {10}^{6}$ |
| ${N}_{\left| v\right| v}$ | $- {2.4684} \times {10}^{5}$ | ${N}_{\left| v\right| r}$ | ${6.5759} \times {10}^{6}$ |
+
+16 6 8 12 14 $x\left( \mathrm{\;m}\right)$ The proposed scheme 14 OSG in [17] Robust contol in [18] 12 10 8 $y\left( \mathrm{\;m}\right)$ 6 4 2 0 -2 -2 0 2 4
+
+Fig. 1. Trajectory of the ship in ${xy}$ -plane.
+
+In this simulation, the desired attitude is set to ${\eta }_{d} =$ $\left\lbrack \begin{array}{lll} 0\mathrm{m}, & 0\mathrm{m}, & 0\mathrm{{deg}} \end{array}\right\rbrack$ . The initial states are set to $\eta \left( 0\right) =$ $\left\lbrack {{12}\mathrm{\;m},{14}\mathrm{\;m},{10}\mathrm{{deg}}}\right\rbrack , v\left( 0\right) = \left\lbrack {0\mathrm{\;m}/\mathrm{s},{14}\mathrm{\;m}/\mathrm{s},{10}\mathrm{{deg}}/\mathrm{s}}\right\rbrack .$ The concrete parameters values setting follows (45). Besides, the RBF-NNs for ${F}_{1}$ and ${F}_{1}$ consisted of 25 nodes with centers spaced in $\left\lbrack {-{2.5}\mathrm{\;m}/\mathrm{s},{2.5}\mathrm{\;m}/\mathrm{s}}\right\rbrack$ for $x, y, u$ and $r$ , $\left\lbrack {-{0.16}\mathrm{\;m}/\mathrm{s},{0.16}\mathrm{\;m}/\mathrm{s}}\right\rbrack$ for $\psi$ and $r$ , respectively. For the comparison algorithms, corresponding parameters refers to [19] and [20].
+
+Fig. 1 exhibits simulation results under the proposed algorithm, OSG and robust control making the ship stay at the desired attitude in the ${xy}$ -plant. It is clear that the proposed algorithm provides a more satisfactory trajectory accuracy compared to the algorithms for comparison. Fig. 2 illustrates that the ship attitude $x, y$ and $\psi$ are stabilized to the desired attitude near the prescribed-time ${T}_{j\upsilon }$ . The proposed scheme achieves faster stabilization compared to the schemes for comparison. The velocities of surge, sway and yaw are showed in Fig. 3. The proposed scheme has a improved convergence performance. Fig. 4 illustrates the fluctuation of the values of the three input signals over time. It is apparent that prior to system stabilization, the proposed scheme exhibits superior convergence performance of ${\tau }_{r}$ compared to other schemes. Once the system has stabilized, the values of ${\tau }_{u}$ and ${\tau }_{v}$ in the proposed scheme converge more rapidly towards zero, further outperforming the other schemes in convergence efficiency. Finally, it can be seen that the constructed new error is successfully confined within the boundaries and converges stably to 0 at the moment of settling time ${T}_{fu},{T}_{fv},{T}_{fr}$ as shown in Fig. 5. Additionally, Fig. 6 and Fig. 7 illustrate the fitting performance between the estimated and true values of the adaptive parameters ${\theta }_{1}$ and ${\theta }_{2}$ , respectively, representing the approximation capability of the RBF-NNs for the system uncertainties terms. It can be observed that, within the permissible margin of error, the RBF-NNs successfully approximate uncertainties terms described by (13) and (17).
+
+10 The proposed scheme OSG in [17] Robust contol in [18] 150 200 250 300 80 150 300 200 Time (s) $x\left( \mathrm{m}\right)$ 50 100 10 $y\;\left( \mathrm{m}\right)$ 100 $\psi$ (deg) 0 100
+
+Fig. 2. Ship’s actual position(x, y)and heading $\psi$ .
+
+0 The proposed scheme OSG in [17] Robust contol in [18] 150 200 250 300 150 200 250 300 150 200 250 300 Time (s) $u\left( {\mathrm{m}/\mathrm{s}}\right)$ -0.2 50 100 $v\;\left( {\mathrm{m}/\mathrm{s}}\right)$ -0.2 -0.4 50 100 0.5 $r\left( {\mathrm{{deg}}/\mathrm{s}}\right)$ 0 -0.5 50 100
+
+Fig. 3. Ship’s surge velocity $u$ , sway velocity $v$ and yaw rate $r$ .
+
+$\times {10}^{7}$ The proposed scheme OSG in [17] Robust contol in [18] 150 300 119.4 119.8 $\times {10}^{4}$ 108.1 108.2 108.3 150 Time (s) Let(N) -5 50 100 $\times {10}^{8}$ p (N) 0 100 $\times {10}^{6}$ ${\tau }_{r}\left( {\mathrm{N} \cdot \mathrm{m}}\right)$ -10 -20
+
+Fig. 4. Ship’s surge force ${\tau }_{u}$ , sway force ${\tau }_{v}$ and yaw force ${\tau }_{r}$ .
+
+${\xi }_{1}\left( x\right) \left( \mathrm{m}\right)$ -0.2 150 200 250 300 _=80s T ${}_{fu}$ =90s 200 250 300 Time (s) 100 ${\xi }_{1}\left( y\right) \left( \mathrm{m}\right)$ 0 -0.5 0.2 ${\xi }_{1}\left( \psi \right) \left( \mathrm{m}\right)$ -0.2 50 100
+
+Fig. 5. The new error ${\xi }_{1}$ for the simulation with the proposed scheme.
+
+In summary, the NNs-based prescribed-time control scheme proposed in this paper demonstrates superior performance and robustness compared to the schemes for comparison. By introducing FTFBs, FTTFBs and constructing new error functions, the control laws are made more concise. Finally, the proposed scheme is validated through simulations to demonstrate its effectiveness on DP ships.
+
+30 ${\theta }_{1}$ ${\widehat{\theta }}_{1}$ 150 200 250 300 Time (s) 25 20 ${\theta }_{1}/{\widehat{\theta }}_{1}$ 15 10 5 0 50 100
+
+Fig. 6. The estimation performance of ${\theta }_{1}$ .
+
+6 $\times {10}^{4}$ ${\theta }_{2}$ ${\widehat{\theta }}_{2}$ 150 200 250 300 Time (s) $\times {10}^{4}$ 2.84 5 2.835 2.83 2.825 4 ${\theta }_{2}/\widehat{{\theta }_{2}}$ 4800 2 4700 4600 4500 1 4400 0 0 50 100
+
+Fig. 7. The estimation performance of ${\theta }_{2}$ .
+
+## VI. CONCLUTION
+
+In this paper, a novel NNs-based control scheme is proposed for the ship DP system under model uncertain and unknown environmental disturbances, making the new dynamic errors converging within fixed boundaries. The prescribed-time performance of the algorithm is validated through by a simulation example and two comparative simulations with satisfactory results. Consequently, the prescribed-time control algorithm proposed in this paper can be applied to ships performing DP tasks, enabling the ship's dynamic system to achieve more precise time-based prescribed performance.
+
+Given the presence of multiple dynamic actuators in engineering practices related to marine equipment, future research on the proposed algorithm could focus on the issue of actuator control allocation. In addition, the integration of event-triggered control, fault-tolerant control, and blind zone constraints could further enhance the development of this control algorithm toward more advanced and precise control techniques.
+
+## REFERENCES
+
+[1] S $\phi$ rensen A J. Propulsion and motion control of ships and ocean structures[J]. Marine Technology Center, Department of Marine Technology. Lecture notes, 2011.
+
+[2] Du J, Hu X, Krstić M, et al. Dynamic positioning of ships with unknown parameters and disturbances[J]. Control Engineering Practice, 2018, 76: 22-30.
+
+[3] Sørensen A J. A survey of dynamic positioning control systems[J]. Annual reviews in control, 2011, 35(1): 123-136.
+
+[4] Do K D. Global robust and adaptive output feedback dynamic positioning of surface ships[J]. Journal of Marine Science and Application, 2011, 10: 325-332.
+
+[5] Park B S, Kwon J W, Kim H. Neural network-based output feedback control for reference tracking of underactuated surface vessels[J]. Au-tomatica, 2017, 77: 353-359.
+
+[6] Yang Y, Guo C, Du J L. Robust adaptive NN-based output feedback control for a dynamic positioning ship using DSC approach[J]. Science China Information Sciences, 2014, 57: 1-13.
+
+[7] Skulstad R, Li G, Zhang H, et al. A neural network approach to control allocation of ships for dynamic positioning[J]. IFAC-PapersOnLine, 2018, 51(29): 128-133.
+
+[8] Liang K, Lin X, Chen Y, et al. Robust adaptive neural networks control for dynamic positioning of ships with unknown saturation and time-delay[J]. Applied Ocean Research, 2021, 110: 102609.
+
+[9] Dai S L, He S, Lin H. Transverse function control with prescribed performance guarantees for underactuated marine surface vehicles[J]. International Journal of Robust and Nonlinear Control, 2019, 29(5): 1577-1596.
+
+[10] Dai S L, He S, Wang M, et al. Adaptive neural control of underactu-ated surface vessels with prescribed performance guarantees[J]. IEEE transactions on neural networks and learning systems, 2018, 30(12): 3686-3698.
+
+[11] Li J, Du J, Hu X. Robust adaptive prescribed performance control for dynamic positioning of ships under unknown disturbances and input constraints[J]. Ocean Engineering, 2020, 206: 107254.
+
+[12] Gong C, Su Y, Zhang D. Variable gain prescribed performance control for dynamic positioning of ships with positioning error constraints[J]. Journal of Marine Science and Engineering, 2022, 10(1): 74.
+
+[13] Li H, Lin X. Robust fault-tolerant control for dynamic positioning of ships with prescribed performance[J]. Ocean Engineering, 2024, 298: 117314.
+
+[14] Fossen T I. Marine control systems. Marine cybernetics[J]. Trondhiem, Norway, 2002.
+
+[15] Li Z, Chen X, Ding S, et al. TCP/AWM network congestion algorithm with funnel control and arbitrary setting time[J]. Applied Mathematics and Computation, 2020, 385: 125410.
+
+[16] Wu J, Chen W, Li J. Global finite-time adaptive stabilization for nonlinear systems with multiple unknown control directions[J]. Automatica, 2016, 69: 298-307.
+
+[17] Xie H, Jing Y, Dimirovski G M, et al. Adaptive fuzzy prescribed time tracking control for nonlinear systems with input saturation[J]. ISA transactions, 2023, 143: 370-384.
+
+[18] X.-K. Zhang. Ship Motion Concise Robust Control. Beijing, China: Sci. Press, 2012.
+
+[19] Zhang G, Cai Y, Zhang W. Robust neural control for dynamic positioning ships with the optimum-seeking guidance[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2016, 47(7): 1500-1509.
+
+[20] Du J, Yang Y, Wang D, et al. A robust adaptive neural networks controller for maritime dynamic positioning system[J]. Neurocomputing, 2013, 110: 128-136.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bKg0I5ZIXm/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bKg0I5ZIXm/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..2257aa83dcd56c70deb9d6a4e5161592e2645553
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bKg0I5ZIXm/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,594 @@
+§ ADAPTIVE PRESCRIBED-TIME CONTROL OF DYNAMIC POSITIONING SHIPS BASED ON NEURAL NETWORKS
+
+${1}^{\text{ st }}$ Yongsheng Dou
+
+College of Navigation
+
+Dalian Maritime University
+
+Dalian, China
+
+dysheng@dlmu.edu.cn
+
+${2}^{\text{ nd }}$ Chenfeng Huang
+
+College of Navigation Dalian Maritime University Dalian, China
+
+chenfengh@dlmu.edu.cn
+
+${3}^{\text{ rd }}\mathrm{{Yi}}$ Zhao
+
+College of Navigation
+
+Dalian Maritime University
+
+Dalian, China
+
+yi_zhao@dlmu.edu.cn
+
+${Abstract}$ -In this paper, a novel controller with prescribed-time performance is designed for dynamic positioning (DP) system of ships with model uncertainty and unknown time-varying disturbances. Initially, an error transformation function with zero initial value is introduced by constructing fixed-time funnel boundaries (FTFBs) and a fixed-time tracking performance function (FTTPF). The proposed controller ensures stable convergence of the new error, maintaining it within fixed upper and lower boundaries. When the prescribed time is reached, the system state will achieve prescribed-time (PT) stability. Secondly, by deploying radial basis function neural networks (RBF-NNs) and dynamic surface control (DSC), adaptive controller with simple forms are rationally applied to Backstepping technology, and the uncertain terms of the system are approximated online, the singularity and complexity explosion problems of the ship control system are also addressed. In addition to that, the stability analysis results of the system prove that all errors of the closed-loop system are semi-global uniformly ultimately bounded (SGUUB) stable. Finally, the simulation results on a DP ship confirm the superiority of the proposed scheme.
+
+Index Terms-Dynamically positioned ships, prescribed-time control, fixed-time funnel boundaries, Backstepping
+
+§ I. INTRODUCTION
+
+In marine engineering, dynamic positioning (DP) systems are critical for maintaining precise ship positions and orientations in the marine environment [1]. These systems enable ships to hold exact positions or follow predetermined paths without anchoring by utilizing thrusters and power systems. This capability is crucial for marine engineering operations, including oil and gas drilling, underwater pipeline installation, and cable laying. As marine operations grow more complex, traditional DP methods encounter significant challenges, such as environmental disturbances, system parameter uncertainties, and operational efficiency concerns [2].
+
+Consequently, researchers are increasingly adopting advanced control strategies to improve the performance and adaptability of DP systems. However, the highly nonlinear terms of ship dynamics and the continuously changing marine environment often cause traditional control methods to struggle under extreme conditions. Furthermore, most existing DP control strategies depend on extended control processes to achieve stability [3], which may not always be the optimal solution. Therefore, developing a control strategy that can respond quickly and complete tasks within a prescribed time is particularly crucial.
+
+With the rapid advancement of control technologies and methods in recent years, DP systems have been more broader application in maritime operations and offshore exploration for ships and drilling platforms. For instance, in the presence of unknown ship parameters, [4] developed a robust adaptive observer for DP systems, capable of estimating ship velocities and unknown parameters under external disturbances. An adaptive observer based on neural networks (NNs) was developed to estimate the velocity data of the unmanned surface vessel (USV) in [5], even though both the system parameters and nonlinearities of the USV were presumed to be uncertain. NNs approximation techniques are used to compensate for uncertainty and unknown external disturbances, removing the prerequisite for a priori knowledge of ship parameters and external disturbances. Meanwhile, MLP technology is employed to address the computational explosion problem [6] [8]. However, in [7], static NNs are used for control force and moment allocation of an over-actuated ship by measuring the thruster force and commands and gathering data for practice of the NNs.
+
+Due to the time-varying boundary functions can achieve prescribed performance of dynamic system on both transient and steady-state phases, [10] proposed a novel boundary function control approach and introduced an error transformation function, showing training stability of the closed-loop systems with prescribed transient and steady-state functions. In the field of marine engineering operations, [11] proposed a robust adaptive prescribed performance control (RAPPC) law by constructing a concise error mapping function and achieved the DP prescribed performance control. To address positioning error constraints, input saturation and unknown external disturbances, [12] proposed a variable gain prescribed performance control law and constructed the error mapping functions to integrate the prescribed performance boundary to the controller design. Soon after that, a robust fault-tolerant control allocation scheme is developed to distribute again the forces among faulty actuators in [13]. Its performance function is united with an auxiliary in-between control technique to create a high-level controller.
+
+Inspired by the above research work, The contributions of this paper are as follows:
+
+1) Building upon the research foundation of reference [11], this article proposes an adaptive prescribed-time control scheme for DP system of ship with model uncertain and unknown environment disturbances. Unlike the reliance on initial conditions discussed in reference [10], the construction of the fixed-time tracking performance function (FTTPF) ensures that the controller's prescribed performance is no longer dependent on initial conditions. Furthermore, the new dynamic errors will deviate from an initial value of 0, remaining consistently confined within the set fixed-time tracking performance function (FTFBs).
+
+2) Based on NNs, unknown functions of the new dynamic error derivative terms and unknown model parameters of the ship are approximated online. In addition, the adaptive parameters based on weight allocation are reduced to two to compensate for the unknown gain function. The dynamic surface control (DSC) filtering technique is introduced to address the complexity explosion problem caused by the differentiation of the virtual controller, thereby reducing the computational burden. Finally, two comparative simulations of a DP ship is executed to demonstrate the effectiveness of the proposed algorithm.
+
+§ II. MATHEMATICAL MODEL OF DYNAMICALLY POSITIONED SHIPS AND PROBLEM FORMULATION
+
+In the design of DP systems, a ship is considered a multi-input multi-output (MIMO) control system that includes dynamics influenced by mass, damping, stiffness, and external disturbances. On the basis of the seakeeping and maneuvering theory, the following three DOF nonlinear mathematical model is used to describe the dynamic behavior of the ship in the presence of disturbances [14]:
+
+$$
+\dot{\eta } = J\left( \psi \right) v \tag{1}
+$$
+
+$$
+M\dot{v} + D\left( v\right) v = \tau + {\tau }_{d} \tag{2}
+$$
+
+where $\eta = {\left\lbrack x,y,\psi \right\rbrack }^{\top } \in {\mathcal{R}}^{3}$ represent the attitude vector including the surge position $x$ , the sway position $\mathrm{y}$ and the heading $\psi \in \left\lbrack {0,{2\pi }}\right\rbrack$ in the earth-fixed coordinate system. $v = {\left\lbrack u,v,r\right\rbrack }^{\top } \in {\mathcal{R}}^{3}$ denotes the velocity vector of the ship in the body-fixed coordinate system, which composed of the surge velocity $u$ , sway velocity $v$ and yaw velocity $r$ , respectively. $J\left( \psi \right)$ is the velocity transformation matrix as
+
+follow:
+
+$$
+J\left( \psi \right) = \left\lbrack \begin{matrix} \cos \left( \psi \right) & - \sin \left( \psi \right) & 0 \\ \sin \left( \psi \right) & \cos \left( \psi \right) & 0 \\ 0 & 0 & 1 \end{matrix}\right\rbrack \tag{3}
+$$
+
+with ${J}^{-1}\left( \psi \right) = {J}^{\top }\left( \psi \right)$ and $\parallel J\left( \psi \right) \parallel = 1$ . Equation (4) gives the specific expression of the positive definite symmetric inertia matrix $M \in {\mathcal{R}}^{3 \times 3}$ , which including additional mass. Equation (5) gives the specific expression of the nonlinear hydrodynamic function $D\left( v\right) v$ .
+
+$$
+M = \left\lbrack \begin{matrix} m - {X}_{\dot{u}} & 0 & 0 \\ 0 & m - {Y}_{\dot{v}} & m{x}_{G} - {X}_{\dot{r}} \\ 0 & m{x}_{G} - {X}_{\dot{r}} & {I}_{z} - {N}_{\dot{r}} \end{matrix}\right\rbrack \tag{4}
+$$
+
+$$
+D\left( v\right) v = \left\lbrack \begin{array}{l} {D}_{1} \\ {D}_{1} \\ {D}_{3} \end{array}\right\rbrack
+$$
+
+$$
+{D}_{1} = - {X}_{u}u - {X}_{\left| u\right| u}\left| u\right| u + {Y}_{\dot{v}}v\left| r\right| + {Y}_{\dot{r}}{rr} \tag{5}
+$$
+
+$$
+{D}_{2} = - {X}_{\dot{u}}{ur} - {Y}_{v}v - {Y}_{r}r - {X}_{\left| v\right| v}\left| v\right| v - {X}_{\left| v\right| r}\left| v\right| r
+$$
+
+$$
+{D}_{3} = \left( {{X}_{\dot{u}} - {Y}_{\dot{v}}}\right) {uv} - {Y}_{\dot{r}}{ur} - {N}_{v}v - {N}_{r}r - {N}_{\left| v\right| v}\left| v\right| v
+$$
+
+$$
+- {N}_{\left| v\right| r}\left| v\right| r
+$$
+
+where $m$ are ship’s mass, ${I}_{z}$ are moment of inertia and ${X}_{u}$ , ${X}_{\left| u\right| u},{Y}_{\dot{v}}$ , etc., are every hydrodynamic force derivatives. It is obvious from the expression in (5) that the nonlinear damping force composed of linear and quadratic terms. In the controller design of this paper, $D\left( v\right) v$ is an uncertain term in which the structure and parameters are unknown and is approximated online using NNs in later section.
+
+$\tau = {\left\lbrack {\tau }_{u},{\tau }_{v},{\tau }_{r}\right\rbrack }^{\top } \in {\mathcal{R}}^{3}$ denotes the control inputs, which are the forces and moments generated by the equipped actuators on the ship consisting of the shaft thruster, the tunnel thruster, and the azimuth thruster. In order to simplify the control inputs, all actuator devices inputs are fused into three degrees of freedom : ${\tau }_{u}$ in surge, ${\tau }_{v}$ in sway and ${\tau }_{r}$ in yaw. ${\tau }_{d} = \left\lbrack {\tau }_{du}\right.$ , ${\left. {\tau }_{dv},{\tau }_{dr}\right\rbrack }^{\top }$ indicates the unknown time-varying environment distraction induced by wind, and waves.
+
+Assumption 1. The environment disturbance ${\tau }_{d\upsilon }$ is bounded in the marine environment, indicating the existence of bounded ${\bar{\tau }}_{d\upsilon } > 0$ for ${\tau }_{d\upsilon }$ . i.e., $\left| {\tau }_{d\upsilon }\right| < {\bar{\tau }}_{d\upsilon }$ .
+
+Remark 1. When modeling ship DP systems, it is often necessary to accurately characterize and predict the effects of environmental disturbances on the ship. In order to simplify the model and to facilitate the design and testing of control algorithms, these environmental disturbances can be approximated and modeled using a sine-cosine function. The frequency, amplitude and phase of the interference can be easily adjusted using the sine-cosine function to simulate different intensities and types of environmental conditions.
+
+In the setting of unknown time-varying disturbances and model uncertainty, the goal of the control is to find a control laws $\tau$ makes the ship’s position(x, y)and heading $\psi$ successfully reach the desired position ${\eta }_{d}$ within the prescribed time. At the same time, the constructed zero-initial-value error function also converges within the set boundaries within the settling time and arbitrarily small errors, and all the errors are bounded all the time.
+
+§ III. FUNNEL CONTROL AND FUNNEL VARIABLE
+
+In the context of advanced control strategies for DP systems, particularly those addressing strict timing requirements, the concepts of FTFBs and FTTPF are integral. These are designed to ensure that the control system adheres to performance metrics strictly within a settling interval, regardless of initial conditions. In this section, the definitions of FTFBs and FTTPF are introduced for the purpose of imposing error bounds on them and constructing new error functions.
+
+§ A.THE DESIGN OF PRESCRIBED-TIME FUNNEL BOUNDARY
+
+Definiion 1. [15] FTFBs define the permissible bounds within which the system's states must remain over time. These boundaries are set to compact over a fixed-time period, ensuring that the system's behavior converges to the desired state within the settling duration. These boundaries are particularly useful in scenarios where rapid and reliable system stabilization is crucial.
+
+Equation (6) is selected as an FTFBs with the following traits: (1) $\Gamma \left( t\right) > 0$ and $\dot{\Gamma }\left( t\right) \leq 0$ ; (2) $\mathop{\lim }\limits_{{t \rightarrow {T}_{j}}}\Gamma \left( t\right) = {\Gamma }_{jT}$ ; (3) $\Gamma \left( t\right) = {\Gamma }_{jT}$ for $\forall t \geq {T}_{j}$ with ${T}_{j}$ being the predefined fixed time after which the boundary ceases to contracting.
+
+$$
+{\Gamma }_{jv} = \left\{ \begin{array}{ll} {\Gamma }_{jv0}\tanh \left( \frac{{\lambda }_{j}t}{t - {T}_{jv}}\right) + {\Gamma }_{jv0} + {\Gamma }_{jvT}, & t \in \left\lbrack {0,{T}_{jv}}\right) \\ {\Gamma }_{jvT}, & t \in \left\lbrack {{T}_{jv},\infty }\right) \end{array}\right. \tag{6}
+$$
+
+where ${\Gamma }_{jv0},{\Gamma }_{jvT}$ and ${\Gamma }_{jvT}$ are the initial and final boundary values, ${\lambda }_{j}$ is the decay rate, $j = 1,2$ and ${T}_{jv}$ is the predefined fixed time after which the boundary ceases to contracting.
+
+Definiion 2. [16] FTTPF is a function designed to evaluate and ensure the system's tracking performance over a fixed time, dictating how the tracking error should decrease over time to meet specific performance criteria by a predefined deadline.
+
+$$
+{\varphi }_{v}\left( t\right) = \left\{ \begin{array}{ll} {e}^{-\frac{{k}_{v}t}{{T}_{fv} - t}}, & t \in \left\lbrack {0,{T}_{fv}}\right) \\ 0, & t \in \left\lbrack {{T}_{fv},\infty }\right) \end{array}\right. \tag{7}
+$$
+
+Equation (7) is concretely constructed as an FTTPF with the following properties : $\left( 1\right) \varphi \left( 0\right) = 1$ ; (2) $\mathop{\lim }\limits_{{t \rightarrow {T}_{fv}}}\varphi \left( t\right) = 0$ and $\varphi \left( t\right) = 0$ for $\forall t \geq {T}_{fv}$ with ${T}_{fv}$ being a prescribed settling time. ${\Gamma }_{jv0},{\Gamma }_{jvT},{\lambda }_{j},{T}_{jv},{T}_{fv}$ and ${k}_{v}$ are positive constant.
+
+§ B. FUNNEL ERROR TRANSFORMATION
+
+In this paper, by embedding FTTPF ${\varphi }_{v}\left( t\right)$ we construct a new error $\chi \left( t\right)$ variable with 0 initial value as in (9).
+
+$$
+{z}_{1} = \eta - {\eta }_{d} \tag{8}
+$$
+
+$$
+\chi \left( t\right) = {z}_{1}\left( t\right) - {z}_{1}\left( 0\right) {\varphi }_{v}\left( t\right) = \eta - {\eta }_{d} - {z}_{1}\left( 0\right) {\varphi }_{v}\left( t\right) \tag{9}
+$$
+
+Then, ${\Gamma }_{j\upsilon },j = 1,2$ , is applied to ensure that the following symmetry performance constraints on $\chi \left( t\right)$ which are satisfied.
+
+$$
+- {\Gamma }_{1v} < \chi \left( t\right) < {\Gamma }_{2v} \tag{10}
+$$
+
+where ${\eta }_{d} = {\left\lbrack \begin{array}{lll} {x}_{d}, & {y}_{d}, & {\psi }_{d} \end{array}\right\rbrack }^{\top }$ represents the desired position of the ship DP system. Besides, to simplify the design of the controller, ${T}_{1v} = {T}_{2v}$ is adopted in this paper. In order to comply with the definition of $\chi \left( t\right)$ and the requirements of (9), $\chi \left( 0\right) = {z}_{1}\left( 0\right) - {z}_{1}\left( 0\right) {\varphi }_{v}\left( 0\right) = 0$ guarantees that the initial state $- {\Gamma }_{1\upsilon }\left( 0\right) < \chi \left( 0\right) < {\Gamma }_{2\upsilon }\left( 0\right) \Leftrightarrow - {\Gamma }_{1\upsilon }\left( 0\right) + {z}_{1}\left( 0\right) < {z}_{1}\left( 0\right) <$ ${\Gamma }_{2v}\left( 0\right) + {z}_{1}\left( 0\right)$ is always satisfied, which implicitly means that ${\Gamma }_{1v}$ and ${\Gamma }_{2v}$ no longer need to be redesigned in order to keep the characteristic that initial value is 0 of the new error.
+
+By introducing the constructed ${\Gamma }_{1v}$ and ${\Gamma }_{2v}$ , the maximum overshoot, settling time, and steady boundaries of $\chi \left( t\right)$ can be determined by $\max \left\{ {{\Gamma }_{1\mathrm{v}0} + {\Gamma }_{1\mathrm{{vT}}},{\Gamma }_{2\mathrm{v}0} + {\Gamma }_{2\mathrm{{vT}}}}\right\} ,{T}_{jv}$ and ${\Gamma }_{jvT}$ , respectively. The changing of ${z}_{1}\left( t\right)$ is required to be preassigned over $\left\lbrack {0,{T}_{fv}}\right)$ due to $- {\Gamma }_{1v}\left( t\right) + {z}_{1}\left( 0\right) {\varphi }_{v}\left( t\right) <$ ${z}_{1}\left( t\right) < {\Gamma }_{2v}\left( t\right) + {z}_{1}\left( 0\right) {\varphi }_{v}\left( t\right)$ for $\forall t \in \left\lbrack {0,{T}_{fv}}\right)$ . From the above analysis, (10) can be reformulated as:
+
+$$
+- {\Gamma }_{1}\left( t\right) < \chi \left( t\right) = {z}_{1}\left( t\right) - {z}_{1}\left( 0\right) {\Phi }_{1} < {\Gamma }_{2}\left( t\right) ,\forall t \geq 0 \tag{11}
+$$
+
+where ${\Gamma }_{1} = {\left\lbrack \begin{array}{lll} {\Gamma }_{1u}, & {\Gamma }_{1v}, & {\Gamma }_{1r} \end{array}\right\rbrack }^{\top },{\Gamma }_{2} = {\left\lbrack \begin{array}{lll} {\Gamma }_{2u}, & {\Gamma }_{2v}, & {\Gamma }_{2r} \end{array}\right\rbrack }^{\top }$ and ${\Phi }_{1} = {\left\lbrack \begin{array}{lll} {\varphi }_{u}, & {\varphi }_{v}, & {\varphi }_{r} \end{array}\right\rbrack }^{\top }$ .
+
+Although the extant FC results can tuned the transient and steady-state responses of ${z}_{1}$ , the corresponding problem is the need to rely on specific initial conditions. To solve this problem, inspiring from [17], we introduce the following variable transformation:
+
+$$
+\vartheta \left( t\right) = \chi \left( t\right) + \mu \left( t\right) \tag{12}
+$$
+
+with
+
+$$
+\mu \left( t\right) = \left( {{\Gamma }_{1}\left( t\right) - {\Gamma }_{2}\left( t\right) }\right) /2,\omega \left( t\right) = \left( {{\Gamma }_{1}\left( t\right) + {\Gamma }_{2}\left( t\right) }\right) /2 \tag{13}
+$$
+
+From (12) and (13), (11) is equivalent to
+
+$$
+- \omega \left( t\right) < \vartheta \left( t\right) < \omega \left( t\right) \tag{14}
+$$
+
+To improve control performance and achieve control objectives, the funnel error transformation as given by equation (15) is applied.
+
+$$
+{\xi }_{1}\left( t\right) = \frac{\vartheta \left( t\right) }{\sqrt{{\omega }^{2}\left( t\right) - {\vartheta }^{2}\left( t\right) }} \tag{15}
+$$
+
+The derivation of (15) yields ${\dot{\xi }}_{1}$
+
+$$
+{\dot{\xi }}_{1}\left( t\right) = {\Phi }_{2}\left( {\dot{\eta } - {\dot{\eta }}_{d} - {z}_{1}\left( 0\right) {\dot{\Phi }}_{1}\left( t\right) + \dot{\mu }\left( t\right) - \vartheta \left( t\right) \dot{\omega }\left( t\right) /\omega \left( t\right) }\right)
+$$
+
+(16)
+
+where ${\Phi }_{2} = {\omega }^{2}\left( t\right) /\sqrt{{\left( {\omega }^{2}\left( t\right) - {\vartheta }^{2}\left( t\right) \right) }^{3}} > 0$ . It should be noted that for complex representations of ${\dot{\xi }}_{1}$ , NNs are employed to approximate the uncertain terms. In the subsequent function formulations, function arguments are omitted to simplify the presentation and improve readability.
+
+§ IV. ADAPTIVE PT FUNNEL CONTROL DESIGN FOR DYNAMIC POSITIONED SHIPS
+
+In this section, adaptive parameters are introduced using NNs for online approximation of the uncertainty terms arising during the controller design process. The Backstepping means is utilized to design the virtual controller ${\alpha }_{v}$ and the control law $\tau$ for the second-order ship motion mathematical model (1) and (2). The DSC technique is applied to address the complexity in deriving ${\alpha }_{v}$ . The controller design procedure consists of two steps for the attitudes and velocity parts. The specific details of the controller design are detailed in IV-A, and the corresponding stability analysis is detailed in IV-B.
+
+§ A. CONTROLLER DESIGN
+
+Step 1: In the ship's DP system, the reference attitude signal ${\eta }_{d}$ is a constant with derivative 0, meaning ${\dot{\eta }}_{d} = 0$ . It is noted that in the derivative form of the boundary transformation error ${\xi }_{1},{\Phi }_{2}\left( {-{z}_{1}\left( 0\right) {\Phi }_{1}\left( t\right) + \dot{\mu }\left( t\right) - \vartheta \left( t\right) \dot{\omega }\left( t\right) /\omega \left( t\right) }\right)$ represents the unknown function vector. It can be obtained as (17) by using RBF-NNs ${F}_{1}\left( \eta \right)$ .
+
+$$
+{F}_{1}\left( \eta \right) = {\Phi }_{2}\left( {-{z}_{1}\left( 0\right) {\Phi }_{1}\left( t\right) + \dot{\mu }\left( t\right) - \vartheta \left( t\right) \dot{\omega }\left( t\right) /\omega \left( t\right) }\right)
+$$
+
+$$
+= {S}_{1}\left( \eta \right) {A}_{1}\eta + {\varepsilon }_{\eta }
+$$
+
+$$
+= \left\lbrack \begin{matrix} {s}_{x}\left( \eta \right) & 0 & 0 \\ 0 & {s}_{y}\left( \eta \right) & 0 \\ 0 & 0 & {s}_{\psi }\left( \eta \right) \end{matrix}\right\rbrack \left\lbrack \begin{array}{l} {A}_{x} \\ {A}_{y} \\ {A}_{\psi } \end{array}\right\rbrack \left\lbrack \begin{array}{l} x \\ y \\ \psi \end{array}\right\rbrack + \left\lbrack \begin{array}{l} {\varepsilon }_{x} \\ {\varepsilon }_{y} \\ {\varepsilon }_{\psi } \end{array}\right\rbrack
+$$
+
+(17)
+
+where ${\varepsilon }_{\eta }$ is corresponding upper bound vector. ${s}_{x}\left( \eta \right) =$ ${s}_{y}\left( \eta \right) = {s}_{\psi }\left( \eta \right)$ due to these RBF functions are with the same input vector $v$ . Let ${\theta }_{1} = {\begin{Vmatrix}{A}_{1}\eta \end{Vmatrix}}^{2}$ , where ${\widehat{\theta }}_{1}$ represents the estimated values of ${\theta }_{1}$ . From the above analysis, the immediate virtual controller ${\alpha }_{v}$ is determined as shown in (18).
+
+$$
+{\alpha }_{v} = - \frac{1}{{\Phi }_{2}J\left( \psi \right) }\left( {{k}_{1}{\xi }_{1} + \frac{{S}_{1}{}^{T}{S}_{1}{\widehat{\theta }}_{1}}{2{a}_{1}{}^{2}}{\xi }_{1} + \frac{1}{4}{\begin{Vmatrix}{\Phi }_{2}\end{Vmatrix}}^{2}{\xi }_{1}}\right) \tag{18}
+$$
+
+where ${k}_{1}$ is a strictly positive diagonal matrix of parameters. the DSC technique, i.e., a first-order low-pass filter (19), is applied here, considering that the derivatives of a are difficult to obtain and complex in form.
+
+$$
+{t}_{v}{\dot{\beta }}_{v} + {\beta }_{v} = {\alpha }_{v},{\beta }_{v}\left( 0\right) = {\alpha }_{v}\left( 0\right) \tag{19}
+$$
+
+${t}_{v}$ is a constant time-related matrix, and the input velocity vector signal ${\alpha }_{v}$ is transformed into the output velocity vector ${\beta }_{v}$ which is the reference vector for the velocity signal in the second step. Defining the error vector ${q}_{v} = {\left\lbrack {q}_{u},{q}_{v},{q}_{r}\right\rbrack }^{\top } =$ ${\alpha }_{v} - {\beta }_{v},{z}_{2} = {\beta }_{v} - v$ , the derivative of ${q}_{v}$ is acquired along with (18) and (19).
+
+$$
+{\dot{q}}_{v} = - {\dot{\beta }}_{v} + {\dot{\alpha }}_{v}
+$$
+
+$$
+= {t}_{v}^{-1}{q}_{v} + {B}_{v}\left( {{z}_{1},{\dot{z}}_{1},\psi ,r,{\widehat{\theta }}_{1},{\dot{\widehat{\theta }}}_{1}}\right) \tag{21}
+$$
+
+where ${B}_{v} = {\left\lbrack {B}_{u}\left( \cdot \right) ,{B}_{v}\left( \cdot \right) ,{B}_{r}\left( \cdot \right) \right\rbrack }^{\top }$ is a vector which includes 3 bounded continuous functions. Otherwise, there are the unknown positive value ${\bar{B}}_{v} = {\left\lbrack {\bar{B}}_{u}\left( \cdot \right) ,{\bar{B}}_{v}\left( \cdot \right) ,{\bar{B}}_{r}\left( \cdot \right) \right\rbrack }^{\top }$ such that $\left| {B}_{v}\right| \leq {\bar{B}}_{v}$ . Then, the dynamic error ${z}_{1}$ can be expressed as (21).
+
+Step 2: Together with the time derivative (19) of ${z}_{2}$ yields the corresponding result as (22).
+
+$$
+{\dot{z}}_{2} = {\dot{\beta }}_{v} - \dot{v} = {M}^{-1}\left( {M{\dot{\beta }}_{v} + D\left( v\right) v - \tau - {\tau }_{d}}\right) \tag{22}
+$$
+
+It is noted that $D\left( v\right) v$ is the uncertain term in the dynamic positioning system. Similar to the treatment of the unknown function vector in the first step, RBF-NNs are used to approximate this uncertainty term as follows:
+
+$$
+{F}_{2}\left( {v,{A}_{2}}\right) = {S}_{2}\left( v\right) {A}_{2}v + {\varepsilon }_{v}
+$$
+
+$$
+= \left\lbrack \begin{matrix} {s}_{u}\left( v\right) & 0 & 0 \\ 0 & {s}_{v}\left( v\right) & 0 \\ 0 & 0 & {s}_{r}\left( r\right) \end{matrix}\right\rbrack \left\lbrack \begin{array}{l} {A}_{u} \\ {A}_{v} \\ {A}_{r} \end{array}\right\rbrack \left\lbrack \begin{array}{l} u \\ v \\ r \end{array}\right\rbrack + \left\lbrack \begin{array}{l} {\varepsilon }_{u} \\ {\varepsilon }_{v} \\ {\varepsilon }_{r} \end{array}\right\rbrack
+$$
+
+(23)
+
+In (23), the output vector ${F}_{2} = \left\lbrack \begin{array}{lll} {f}_{2}\left( u\right) , & {f}_{2}\left( v\right) , & {f}_{2}\left( r\right) \end{array}\right\rbrack$ contains three components correspond to the $u,v,r$ component velocities. Let ${\theta }_{2} = {\begin{Vmatrix}{A}_{2}v\end{Vmatrix}}^{2}$ , where ${\widehat{\theta }}_{2}$ represents the estimated values of ${\theta }_{2}$ . The application of RBF-NNs simplifies the design of subsequent controller and adaptive laws, while reducing the computational complexity of the algorithms to enhance control performance.
+
+In the derivation of formulas involving NNs, three key applications of Youngs inequality are highlighted below:
+
+$$
+{\Phi }_{2}{\xi }_{1}{F}_{1} \leq {\xi }_{1}\left( {{S}_{1}{A}_{1}\eta + {\varepsilon }_{\eta }}\right)
+$$
+
+$$
+\leq \frac{{S}_{1}{}^{T}{S}_{1}{\begin{Vmatrix}{A}_{1}\eta \end{Vmatrix}}^{2}}{2{a}_{1}{}^{2}}{\xi }_{1}{}^{2} + \frac{1}{2}{a}_{1}{}^{2} + {\xi }_{1}{}^{2} + \frac{1}{4}{\varepsilon }_{\eta }{}^{2}
+$$
+
+(24)
+
+$$
+{z}_{2}{F}_{2} \leq {z}_{2}\left( {{S}_{2}{A}_{2}v + {\varepsilon }_{v}}\right)
+$$
+
+$$
+\leq \frac{{S}_{2}{}^{T}{S}_{2}{\begin{Vmatrix}{A}_{2}v\end{Vmatrix}}^{2}}{2{a}_{2}{}^{2}}{z}_{2}{}^{2} + \frac{1}{2}{a}_{2}{}^{2} + {z}_{2}{}^{2} + \frac{1}{2}{\varepsilon }_{v}{}^{2}
+$$
+
+(25)
+
+$$
+- {\Phi }_{2}J\left( \psi \right) {\xi }_{1}{q}_{v} \leq {\begin{Vmatrix}{q}_{v}\end{Vmatrix}}^{2} + \frac{1}{4}{\begin{Vmatrix}{\Phi }_{2}\end{Vmatrix}}^{2}{\xi }_{1}{}^{2} \tag{26}
+$$
+
+Based on the above analysis, (27) is chosen as the control input $\tau$ for the ship dynamic positioning system in this paper. Equation (28),(29) are the expression for the adaptive rate ${\widehat{\theta }}_{1}$ , ${\widehat{\theta }}_{2}$ .
+
+$$
+\tau = {k}_{2}{z}_{2} + {\dot{\beta }}_{v} + \frac{{S}_{2}{}^{T}{S}_{2}{\widehat{\theta }}_{2}}{2{a}_{2}{}^{2}}{z}_{2} + {\Phi }_{2}J\left( \psi \right) {\xi }_{1} \tag{27}
+$$
+
+$$
+{\dot{\widehat{\theta }}}_{1} = \frac{{\gamma }_{1}{S}_{1}^{T}{S}_{1}{\widehat{\theta }}_{1}}{2{a}_{1}{}^{2}} - {\varsigma }_{1}{\widehat{\theta }}_{1} \tag{28}
+$$
+
+$$
+{\dot{\widehat{\theta }}}_{2} = \frac{{\gamma }_{2}{S}_{2}{}^{T}{S}_{2}{\widehat{\theta }}_{2}}{2{a}_{2}{}^{2}} - {\varsigma }_{2}{\widehat{\theta }}_{2} \tag{29}
+$$
+
+where ${k}_{2}$ is a strictly negative diagonal parameter matrix, ${a}_{1}$ , ${a}_{2},{\gamma }_{1},{\gamma }_{2},{\zeta }_{1}$ and ${\zeta }_{2}$ is positive design constants. It can be obviously observed that the designed controller has a very simple form, which significantly reduces the computational load and memory usage. Next, the semi-global uniformly ultimately bounded (SGUUB) stability of the DP system is demonstrated after incorporating the proposed algorithm, through a stability analysis.
+
+§ B. STABILITY ANALYSIS
+
+Select the Lyapunov function as following:
+
+$$
+V = \frac{1}{2}{\xi }_{1}^{\top }{\xi }_{1} + \frac{1}{2}{z}_{2}^{\top }M{z}_{2} + \frac{1}{2}{q}_{v}^{\top }{q}_{v} + \frac{1}{2{\gamma }_{1}}{\widetilde{\theta }}_{1}^{\top }{\widetilde{\theta }}_{1} + \frac{1}{2{\gamma }_{2}}{\widetilde{\theta }}_{1}^{\top }{\widetilde{\theta }}_{1}
+$$
+
+(30)where ${\widetilde{\theta }}_{1} = {\widehat{\theta }}_{1} - {\theta }_{1}$ , and ${\widetilde{\theta }}_{2} = {\widehat{\theta }}_{2} - {\theta }_{2}$ . By considering $\vartheta \left( t\right) /\sqrt{{\omega }^{2}\left( t\right) - {\vartheta }^{2}\left( t\right) }$ and ${z}_{2} = {\beta }_{v} - v$ , the time derivative of $V$ is expressed as:
+
+$$
+\dot{V} = {\xi }_{1}^{\top }{\dot{\xi }}_{1} + {z}_{2}{}^{T}M{\dot{z}}_{2} + {q}_{v}{}^{T}{\dot{q}}_{v} + \frac{1}{{\gamma }_{1}}{\widetilde{\theta }}_{1}^{T}{\dot{\widehat{\theta }}}_{1} + \frac{1}{{\gamma }_{1}}{\widetilde{\theta }}_{2}^{T}{\dot{\widehat{\theta }}}_{2} \tag{31}
+$$
+
+According to (24),(26), $\parallel J\left( \psi \right) \parallel = 1$ and Young’s inequality, it is obtained that
+
+$$
+{\xi }_{1}^{\top }{\dot{\xi }}_{1} = {\xi }_{1}^{\top }\left\lbrack {{\Phi }_{2}J\left( \psi \right) \left( {{\alpha }_{v} - \left( {{z}_{2} - {q}_{v}}\right) }\right) + {S}_{1}{A}_{1}\eta + {\varepsilon }_{\eta }}\right\rbrack
+$$
+
+$$
+= {\xi }_{1}^{\top }\left\{ {{\Phi }_{2}J\left( \psi \right) \left( {-\frac{1}{{\Phi }_{2}}J{\left( \psi \right) }^{-1}\left( {{k}_{1}{\xi }_{1} + \frac{{S}_{1}^{\top }{S}_{1}{\widehat{\theta }}_{1}}{2{a}_{1}{}^{2}}{\xi }_{1}}\right. }\right. }\right.
+$$
+
+$$
+\left. \left. {\left. {+\frac{1}{4}{\begin{Vmatrix}{\Phi }_{2}\end{Vmatrix}}^{2}{\xi }_{1}}\right) - \left( {{z}_{2} - {q}_{v}}\right) }\right) \right\} + {\xi }_{1}^{\top }{S}_{1}{A}_{1}\eta + {\xi }_{1}^{\top }{\varepsilon }_{\eta }
+$$
+
+$$
+\leq {\xi }_{1}{}^{T}\left\{ {-{k}_{1}{\xi }_{1} - \frac{{S}_{1}{}^{T}{S}_{1}{\widehat{\theta }}_{1}}{2{a}_{1}{}^{2}}{\xi }_{1} - \frac{1}{4}{\begin{Vmatrix}{\Phi }_{2}\end{Vmatrix}}^{2}{\xi }_{1}}\right\}
+$$
+
+$$
+- {\xi }_{1}^{\top }{\Phi }_{2}J\left( \psi \right) {z}_{2} - {\xi }_{1}^{\top }{\Phi }_{2}J\left( \psi \right) {q}_{v} + \frac{{S}_{1}^{\top }{S}_{1}{\begin{Vmatrix}{A}_{1}\eta \end{Vmatrix}}^{2}}{2{a}_{1}{}^{2}}{\xi }_{1}{}^{\top }{\xi }_{1}
+$$
+
+$$
++ \frac{1}{2}{a}_{1}{}^{2} + {\xi }_{1}{}^{\top }{\xi }_{1} + \frac{1}{4}{\varepsilon }_{\eta }{}^{2}
+$$
+
+$$
+\leq - {k}_{1}{\xi }_{1}^{\top }{\xi }_{1} + \frac{{S}_{1}^{\top }{S}_{1}\left( {{\theta }_{1} - {\widehat{\theta }}_{1}}\right) }{2{a}_{1}{}^{2}}{\xi }_{1}{}^{\top }{\xi }_{1} - \frac{1}{4}{\begin{Vmatrix}{\Phi }_{2}\end{Vmatrix}}^{2}{\xi }_{1}{}^{\top }{\xi }_{1}
+$$
+
+$$
+- {\xi }_{1}^{\top }{\Phi }_{2}J\left( \psi \right) {z}_{2} + {\begin{Vmatrix}{q}_{v}\end{Vmatrix}}^{2} + \frac{1}{4}{\begin{Vmatrix}{\Phi }_{2}\end{Vmatrix}}^{2}{\xi }_{1}^{\top }{\xi }_{1} + {\xi }_{1}^{\top }{\xi }_{1}
+$$
+
+$$
++ \frac{1}{4}{\varepsilon }_{\eta }{}^{2} + \frac{1}{2}{a}_{1}{}^{2}
+$$
+
+$$
+\leq - {k}_{1}{\xi }_{1}^{\top }{\xi }_{1} - \frac{{S}_{1}{}^{T}{S}_{1}{\widetilde{\theta }}_{1}}{2{a}_{1}{}^{2}}{\xi }_{1}{}^{\top }{\xi }_{1} - {\xi }_{1}{}^{\top }{\Phi }_{2}J\left( \psi \right) {z}_{2}
+$$
+
+$$
++ {\begin{Vmatrix}{q}_{v}\end{Vmatrix}}^{2} + {\xi }_{1}^{\top }{\xi }_{1} + \frac{1}{4}{\varepsilon }_{\eta }{}^{2} + \frac{1}{2}{a}_{1}{}^{2} \tag{32}
+$$
+
+In view of (22),(23),(25) and (27), $\parallel J\left( \psi \right) \parallel = 1$ and Young's inequality, it follows that
+
+$$
+{z}_{2}^{\top }M{\dot{z}}_{2} = {z}_{2}^{\top }M\left\lbrack {{M}^{-1}\left( {M{\dot{\beta }}_{v} + {Dv} - \tau - {\tau }_{d}}\right) }\right\rbrack
+$$
+
+$$
+= {z}_{2}^{\top }\left\lbrack {M{\dot{\beta }}_{v} + {F}_{2} - \left( {{k}_{2}{z}_{2} + {\dot{\beta }}_{v} + \frac{{S}_{2}^{T}{S}_{2}{\widehat{\theta }}_{2}}{2{a}_{2}{}^{2}}{z}_{2}}\right) }\right.
+$$
+
+$$
+\left. {-{\Phi }_{2}R\left( \psi \right) {\xi }_{1} - {\tau }_{d}}\right\rbrack
+$$
+
+$$
+\leq {z}_{2}^{\top }\left( {M - I}\right) {\dot{\beta }}_{v} + \frac{{S}_{2}^{\top }{S}_{2}{\widehat{\theta }}_{2}}{2{a}_{2}{}^{2}}{z}_{2}{}^{\top }{z}_{2} + \frac{1}{2}{a}_{2}{}^{2} + {z}_{2}{}^{\top }{z}_{2}
+$$
+
+$$
++ \frac{1}{2}{\varepsilon }_{v}{}^{2} - {k}_{2}{z}_{2}{}^{\top }{z}_{2} - \frac{{S}_{2}{}^{\top }{S}_{2}{\widehat{\theta }}_{2}}{2{a}_{2}{}^{2}}{z}_{2} + {\Phi }_{2}R\left( \psi \right) {z}_{2}{}^{\top }{\xi }_{1} - {z}_{2}{}^{\top }{\tau }_{d}
+$$
+
+$$
+\leq {z}_{2}^{\top }\left( {M - I}\right) {\dot{\beta }}_{v} + \frac{{S}_{2}^{\top }{S}_{2}{\widetilde{\theta }}_{2}}{2{a}_{2}{}^{2}}{z}_{2}{}^{\top }{z}_{2} + \frac{1}{2}{a}_{2}{}^{2} + {z}_{2}{}^{\top }{z}_{2}
+$$
+
+$$
++ \frac{1}{2}{\varepsilon }_{v}{}^{2} - {k}_{2}{z}_{2}{}^{\top }{z}_{2} + {\Phi }_{2}R\left( \psi \right) {z}_{2}{}^{\top }{\xi }_{1} - {z}_{2}{}^{\top }{\tau }_{d} \tag{33}
+$$
+
+It is worth noticing that
+
+$$
+{z}_{2}\left( {M - I}\right) {\dot{\beta }}_{v} \leq {\begin{Vmatrix}\left( M - I\right) {t}_{v}{}^{-1}\end{Vmatrix}}_{F}^{2}{\begin{Vmatrix}{z}_{2}\end{Vmatrix}}^{2} + \frac{1}{4}{\begin{Vmatrix}{q}_{v}\end{Vmatrix}}^{2} \tag{34}
+$$
+
+$$
+- {z}_{2}{\tau }_{d} \leq {z}_{2}^{\top }{z}_{2} + \frac{{\tau }_{d}^{\top }{\tau }_{d}}{4} \tag{35}
+$$
+
+Note that $I$ is the identity matrix. Then (33) becomes
+
+$$
+{z}_{2}^{\top }M{\dot{z}}_{2} \leq {\begin{Vmatrix}\left( M - I\right) {t}_{v}{}^{-1}\end{Vmatrix}}_{F}^{2}{\begin{Vmatrix}{z}_{2}\end{Vmatrix}}^{2} + 2{z}_{2}{}^{\top }{z}_{2} - {k}_{2}{z}_{2}{}^{\top }{z}_{2}
+$$
+
+$$
+- \frac{{S}_{2}^{\top }{S}_{2}{\widetilde{\theta }}_{2}}{2{a}_{2}{}^{2}}{z}_{2}{}^{\top }{z}_{2} + \frac{1}{4}{\begin{Vmatrix}{q}_{v}\end{Vmatrix}}^{2} + {\Phi }_{2}R\left( \psi \right) {z}_{2}{}^{\top }{\xi }_{1}
+$$
+
+$$
++ \frac{{\tau }_{d}{}^{\top }{\tau }_{d}}{4} + \frac{1}{2}{a}_{2}{}^{2} + \frac{1}{2}{\varepsilon }_{v}{}^{2} \tag{36}
+$$
+
+Incorporating adaptive law (28) and ${\widetilde{\theta }}_{2} = {\widehat{\theta }}_{2} - {\theta }_{2}$ ,(37) and (38) is obtained.
+
+$$
+\frac{1}{{\gamma }_{1}}{\widetilde{\theta }}_{1}^{\top }{\dot{\widehat{\theta }}}_{1} \leq \frac{1}{{\gamma }_{1}}{\widetilde{\theta }}_{1}^{\top }\left( {\frac{{\gamma }_{1}{S}_{1}^{\top }{S}_{1}{\xi }_{1}^{\top }{\xi }_{1}}{2{a}_{1}{}^{2}} - {\varsigma }_{1}{\widehat{\theta }}_{1}}\right)
+$$
+
+$$
+\leq \frac{{S}_{1}^{\top }{S}_{1}{\widetilde{\theta }}_{1}^{\top }{\xi }_{1}^{\top }{\xi }_{1}}{2{a}_{1}{}^{2}} - \frac{{\varsigma }_{1}{\widetilde{\theta }}_{1}^{\top }{\widehat{\theta }}_{1}}{{\gamma }_{1}} \tag{37}
+$$
+
+$$
+{\widetilde{\theta }}_{1}^{\top }{\widehat{\theta }}_{1} \leq {\widetilde{\theta }}_{1}^{\top }\left( {{\widetilde{\theta }}_{1} + {\theta }_{1}}\right)
+$$
+
+$$
+\leq {\widetilde{\theta }}_{1}^{\top }{\widetilde{\theta }}_{1} + {\widetilde{\theta }}_{1}^{\top }{\widetilde{\theta }}_{1}^{\top }
+$$
+
+$$
+\leq 2{\widetilde{\theta }}_{1}^{\top }{\widetilde{\theta }}_{1} + \frac{1}{4}{\theta }_{1}^{2} \tag{38}
+$$
+
+Substituting (38) into (37), one gets
+
+$$
+\frac{1}{{\gamma }_{1}}{\widetilde{\theta }}_{1}^{\top }{\dot{\widehat{\theta }}}_{1} \leq \frac{{S}_{1}^{\top }{S}_{1}{\widetilde{\theta }}_{1}^{\top }{\xi }_{1}^{\top }{\xi }_{1}}{2{a}_{1}{}^{2}} - \frac{2{\varsigma }_{1}}{{\gamma }_{1}}{\widetilde{\theta }}_{1}^{\top }{\widetilde{\theta }}_{1} - \frac{{\varsigma }_{1}}{4{\gamma }_{1}}{\theta }_{1}{}^{2} \tag{39}
+$$
+
+As the same as above steps, another gets:
+
+$$
+\frac{1}{{\gamma }_{2}}{\widetilde{\theta }}_{2}^{\top }{\dot{\widehat{\theta }}}_{2} \leq \frac{{S}_{2}{}^{\top }{S}_{2}{\widetilde{\theta }}_{2}^{\top }{z}_{2}{}^{\top }{z}_{2}}{2{a}_{2}{}^{2}} - \frac{2{\varsigma }_{2}}{{\gamma }_{2}}{\widetilde{\theta }}_{2}^{\top }{\widetilde{\theta }}_{2} - \frac{{\varsigma }_{2}}{4{\gamma }_{2}}{\theta }_{2}{}^{2} \tag{40}
+$$
+
+Using (21) and Young’s inequality, ${q}_{v}^{\top }{\dot{q}}_{v}$ follows that
+
+$$
+{q}_{v}^{\top }{\dot{q}}_{v} \leq - \mathop{\sum }\limits_{{i = u,v,r}}\left( {\frac{{q}_{i}{}^{2}}{{t}_{i}} - \frac{{B}_{i}^{2}{q}_{i}{\bar{B}}_{i}^{2}}{{2b}{\bar{B}}_{i}} - \frac{b}{2}}\right)
+$$
+
+$$
+\leq - \mathop{\sum }\limits_{{i = u,v,r}}\left\lbrack {\left( {\frac{1}{{t}_{i}} - \frac{{\bar{B}}_{i}^{2}}{2b}}\right) {q}_{i}{}^{2} + \left( {1 - \frac{{B}_{i}^{2}}{{\bar{B}}_{i}^{2}}}\right) \frac{{\bar{B}}_{i}^{2}{q}_{i}{}^{2}}{2b} - \frac{b}{2}}\right\rbrack
+$$
+
+$$
+\leq - \mathop{\sum }\limits_{{i = u,v,r}}\left\lbrack {\left( {\frac{1}{{t}_{i}} - \frac{{\bar{B}}_{i}^{2}}{2b}}\right) {q}_{i}{}^{2}}\right\rbrack + \frac{3b}{2} \tag{41}
+$$
+
+Submitting (32) (36) (39) (40) and (41) into (31), the time
+
+$$
+\dot{V} \leq - \left( {{k}_{1} - I}\right) {\xi }_{1}^{\top }{\xi }_{1} - \left( {{k}_{2} - {2I}}\right) {z}_{2}^{\top }{z}_{2} + {\begin{Vmatrix}\left( M - I\right) {t}_{v}{}^{-1}\end{Vmatrix}}_{F}^{2}{\begin{Vmatrix}{z}_{2}\end{Vmatrix}}^{2}
+$$
+
+$$
++ \frac{5}{4}{\begin{Vmatrix}{q}_{v}\end{Vmatrix}}^{2} - \mathop{\sum }\limits_{{i = u,v,r}}\left\lbrack {\left( {\frac{1}{{t}_{i}} - \frac{{\bar{B}}_{i}^{2}}{2b}}\right) {q}_{i}{}^{2}}\right\rbrack - \frac{2{\varsigma }_{1}}{{\gamma }_{1}}{\widetilde{\theta }}_{1}^{\top }{\widetilde{\theta }}_{1}
+$$
+
+$$
+- \frac{2{\varsigma }_{2}}{{\gamma }_{2}}{\widetilde{\theta }}_{2}^{\top }{\widetilde{\theta }}_{2} - {\xi }_{1}^{\top }{\Phi }_{2}J\left( \psi \right) {z}_{2} + {\xi }_{1}{\Phi }_{2}R\left( \psi \right) {z}_{2}^{\top } - \frac{{S}_{1}^{\top }{S}_{1}{\widetilde{\theta }}_{1}}{2{a}_{1}{}^{2}}{\xi }_{1}{}^{\top }{\xi }_{1}
+$$
+
+$$
++ \frac{{S}_{1}^{\top }{S}_{1}{\widetilde{\theta }}_{1}^{\top }{\xi }_{1}^{\top }{\xi }_{1}}{2{a}_{1}{}^{2}} - \frac{{S}_{2}{}^{T}{S}_{2}{\widetilde{\theta }}_{2}}{2{a}_{2}{}^{2}}{z}_{2}{}^{\top }{z}_{2} + \frac{{S}_{2}{}^{T}{S}_{2}{\widetilde{\theta }}_{2}^{\top }{\xi }_{2}{}^{\top }{\xi }_{2}}{2{a}_{2}{}^{2}}
+$$
+
+$$
++ \frac{1}{4}{\varepsilon }_{\eta }{}^{2} + \frac{1}{2}{a}_{1}{}^{2} + \frac{{\tau }_{d}{}^{\top }{\tau }_{d}}{4} + \frac{1}{2}{a}_{2}{}^{2} + \frac{1}{2}{\varepsilon }_{\upsilon }{}^{2} - \frac{{\varsigma }_{1}}{4{\gamma }_{1}}{\theta }_{1}{}^{2}
+$$
+
+$$
+- \frac{{\varsigma }_{2}}{4{\gamma }_{2}}{\theta }_{2}{}^{2} + \frac{3b}{2}
+$$
+
+$$
+\leq - {2aV} + \varrho \tag{42}
+$$
+
+where $a = {\lambda }_{\min }\left\{ {-\left( {{k}_{1} - I}\right) , - \left( {{k}_{2} - {2I}}\right) + {\begin{Vmatrix}\left( M - I\right) {t}^{-1}{}_{v}\end{Vmatrix}}_{F}}\right.$ , $\left\{ {5/4\underset{i = u,v,r}{-\sum }\left\lbrack {1/{t}_{i} - {\bar{B}}_{i}/{2b}}\right\rbrack }\right\} ,\left. {-2{\varsigma }_{1}/{\gamma }_{2} - 2{\varsigma }_{2}/{\gamma }_{2}}\right\} ,\varrho = 1/\left( {4{\varepsilon }_{\eta }^{2}}\right)$ $+ 1/\left( {2{a}_{1}^{2}}\right) + {\tau }_{d}^{\top }{\tau }_{d}/4 + 1/\left( {2{a}_{2}^{2}}\right) + 1/\left( {2{\varepsilon }_{v}^{2}}\right) - {\varsigma }_{1}{\theta }_{1}^{2}/\left( {4{\gamma }_{1}}\right)$ $- {\varsigma }_{2}{\theta }_{2}^{2}/\left( {4{\gamma }_{2}}\right) + {3b}/2$ .
+
+By integrating both sides of equation (42), we obtain:
+
+$$
+V\left( t\right) \leq \left( {V\left( 0\right) - \frac{\varrho }{2a}}\right) {e}^{\left( -2at\right) } + \frac{\varrho }{2a} \tag{43}
+$$
+
+According to the closed-loop gain shaping algorithm [18], all errors variables in closed-loop system decrease to the compact set $\Omega \mathrel{\text{ := }} \left\{ {\left. \left( {{\xi }_{1},{z}_{2},{q}_{v},{\widetilde{\theta }}_{1},{\widehat{\widetilde{\theta }}}_{2}}\right) \right| \;\parallel {\xi }_{1}\parallel \leq {C}_{0},{C}_{0} > \sqrt{\varrho /a}}\right\}$ as $t \rightarrow \infty$ by choosing appropriate parameters. ${C}_{0} > \sqrt{\varrho /a}$ is a positive constant. Thus, the closed-loop control system is SGUUB stable under the proposed control scheme, given the positive constant ${C}_{0}$ , where all signal errors in the closed-loop system can be made arbitrarily small.
+
+§ V. SIMULATION
+
+In this section, to verify the effectiveness of the proposed prescribed-time algorithm, a simulation example for a supply ship (length: ${76.2}\mathrm{\;m}$ , mass: ${4.591} \times {10}^{6}\mathrm{\;{kg}}$ ) equipped with a DP system is executed and compared with the Optimum-seeking Guidance scheme (OSG) in [19] and robust control scheme in [20]. The ship mathematic model parameters are presented in TABLE I. In the modeling of ship Dynamic Positioning (DP) systems, it is essential to precisely characterize and predict the impacts of environmental disturbances, such as wind, waves, and ocean currents on the ship's performance. For the sake of simplifying the ship model and facilitating the design and testing of control algorithms, these environmental disturbances are approximated and modeled using a sine-cosine function (44).
+
+$$
+\left\{ \begin{array}{l} {\tau }_{du} = 2\left( {1 + {35}\sin \left( {{0.2t} + {15}\cos \left( {0.5t}\right) }\right) }\right) \left( \mathrm{N}\right) \\ {\tau }_{dv} = 2\left( {1 + {30}\cos \left( {{0.4t} + {20}\cos \left( {0.1t}\right) }\right) }\right) \left( \mathrm{N}\right) \\ {\tau }_{dr} = 3\left( {1 + {30}\cos \left( {{0.3t} + {10}\sin \left( {0.5t}\right) }\right) }\right) \left( {\mathrm{N} \cdot \mathrm{r}}\right. \end{array}\right. \tag{44}
+$$
+
+$$
+{k}_{1} = \operatorname{diag}\left\lbrack {{0.2},{0.38},{0.20}}\right\rbrack ,{k}_{2} = \operatorname{diag}\left\lbrack {{44},{12.8},{78.1}}\right\rbrack ;
+$$
+
+$$
+{t}_{v} = {0.05} \times I;{a}_{1} = {a}_{2} = {80};{\gamma }_{1} = {\gamma }_{2} = {0.5};{\varsigma }_{1} = {\varsigma }_{2} = {0.5}\text{ ; }
+$$
+
+$$
+{T}_{j\upsilon } = \left\lbrack {{T}_{ju},{T}_{jv},{T}_{jr}}\right\rbrack = \left\lbrack {{80s},{80s},{90s}}\right\rbrack ;
+$$
+
+$$
+{T}_{fv} = \left\lbrack {{T}_{fu},{T}_{fv},{T}_{fr}}\right\rbrack = \left\lbrack {{80s},{80s},{90s}}\right\rbrack ; \tag{45}
+$$
+
+TABLE I
+
+MODEL PARAMETERS
+
+max width=
+
+Indexes Values Indexes Values
+
+1-4
+${X}_{\dot{u}}$ $- {0.72} \times {10}^{6}$ ${X}_{\dot{u}}$ ${5.0242} \times {10}^{4}$
+
+1-4
+${Y}_{v}$ $- {3.6921} \times {10}^{6}$ ${Y}_{v}$ ${2.7229} \times {10}^{6}$
+
+1-4
+${Y}_{\dot{r}}$ $- {1.0234} \times {10}^{6}$ ${Y}_{r}$ $- {4.3933} \times {10}^{6}$
+
+1-4
+${I}_{z} - {N}_{\dot{r}}$ ${3.7454} \times {10}^{9}$ ${Y}_{\left| v\right| v}$ ${1.7860} \times {10}^{4}$
+
+1-4
+${X}_{\left| u\right| u}$ ${1.0179} \times {10}^{3}$ ${Y}_{\left| v\right| r}$ $- {3.0068} \times {10}^{5}$
+
+1-4
+${N}_{v}$ $- {4.3821} \times {10}^{6}$ ${N}_{r}$ ${4.1894} \times {10}^{6}$
+
+1-4
+${N}_{\left| v\right| v}$ $- {2.4684} \times {10}^{5}$ ${N}_{\left| v\right| r}$ ${6.5759} \times {10}^{6}$
+
+1-4
+
+ < g r a p h i c s >
+
+Fig. 1. Trajectory of the ship in ${xy}$ -plane.
+
+In this simulation, the desired attitude is set to ${\eta }_{d} =$ $\left\lbrack \begin{array}{lll} 0\mathrm{m}, & 0\mathrm{m}, & 0\mathrm{{deg}} \end{array}\right\rbrack$ . The initial states are set to $\eta \left( 0\right) =$ $\left\lbrack {{12}\mathrm{\;m},{14}\mathrm{\;m},{10}\mathrm{{deg}}}\right\rbrack ,v\left( 0\right) = \left\lbrack {0\mathrm{\;m}/\mathrm{s},{14}\mathrm{\;m}/\mathrm{s},{10}\mathrm{{deg}}/\mathrm{s}}\right\rbrack .$ The concrete parameters values setting follows (45). Besides, the RBF-NNs for ${F}_{1}$ and ${F}_{1}$ consisted of 25 nodes with centers spaced in $\left\lbrack {-{2.5}\mathrm{\;m}/\mathrm{s},{2.5}\mathrm{\;m}/\mathrm{s}}\right\rbrack$ for $x,y,u$ and $r$ , $\left\lbrack {-{0.16}\mathrm{\;m}/\mathrm{s},{0.16}\mathrm{\;m}/\mathrm{s}}\right\rbrack$ for $\psi$ and $r$ , respectively. For the comparison algorithms, corresponding parameters refers to [19] and [20].
+
+Fig. 1 exhibits simulation results under the proposed algorithm, OSG and robust control making the ship stay at the desired attitude in the ${xy}$ -plant. It is clear that the proposed algorithm provides a more satisfactory trajectory accuracy compared to the algorithms for comparison. Fig. 2 illustrates that the ship attitude $x,y$ and $\psi$ are stabilized to the desired attitude near the prescribed-time ${T}_{j\upsilon }$ . The proposed scheme achieves faster stabilization compared to the schemes for comparison. The velocities of surge, sway and yaw are showed in Fig. 3. The proposed scheme has a improved convergence performance. Fig. 4 illustrates the fluctuation of the values of the three input signals over time. It is apparent that prior to system stabilization, the proposed scheme exhibits superior convergence performance of ${\tau }_{r}$ compared to other schemes. Once the system has stabilized, the values of ${\tau }_{u}$ and ${\tau }_{v}$ in the proposed scheme converge more rapidly towards zero, further outperforming the other schemes in convergence efficiency. Finally, it can be seen that the constructed new error is successfully confined within the boundaries and converges stably to 0 at the moment of settling time ${T}_{fu},{T}_{fv},{T}_{fr}$ as shown in Fig. 5. Additionally, Fig. 6 and Fig. 7 illustrate the fitting performance between the estimated and true values of the adaptive parameters ${\theta }_{1}$ and ${\theta }_{2}$ , respectively, representing the approximation capability of the RBF-NNs for the system uncertainties terms. It can be observed that, within the permissible margin of error, the RBF-NNs successfully approximate uncertainties terms described by (13) and (17).
+
+ < g r a p h i c s >
+
+Fig. 2. Ship’s actual position(x, y)and heading $\psi$ .
+
+ < g r a p h i c s >
+
+Fig. 3. Ship’s surge velocity $u$ , sway velocity $v$ and yaw rate $r$ .
+
+ < g r a p h i c s >
+
+Fig. 4. Ship’s surge force ${\tau }_{u}$ , sway force ${\tau }_{v}$ and yaw force ${\tau }_{r}$ .
+
+ < g r a p h i c s >
+
+Fig. 5. The new error ${\xi }_{1}$ for the simulation with the proposed scheme.
+
+In summary, the NNs-based prescribed-time control scheme proposed in this paper demonstrates superior performance and robustness compared to the schemes for comparison. By introducing FTFBs, FTTFBs and constructing new error functions, the control laws are made more concise. Finally, the proposed scheme is validated through simulations to demonstrate its effectiveness on DP ships.
+
+ < g r a p h i c s >
+
+Fig. 6. The estimation performance of ${\theta }_{1}$ .
+
+ < g r a p h i c s >
+
+Fig. 7. The estimation performance of ${\theta }_{2}$ .
+
+§ VI. CONCLUTION
+
+In this paper, a novel NNs-based control scheme is proposed for the ship DP system under model uncertain and unknown environmental disturbances, making the new dynamic errors converging within fixed boundaries. The prescribed-time performance of the algorithm is validated through by a simulation example and two comparative simulations with satisfactory results. Consequently, the prescribed-time control algorithm proposed in this paper can be applied to ships performing DP tasks, enabling the ship's dynamic system to achieve more precise time-based prescribed performance.
+
+Given the presence of multiple dynamic actuators in engineering practices related to marine equipment, future research on the proposed algorithm could focus on the issue of actuator control allocation. In addition, the integration of event-triggered control, fault-tolerant control, and blind zone constraints could further enhance the development of this control algorithm toward more advanced and precise control techniques.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bmvHIfgK1y/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bmvHIfgK1y/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..450681637b2dfa001f6a89c93fe126bb70bd2c9b
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bmvHIfgK1y/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,319 @@
+# Modeling and analysis of UAV charging scheduling in fixed/mobile charging station systems
+
+${1}^{\text{st }}$ Zeyu Guo
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+eyuguoll@163.com
+
+${2}^{\text{nd }}$ Sining Zhang
+
+China north vehicle research institute
+
+Beijing, China
+
+13426157603@163.com
+
+${3}^{\text{rd }}$ Jiahe Wang
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+213220649@seu.edu.cn
+
+${4}^{\text{th }}$ Xinyuan Huang
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+213220199@seu.edu.cn
+
+${5}^{\text{th }}$ Ruixu Hu
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+220231953@seu.edu.cn
+
+${6}^{\text{th }}$ Wenying Xu
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+wyxu@seu.edu.cn
+
+${Abstract}$ -This paper proposes two novel mathematical models for optimizing the charging schedules of Unmanned Aerial Vehicles (UAVs) within systems featuring either fixed or mobile charging stations. The primary objective is to minimize the total charging time for all UAVs. Initially, the fixed charging station (FCS) system is modeled, followed by a comparison of four different algorithms. Subsequently, the model is extended to consider a mobile charging station (MCS) system, where the station can relocate as necessary. In this scenario, an algorithm is proposed to optimize the charging station's position to enhance charging efficiency. Finally, a numerical example is presented to compare the performance of different algorithms in both fixed and mobile charging station systems. The simulation results demonstrate that the proposed algorithm effectively improves charging efficiency by strategically positioning the charging station.
+
+Index Terms-Unmanned aerial vehicles, charging scheduling, fixed charging station, mobile charging station
+
+## I. INTRODUCTION
+
+In recent years, unmanned aerial vehicles have been widely used in military, civil and commercial fields, showing great potential for application. In the military field, UAVs are widely used for tasks such as dangerous reconnaissance [1], search and rescue [2], and logistics distribution [3] due to their flexible deployment capability. Compared with ordinary vehicles, UAVs can quickly reach areas with complex road conditions, greatly improving the efficiency of rescue and cargo transportation missions [4]. Due to the limitations of battery technology and longer mission times, the endurance issue of UAVs has become a major barrier to their development. To overcome this problem, there has been some research to improve the battery technology of UAVs. Reference [5] summarizes recent advancements in battery technology. Another way to enhance endurance is to plan the scheduling of UAVs such as path planning and task assignment, which is a strategy used in many studies [6]-[9]. However, these studies have some shortcomings. On one hand, some research imposes restrictions on the charging scenarios, such as requiring mobile charging vehicles to travel to the drones for power supply [8]. In certain situations, UAVs may need to perform hazardous tasks in remote areas where charging vehicles cannot reach. This requires UAVs to travel to the charging vehicle location for energy replenishment. On the other hand, these studies do not pay enough attention to the queuing problem caused by the increased demand for charging, as their focus is on the path and task planning of drones. Therefore, it is essential to study the queuing problem in the scenario where UAVs need to travel to charging stations to recharge their batteries.
+
+The charging scheduling problem for commercial electric vehicles shares similarities with that of drones [10]. The former has conducted thorough research on the queuing problem in the charging process, providing methods and insights for the latter. Based on the mobility of charging stations, these studies can be categorized into fixed charging stations (FCS) and mobile charging stations (MCS). For FCS, the studies mainly address the layout of charging stations and the scheduling and management of charging vehicles. Zhu et al. [11] proposed a charging scheduling strategy that determines the charging sequence based on the electric vehicles' charging needs rather than their arrival times at the charging stations. Hamed et al. [12] considered different charging needs (day and night) and various charging scenarios, formulating the problem as a mixed-integer linear problem with multiple constraints. To address the shortcomings of FCS, such as range anxiety and lengthy charging times, some studies have begun to utilize MCS [13], [14]. Li et al. [13] proposed a framework for optimizing mobile charging vehicle operations and developed a variant of a Mixed-Integer Linear Programming (MILP) model. Inspired by the above research, we consider both fixed and mobile scenarios of charging stations, and build the FCS and MCS systems based on the scenario in which UAVs need to travel to charging stations to charge.
+
+The contributions of this paper are summarized as follows.
+
+- Considering the scenario of UAVs traveling to charging stations and the queuing problem caused by limited charging capacity, we model the fixed charging station system and use four different algorithms to solve the problem.
+
+- Based on the FCS system model, the MCS system model is established by adding the mobility of the charging station, and an algorithm to optimize the location of the charging station is proposed. Simulation results indicate that, compared to the FCS, the MCS system model effectively reduces the total flight distance of drones caused by charging.
+
+The rest of the paper is organized as follows. Section II introduces the system model. In Section III, optimization algorithms are introduced. Numerical experiments and analysis are conducted in Section IV. Finally, Section V concludes this paper.
+
+## II. System Model
+
+In this section, a fixed charging station system is presented to solve the queuing problem encountered in UAV charging. Then, considering the practical needs of mobile charging stations, this paper establishes a mobile charging station system. This system involves the selection of charging station location and the scheduling of UAVs.
+
+## A. Fixed Charging Station System Model
+
+1) FCS System: The fixed charging station system consists of a fixed charging station and several UAVs operating tasks in the vicinity of the FCS. Each UAV is designated by the index $i, i \in \{ 1,2,3,\ldots , N\}$ . The UAVs can communicate with the FCS, sending their current data to the FCS. The FCS is equipped with the capability to collect and process various data from UAVs, enabling it to calculate charging sequence schemes. Data related to UAVs and the charging station are presented in Table I. The FCS has sufficient energy but a fixed number of charging ports. When all charging ports are occupied by UAVs, the remaining UAVs need to queue up in the charging sequence and wait their turn. The number of charging ports is denoted as $M$ and the coordinates of the FCS are located at the origin.
+
+The scheduling process of the FCS system is shown in Fig. 1. A complete charging scheduling process involves: at regular time intervals $T$ , the FCS collects relevant data from each UAV, confirms the set of UAVs capable of reaching the FCS, and proceeds with the solution of the scheduling scheme. UAVs unable to reach the FCS do not participate in the schedule and terminate their tasks. At the same time, the participating UAVs move to the FCS for charging. Upon arrival at the station, each UAV lines up at the charging port in the charging sequence provided by the FCS. After finish charging, they will autonomously return to the mission site to continue their missions. When all the UAVs involved in the scheduling return to their mission sites, the scheduling process for this round ends.
+
+TABLE I: Description of the symbol
+
+| System Model | The number of UAVs | Calculation duration | Scheduling duration | The total distance traveled by the UAV | Location coordinates of the charging vehicle |
| FCS | 10 | 1.604189s | ${1.5147}\mathrm{\;h}$ | ${48.1646}\mathrm{\;{km}}$ | $\left\lbrack {0,0}\right\rbrack$ |
| 15 | 3.705172s | ${3.5737}\mathrm{\;h}$ | 62.4658km | $\left\lbrack {0,0}\right\rbrack$ |
| 20 | 5.776012s | 2.2079h | 99.3726km | $\left\lbrack {0,0}\right\rbrack$ |
| 25 | 9.441099s | 3.7836h | 93.0275km | $\left\lbrack {0,0}\right\rbrack$ |
| 30 | 12.593925s | 3.1608h | 144.8695km | $\left\lbrack {0,0}\right\rbrack$ |
| $\mathbf{{MCS}}$ | 10 | 1.815212s | 1.4937h | 34.6202km | $\left\lbrack {-4,4}\right\rbrack$ |
| 15 | 3.766389s | 3.4914h | ${33.117}\mathrm{\;{km}}$ | [3,2] |
| 20 | 6.124744s | 1.9614h | 77.5759km | [4, 4] |
| 25 | 9.470387s | ${3.7803}\mathrm{\;h}$ | 73.9708km | $\left\lbrack {0,2}\right\rbrack$ |
| 30 | 13.224203s | 3.1273h | 119.9724km | $\left\lbrack {3,2}\right\rbrack$ |
+
+The total distance traveled by the UAV (km) 160 The number of UAVs (a) The total distance traveled by UAVs 20 25 30 The number of UAVs (b) Scheduling duration FCS 140 MCS 120 100 80 60 20 15 FCS MCS 3.5 Scheduling duration (hours) 2.5 15
+
+Fig. 5: Comparison of UAV distance traveled and scheduling duration for FCS and MCS
+
+To simplify the problem, only integer points on a coordinate grid (points where both the horizontal and vertical coordinates are integers) are considered. Initially, set the value of each integer point to 0 . Each time a circular area covers a point, increment the value of that point by 1 . By selecting the integer points with the highest values, a finite set of integer points can be obtained as candidate locations. We designate this set as $G$ , where the elements in $G$ are the coordinates of points, with elements in the form such as(1,2). Subsequently, by traversing and solving or employing a heuristic algorithm, the location $L$ can be determined.
+
+The pseudo code for the algorithm addressing this problem is provided in Algorithm 1.
+
+## IV. NUMERICAL EXPERIMENTS AND ANALYSIS
+
+For testing the FCS system model, we set the parameter $M = 2$ . Based on simulation experiments, it is found that, even with 30 UAVs, the scheduling time remains within 4 hours. Therefore, we set the scheduling cycle parameter $T = 4$ (which can be adjusted according to the number of drones and the number of charging ports at the charging station). Since the optimal solution can be obtained through exhaustive search when the number of UAVs is 10 or fewer, this part compares the relative error of the solution results of four heuristic algorithms against the optimal solution with the number of UAVs ranging from 3 to 10 . The relative error is defined as the difference between the solution obtained by the algorithm and the optimal solution, divided by the optimal solution. The algorithm parameters are set as shown in Table II.
+
+The comparison results of the algorithms are shown in Fig. 3, where the relative errors of all four heuristic algorithms are less than ${0.9}\%$ . This indicates the effectiveness of the heuristic algorithms in solving this problem. It is noteworthy that the relative error of SA is generally lower. To compare the performance of heuristic algorithms with a larger number of UAVs, we test the four algorithms with 10,20, and 30 UAVs, as shown in Fig. 4. Fig. 4 shows that SA consistently outperforms the other three algorithms. This is the reason for selecting SA for this problem.
+
+For testing the MCS system model, we set the parameter $M = 4$ and employ SA for solution. We compare the results of the MCS and FCS systems under the same drone data, as shown in Table III and Fig. 5. It can be seen that, compared to the FCS system, the MCS system results in shorter scheduling times and a significant reduction in the total distance traveled by UAVs. The time required to solve the MCS system is similar to that of the FCS system. This indicates that the MCS system provides a better charging scheduling solution while maintaining similar solution speed, demonstrating the effectiveness and superiority of this model.
+
+## V. Conclusion
+
+A scheduling system with fixed charging station is described. Through simulation, it is concluded that the simulated annealing algorithm is the most suitable method among the four algorithms for solving this problem. Compared to GA, PSO, and TS, SA demonstrates superior performance in solving large-scale drone problems. Based on FCS system, a scheduling system is proposed that replaces fixed charging station with a mobile charging station. For the MCS system, a two-stage optimization model is established, including the selection of MCS location and the scheduling of drone charging. By comparing the results of two models, it is validated that the MCS system model can provide a more optimal scheduling solution to meet practical needs. The future work is to study large scheduling systems with relay charging platforms and to design efficient algorithms for determining charging locations.
+
+## ACKNOWLEDGMENT
+
+This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 62173087. Corresponding author: Zeyu Guo.
+
+## REFERENCES
+
+[1] R. Masroor, M. Naeem, and W. Ejaz, "Efficient deployment of UAVs for disaster management: A multi-criterion optimization approach," Comput. Commun., vol. 177, pp. 185-194, September 2021.
+
+[2] B. Lee, S. Kwon, P. Park, et al., "Active power management system for an unmanned aerial vehicle powered by solar cells, a fuel cell, and batteries," IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 4, pp. 3167- 3177, October 2014.
+
+[3] G. Wang and X. Bai, "Comparation of UAV path planning for logistics distribution," in Proc. Int. Conf. Intelligent Transportation Engineering, Beijing, China, October 2021, pp. 223-238.
+
+[4] Y. Chen, M. Chen, Z. Chen, L. Cheng, Y. Yang, and H. Li, "Delivery path planning of heterogeneous robot system under road network constraints," Comput. Electr. Eng., vol. 92, p. 107197, June 2021.
+
+[5] N. A. Khofiyah, W. Sutopo, and B. D. A. Nugroho, "Technical feasibility of lithium battery to support unmanned aerial vehicle (UAV): A technical review," in Proc. Int. Conf. Industrial Engineering and Operations Management, Bangkok, Thailand, March 2019, pp. 3591-3601.
+
+[6] K. Yu, A. K. Budhiraja, and P. Tokekar, "Algorithms for routing of unmanned aerial vehicles with mobile recharging stations," in Proc. 2018 IEEE Int. Conf. Robotics and Automation (ICRA), Brisbane, QLD, Australia, May 2018, pp. 5720-5725.
+
+[7] B. Li, S. Patankar, B. Moridian, et al., "Planning large-scale search and rescue using team of UAVs and charging stations," in Proc. 2018 IEEE Int. Symp. Safety, Security, and Rescue Robotics (SSRR), Philadelphia, PA, USA, August 2018, pp. 1-8.
+
+[8] W. Qin, T. Zhang, Z. Shi, H. Huang, H. He, and W. Li, "Scheduling and routing of mobile charging vehicles for unmanned aerial vehicles charging," in Proc. 2021 IEEE 5th Conf. Energy Internet Energy System Integr. (EI2), Chengdu, China, October 2021, pp. 715-720.
+
+[9] Y. Wang and Z. Su, "An envy-free online UAV charging scheme with vehicle-mounted mobile wireless chargers," China Commun., vol. 20, no. 8, pp. 89-102, August 2023.
+
+[10] T. Erdelić and T. Carić, "A survey on the electric vehicle routing problem: variants and solution approaches," J. Adv. Transp., vol. 2019, no. 1, pp. 5075671, May 2019.
+
+[11] M. Zhu, X. Y. Liu, L. Kong, R. Shen, W. Shu, and M. Y. Wu, "The charging-scheduling problem for electric vehicle networks," in Proc. 2014 IEEE Wireless Communications and Networking Conf. (WCNC), Istanbul, Turkey, April 2014, pp. 3178-3183.
+
+[12] M. M. Hamed, D. M. Kabtawi, A. Al-Assaf, O. Albatayneh, and E. S. Gharaibeh, "Random parameters modeling of charging-power demand for the optimal location of electric vehicle charge facilities," J. Clean. Prod., p. 136022, February 2023.
+
+[13] H. Li, D. Son, and B. Jeong, "Electric vehicle charging scheduling with mobile charging stations," J. Clean. Prod., vol. 434, p. 140162, January 2024.
+
+[14] S. Afshar, P. Macedo, F. Mohamed, and V. Disfani, "Mobile charging stations for electric vehicles-A review," Renewable and Sustainable Energy Reviews, vol. 152, pp. 111654, December 2021.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bmvHIfgK1y/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bmvHIfgK1y/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b8209fe69aafb3707f3c4637db2d1166847bd31d
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/bmvHIfgK1y/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,366 @@
+§ MODELING AND ANALYSIS OF UAV CHARGING SCHEDULING IN FIXED/MOBILE CHARGING STATION SYSTEMS
+
+${1}^{\text{ st }}$ Zeyu Guo
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+eyuguoll@163.com
+
+${2}^{\text{ nd }}$ Sining Zhang
+
+China north vehicle research institute
+
+Beijing, China
+
+13426157603@163.com
+
+${3}^{\text{ rd }}$ Jiahe Wang
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+213220649@seu.edu.cn
+
+${4}^{\text{ th }}$ Xinyuan Huang
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+213220199@seu.edu.cn
+
+${5}^{\text{ th }}$ Ruixu Hu
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+220231953@seu.edu.cn
+
+${6}^{\text{ th }}$ Wenying Xu
+
+School of Mathematics
+
+Southeast University
+
+Nanjing, China
+
+wyxu@seu.edu.cn
+
+${Abstract}$ -This paper proposes two novel mathematical models for optimizing the charging schedules of Unmanned Aerial Vehicles (UAVs) within systems featuring either fixed or mobile charging stations. The primary objective is to minimize the total charging time for all UAVs. Initially, the fixed charging station (FCS) system is modeled, followed by a comparison of four different algorithms. Subsequently, the model is extended to consider a mobile charging station (MCS) system, where the station can relocate as necessary. In this scenario, an algorithm is proposed to optimize the charging station's position to enhance charging efficiency. Finally, a numerical example is presented to compare the performance of different algorithms in both fixed and mobile charging station systems. The simulation results demonstrate that the proposed algorithm effectively improves charging efficiency by strategically positioning the charging station.
+
+Index Terms-Unmanned aerial vehicles, charging scheduling, fixed charging station, mobile charging station
+
+§ I. INTRODUCTION
+
+In recent years, unmanned aerial vehicles have been widely used in military, civil and commercial fields, showing great potential for application. In the military field, UAVs are widely used for tasks such as dangerous reconnaissance [1], search and rescue [2], and logistics distribution [3] due to their flexible deployment capability. Compared with ordinary vehicles, UAVs can quickly reach areas with complex road conditions, greatly improving the efficiency of rescue and cargo transportation missions [4]. Due to the limitations of battery technology and longer mission times, the endurance issue of UAVs has become a major barrier to their development. To overcome this problem, there has been some research to improve the battery technology of UAVs. Reference [5] summarizes recent advancements in battery technology. Another way to enhance endurance is to plan the scheduling of UAVs such as path planning and task assignment, which is a strategy used in many studies [6]-[9]. However, these studies have some shortcomings. On one hand, some research imposes restrictions on the charging scenarios, such as requiring mobile charging vehicles to travel to the drones for power supply [8]. In certain situations, UAVs may need to perform hazardous tasks in remote areas where charging vehicles cannot reach. This requires UAVs to travel to the charging vehicle location for energy replenishment. On the other hand, these studies do not pay enough attention to the queuing problem caused by the increased demand for charging, as their focus is on the path and task planning of drones. Therefore, it is essential to study the queuing problem in the scenario where UAVs need to travel to charging stations to recharge their batteries.
+
+The charging scheduling problem for commercial electric vehicles shares similarities with that of drones [10]. The former has conducted thorough research on the queuing problem in the charging process, providing methods and insights for the latter. Based on the mobility of charging stations, these studies can be categorized into fixed charging stations (FCS) and mobile charging stations (MCS). For FCS, the studies mainly address the layout of charging stations and the scheduling and management of charging vehicles. Zhu et al. [11] proposed a charging scheduling strategy that determines the charging sequence based on the electric vehicles' charging needs rather than their arrival times at the charging stations. Hamed et al. [12] considered different charging needs (day and night) and various charging scenarios, formulating the problem as a mixed-integer linear problem with multiple constraints. To address the shortcomings of FCS, such as range anxiety and lengthy charging times, some studies have begun to utilize MCS [13], [14]. Li et al. [13] proposed a framework for optimizing mobile charging vehicle operations and developed a variant of a Mixed-Integer Linear Programming (MILP) model. Inspired by the above research, we consider both fixed and mobile scenarios of charging stations, and build the FCS and MCS systems based on the scenario in which UAVs need to travel to charging stations to charge.
+
+The contributions of this paper are summarized as follows.
+
+ * Considering the scenario of UAVs traveling to charging stations and the queuing problem caused by limited charging capacity, we model the fixed charging station system and use four different algorithms to solve the problem.
+
+ * Based on the FCS system model, the MCS system model is established by adding the mobility of the charging station, and an algorithm to optimize the location of the charging station is proposed. Simulation results indicate that, compared to the FCS, the MCS system model effectively reduces the total flight distance of drones caused by charging.
+
+The rest of the paper is organized as follows. Section II introduces the system model. In Section III, optimization algorithms are introduced. Numerical experiments and analysis are conducted in Section IV. Finally, Section V concludes this paper.
+
+§ II. SYSTEM MODEL
+
+In this section, a fixed charging station system is presented to solve the queuing problem encountered in UAV charging. Then, considering the practical needs of mobile charging stations, this paper establishes a mobile charging station system. This system involves the selection of charging station location and the scheduling of UAVs.
+
+§ A. FIXED CHARGING STATION SYSTEM MODEL
+
+1) FCS System: The fixed charging station system consists of a fixed charging station and several UAVs operating tasks in the vicinity of the FCS. Each UAV is designated by the index $i,i \in \{ 1,2,3,\ldots ,N\}$ . The UAVs can communicate with the FCS, sending their current data to the FCS. The FCS is equipped with the capability to collect and process various data from UAVs, enabling it to calculate charging sequence schemes. Data related to UAVs and the charging station are presented in Table I. The FCS has sufficient energy but a fixed number of charging ports. When all charging ports are occupied by UAVs, the remaining UAVs need to queue up in the charging sequence and wait their turn. The number of charging ports is denoted as $M$ and the coordinates of the FCS are located at the origin.
+
+The scheduling process of the FCS system is shown in Fig. 1. A complete charging scheduling process involves: at regular time intervals $T$ , the FCS collects relevant data from each UAV, confirms the set of UAVs capable of reaching the FCS, and proceeds with the solution of the scheduling scheme. UAVs unable to reach the FCS do not participate in the schedule and terminate their tasks. At the same time, the participating UAVs move to the FCS for charging. Upon arrival at the station, each UAV lines up at the charging port in the charging sequence provided by the FCS. After finish charging, they will autonomously return to the mission site to continue their missions. When all the UAVs involved in the scheduling return to their mission sites, the scheduling process for this round ends.
+
+TABLE I: Description of the symbol
+
+max width=
+
+Symbol Description Unit
+
+1-3
+${C}_{i}$ The battery capacity of the UAV $i$ Wh
+
+1-3
+${I}_{i}$ The initial charge of the UAV $i$ Wh
+
+1-3
+${v}_{i}$ The flying speed of the UAV $i$ km/h
+
+1-3
+${x}_{i}$ The abscissa of the UAV $i$ $\mathrm{{km}}$
+
+1-3
+${y}_{i}$ The ordinate of the UAV $i$ km
+
+1-3
+$P{f}_{i}$ The mobile power of the UAV $i$ W
+
+1-3
+$P{c}_{i}$ The charging power of the UAV $i$ W
+
+1-3
+$N$ The total number of UAVs /
+
+1-3
+$M$ The number of charging ports for the charging station /
+
+1-3
+
+ < g r a p h i c s >
+
+Fig. 1: Scheduling process of FCS system
+
+During the simulation, it is observed that the time required to solve the scheduling solution is significantly shorter than the arrival time of the UAVs. Therefore, it is reasonable to allow the UAVs to travel to the FCS at the beginning of the scheduling process.
+
+2) Problem Formulation: We choose the optimization objective as the total duration from the start of the charging schedule until all UAVs return to their respective task points to resume task execution. This aims to minimize the time wasted by UAVs due to charging.
+
+To calculate the charging sequence, the first step is to assess the eligibility of UAVs for scheduling. The time at which UAV $i$ arrives at the charging location is represented as:
+
+$$
+{t}_{i} = \frac{{S}_{i}}{{v}_{i}} \tag{1}
+$$
+
+where ${v}_{i}$ represents the flying speed of UAV $i.{S}_{i}$ is the distance from UAV $i$ to the charging station, where ${S}_{i} =$ $\sqrt{{x}_{i}^{2} + {y}_{i}^{2}}$ . The condition that the remaining battery level of UAV $i$ needs to satisfy upon reaching the charging location can be represented as:
+
+$$
+{I}_{i} - {t}_{i}P{f}_{i} \geq 0, \tag{2}
+$$
+
+where ${I}_{i}$ is the initial battery level of UAV $i$ and $P{f}_{i}$ is the flight power of UAV $i$ .If condition (2) is satisfied, UAV $i$ participates in the current round of scheduling; otherwise, it does not participate. Then, update the value of parameter $N$ to represent the total number of UAVs participating in the scheduling.
+
+At each moment, the number of UAVs currently charging must not exceed the number of charging ports available at the charging point. This constraint can be represented as:
+
+$$
+\mathop{\sum }\limits_{{i = 1}}^{N}{\chi }_{i} \leq M \tag{3}
+$$
+
+where ${\chi }_{i}$ is a binary function. When UAV $i$ is charging, ${\chi }_{i}$ equals 1 ; otherwise, it equals 0 .
+
+The scheduling solution provided by the model is the charging sequence of the UAVs, which is a permutation of the numbers from 1 to $N$ .This permutation is denoted as $p$ , $p \in {P}_{N}.{P}_{N}$ is the set consisting of all permutations containing the numbers 1 to $N.p\left\lbrack j\right\rbrack$ is the $j$ -th element in the permutation $p$ , representing the UAV index positioned at the $j$ -th slot in the permutation. This implies that for $p\left\lbrack j\right\rbrack = k,j$ represents the charging order of the UAV with index $k$ . For consistency, we denote in the permutation $p$ that $p\left\lbrack {j}_{i}\right\rbrack = i$ , where ${j}_{i}$ represents the charging order of the UAV with index $i$ . After UAVs reach the charging location, they charge sequentially according to the charging order. If there are no available charging ports, they have to wait. $T{w}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right)$ is the waiting time of UAV $i$ in the permutation $p$ . Note that this refers solely to the waiting time of UAV $i$ in the context of permutation $p$ , and not as a complex function. Similarly, $T{c}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right)$ represents the charging time of UAV $i$ in the permutation $p.{t}_{i}$ is denoted as ${t}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right)$ for consistency. The total duration of charging and movement for UAV $i$ in the permutation $p$ can be represented as:
+
+$$
+{t}_{{sum}_{i}}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right) = 2{t}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right) + T{w}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right) + T{c}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right) , \tag{4}
+$$
+
+where ${t}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right)$ needs to be calculated twice because the UAV needs to travel to the charging location and then return to the task location. After determining the charging schedule time for each UAV, taking the maximum value gives the total duration of the scheduling, denoted as:
+
+$$
+{TI}\left( p\right) = \mathop{\max }\limits_{{i \in N}}{t}_{{su}{m}_{i}}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right) , \tag{5}
+$$
+
+where ${TI}\left( p\right)$ is the total duration of the scheduling under the permutation $p$ . The UAV charging scheduling problem can be formulated as:
+
+$$
+\min \;{TI}\left( p\right)
+$$
+
+$$
+\text{ s.t. }\;{C1} : \;{I}_{i} - {t}_{i}\left( {p\left\lbrack {j}_{i}\right\rbrack }\right) P{f}_{i} \geq 0
+$$
+
+$$
+{C2} : \;\mathop{\sum }\limits_{{i = 1}}^{N}{\chi }_{i} \leq M \tag{6}
+$$
+
+$$
+{C3} : \;p \in {P}_{N},
+$$
+
+where ${C1}$ requires that the UAV $i$ must have sufficient power to reach the charging location. ${C2}$ requires that the number of UAVs charging simultaneously does not exceed the number of charging ports available at the charging location. ${C3}$ represents the solution of the problem as a charging order arrangement.
+
+§ B. MOBILE CHARGING STATION SYSTEM MODEL
+
+1) MCS System: The mobile charging station system consists of a mobile charging station and several UAVs operating tasks in the vicinity of the MCS. The only difference between the FCS and MCS systems is that the charging location in the former is fixed, while that in the latter is mobile. Therefore, the scheduling problem is divided into two parts: the selection of MCS location and the scheduling of UAVs.
+
+ < g r a p h i c s >
+
+Fig. 2: Scheduling process of MCS system
+
+The scheduling process of the MCS system is shown in Fig. 2. A complete charging scheduling process for the MCS system involves: at regular time intervals $T$ , the MCS gathers relevant data from the UAVs to determine the MCS location. Once the location $L$ for the MCS is determined, both the MCS and UAVs capable of reaching location $L$ proceed to $L$ simultaneously. The charging sequence scheme is calculated immediately after selecting location $L$ . The round of charging scheduling process is considered complete when all UAVs involved in the charging schedule have returned to their respective mission points. At the beginning of each scheduling round, the coordinates of the MCS are initialized to the origin.
+
+2) Problem Formulation: For selecting the MCS location, the ideal location $L$ is characterized by the charging station being able to accommodate more UAVs for scheduling, while the total distance for UAVs to reach the location is shorter. We assume that the selected location is $L = \left( {{x}_{L},{y}_{L}}\right)$ ; the distance from UAV $i$ to location $L$ is ${S}_{i}$ , where ${S}_{i} =$ $\sqrt{{\left( {x}_{i} - {x}_{L}\right) }^{2} + {\left( {y}_{i} - {y}_{L}\right) }^{2}}$ . For the FCS system, the values of ${x}_{L}$ and ${y}_{L}$ are always 0 . Define the variable ${\beta }_{i}$ to indicate whether UAV $i$ participates in scheduling, as follows:
+
+$$
+{\beta }_{i} = \left\{ \begin{array}{ll} 1 & \text{ if }{I}_{i} - {t}_{i}P{f}_{i} \geq 0 \\ 0 & \text{ otherwise, } \end{array}\right. \tag{7}
+$$
+
+where ${I}_{i} - {t}_{i}P{f}_{i} \geq 0$ indicates that UAV $i$ has enough power to reach the charging location. Then, the location selection problem can be formulated as:
+
+$$
+\mathop{\min }\limits_{{{x}_{L},{y}_{L}}} - \mathop{\sum }\limits_{{i = 1}}^{N}{\beta }_{i} + \alpha \mathop{\sum }\limits_{{i = 1}}^{N}{S}_{i}{\beta }_{i} \tag{8}
+$$
+
+where $\alpha = {0.01},\mathop{\sum }\limits_{{i = 1}}^{N}{\beta }_{i}$ and $\mathop{\sum }\limits_{{i = 1}}^{N}{S}_{i}{\beta }_{i}$ represent the total number of UAVs participating in scheduling and the sum of the distances from these UAVs to the charging location. Setting a small value for $\alpha$ ensures that the number of UAVs participating in scheduling is prioritized before optimizing the total distance.
+
+The scheduling after selecting the charging station location is the same as that in the FCS system. Therefore, it will not be elaborated upon further.
+
+Algorithm 1 Selection of MCS location
+
+§ INITIALIZATION:
+
+1: The abscissa of the UAV $i{x}_{i}$ , the ordinate of the UAV
+
+ $i{y}_{i}$ , the movement speed of the UAV $i{v}_{i}$ , the battery
+
+ capacity of the UAV $i{C}_{i}$ , the initial charge of the UAV $i$
+
+ ${I}_{i}$ , the mobile power of the UAV ${iP}{f}_{i}$
+
+§ ITERATION:
+
+ 1: Through the battery capacity, the movement speed, the
+
+ battery capacity, the initial charge and the mobile power
+
+ of UAVs, calculate the movement radius ${r}_{i}$ of UAVs
+
+$$
+{r}_{i} = \frac{{C}_{i} - {I}_{i}}{P{f}_{i}}{v}_{i}
+$$
+
+2: Take the coordinate $\left( {{x}_{i},{y}_{i}}\right)$ of UAVs as the center of the
+
+ circle, and the movement radius ${r}_{i}$ as the radius of the
+
+ circle to make a circular area
+
+3: Find the area with the most overlapping regions to get the
+
+ set of candidate coordinates
+
+ 4: Through traversal or heuristics, select the coordinate point
+
+ that makes the total distance of the objective function the
+
+ shortest $\mathop{\min }\limits_{{{x}_{L},{y}_{L}}} - \mathop{\sum }\limits_{{i = 1}}^{N}{\beta }_{i} + \alpha \mathop{\sum }\limits_{{i = 1}}^{N}{S}_{i}{\beta }_{i}$
+
+Output: The location $L$
+
+§ C. ASSUMPTIONS
+
+1) Measurements of the initial power of the UAV, the coordinates of the UAV relative to the FCS/MCS, and the power consumed by the UAV are accurate.
+
+2) The communication time between the drone and the FCS/MCS is neglected.
+
+3) The time required to change the charging interface for the UAV is neglected.
+
+4) The time for the UAV to reach the charging location is always greater than the time required by the FCS/MCS to determine the scheduling plan.
+
+5) After selecting the charging location, the MCS always arrives at the charging site before the UAVs.
+
+Assumptions 1)-3) are common and weak assumptions on the UAV charging service problem. Simulation results indicate that the computation time for the MCS/FCS scheduling scheme ranges from a few seconds to several minutes, which is significantly shorter than the time required for the drone to reach the charging station. Since the MCS is located roughly in the center of the swarm, the charging location will be relatively close to the initial MCS location. The MCS moves faster than UAVs, which ensures that it reaches the charging location faster. Therefore, Assumptions 4)-5) are reasonable.
+
+§ III. OPTIMIZATION ALGORITHMS
+
+§ A. SCHEDULING OF UAVS
+
+It can be seen that the optimization problem we have established is a variant of the traveling salesman problem. Therefore, heuristic algorithms can be used to solve it. We select simulated annealing algorithm (SA), genetic algorithm (GA), tabu search algorithm (TS), and particle swarm optimization algorithm (PSO). By comparing the performance of four heuristic algorithms, we chose the simulated annealing algorithm. Details can be found in Section IV.
+
+TABLE II: The parameter settings for each algorithm
+
+max width=
+
+Heuristics Parameter settings
+
+1-2
+PSO The individual learning factor is 0.5 ; the social learning factor is 0.3 ; the inertia factor is 1 ; the number of particles is 30 ; the maximum inertia factor is 1 ; the minimum inertia factor is 0.8 .
+
+1-2
+GA The number of immunized individuals is 30 ; the crossover probability is 0.95 ; the mutation probability is 0.1 .
+
+1-2
+TS The number of neighborhood solutions: $N \times \left( {N - 1}\right) /2$ for $N \leq {10},{50}$ otherwise; the number of candidate solutions is 25; the taboo length is ${\left( N \times \left( N - 1\right) /2\right) }^{0.5}$ .
+
+1-2
+SA The initial temperature is ${33} \times N$ ; the number of cycles in the inner layer is ${26} \times N$ ; the temperature drop rate is 0.98 ; accept the new solution with probability ${e}^{-{18} \times {\Delta E}/{T}_{\text{ init }}}$ .
+
+1-2
+
+§ B. SELECTION OF MCS LOCATION
+
+Based on the objectives of maximizing the number of UAVs involved in scheduling and minimizing the total distance traveled by the UAVs, an algorithm can be designed to solve the problem. First, the circles representing the reachable areas of UAVs are calculated based on UAVs data. Then, the region with the highest overlap of these circles is identified, representing the area accessible to all UAVs. Within this area, the point that minimizes the total distance for UAVs to reach the charging location is determined as location $L$ .
+
+ < g r a p h i c s >
+
+Fig. 3: Comparison of relative errors among four algorithms
+
+ < g r a p h i c s >
+
+Fig. 4: Comparison of scheduling duration for SA, PSO, TS, and GA with different numbers of UAVs
+
+TABLE III: Comparison of results between FCS and MCS
+
+max width=
+
+System Model The number of UAVs Calculation duration Scheduling duration The total distance traveled by the UAV Location coordinates of the charging vehicle
+
+1-6
+5*FCS 10 1.604189s ${1.5147}\mathrm{\;h}$ ${48.1646}\mathrm{\;{km}}$ $\left\lbrack {0,0}\right\rbrack$
+
+2-6
+ 15 3.705172s ${3.5737}\mathrm{\;h}$ 62.4658km $\left\lbrack {0,0}\right\rbrack$
+
+2-6
+ 20 5.776012s 2.2079h 99.3726km $\left\lbrack {0,0}\right\rbrack$
+
+2-6
+ 25 9.441099s 3.7836h 93.0275km $\left\lbrack {0,0}\right\rbrack$
+
+2-6
+ 30 12.593925s 3.1608h 144.8695km $\left\lbrack {0,0}\right\rbrack$
+
+1-6
+5*$\mathbf{{MCS}}$ 10 1.815212s 1.4937h 34.6202km $\left\lbrack {-4,4}\right\rbrack$
+
+2-6
+ 15 3.766389s 3.4914h ${33.117}\mathrm{\;{km}}$ [3,2]
+
+2-6
+ 20 6.124744s 1.9614h 77.5759km [4, 4]
+
+2-6
+ 25 9.470387s ${3.7803}\mathrm{\;h}$ 73.9708km $\left\lbrack {0,2}\right\rbrack$
+
+2-6
+ 30 13.224203s 3.1273h 119.9724km $\left\lbrack {3,2}\right\rbrack$
+
+1-6
+
+ < g r a p h i c s >
+
+Fig. 5: Comparison of UAV distance traveled and scheduling duration for FCS and MCS
+
+To simplify the problem, only integer points on a coordinate grid (points where both the horizontal and vertical coordinates are integers) are considered. Initially, set the value of each integer point to 0 . Each time a circular area covers a point, increment the value of that point by 1 . By selecting the integer points with the highest values, a finite set of integer points can be obtained as candidate locations. We designate this set as $G$ , where the elements in $G$ are the coordinates of points, with elements in the form such as(1,2). Subsequently, by traversing and solving or employing a heuristic algorithm, the location $L$ can be determined.
+
+The pseudo code for the algorithm addressing this problem is provided in Algorithm 1.
+
+§ IV. NUMERICAL EXPERIMENTS AND ANALYSIS
+
+For testing the FCS system model, we set the parameter $M = 2$ . Based on simulation experiments, it is found that, even with 30 UAVs, the scheduling time remains within 4 hours. Therefore, we set the scheduling cycle parameter $T = 4$ (which can be adjusted according to the number of drones and the number of charging ports at the charging station). Since the optimal solution can be obtained through exhaustive search when the number of UAVs is 10 or fewer, this part compares the relative error of the solution results of four heuristic algorithms against the optimal solution with the number of UAVs ranging from 3 to 10 . The relative error is defined as the difference between the solution obtained by the algorithm and the optimal solution, divided by the optimal solution. The algorithm parameters are set as shown in Table II.
+
+The comparison results of the algorithms are shown in Fig. 3, where the relative errors of all four heuristic algorithms are less than ${0.9}\%$ . This indicates the effectiveness of the heuristic algorithms in solving this problem. It is noteworthy that the relative error of SA is generally lower. To compare the performance of heuristic algorithms with a larger number of UAVs, we test the four algorithms with 10,20, and 30 UAVs, as shown in Fig. 4. Fig. 4 shows that SA consistently outperforms the other three algorithms. This is the reason for selecting SA for this problem.
+
+For testing the MCS system model, we set the parameter $M = 4$ and employ SA for solution. We compare the results of the MCS and FCS systems under the same drone data, as shown in Table III and Fig. 5. It can be seen that, compared to the FCS system, the MCS system results in shorter scheduling times and a significant reduction in the total distance traveled by UAVs. The time required to solve the MCS system is similar to that of the FCS system. This indicates that the MCS system provides a better charging scheduling solution while maintaining similar solution speed, demonstrating the effectiveness and superiority of this model.
+
+§ V. CONCLUSION
+
+A scheduling system with fixed charging station is described. Through simulation, it is concluded that the simulated annealing algorithm is the most suitable method among the four algorithms for solving this problem. Compared to GA, PSO, and TS, SA demonstrates superior performance in solving large-scale drone problems. Based on FCS system, a scheduling system is proposed that replaces fixed charging station with a mobile charging station. For the MCS system, a two-stage optimization model is established, including the selection of MCS location and the scheduling of drone charging. By comparing the results of two models, it is validated that the MCS system model can provide a more optimal scheduling solution to meet practical needs. The future work is to study large scheduling systems with relay charging platforms and to design efficient algorithms for determining charging locations.
+
+§ ACKNOWLEDGMENT
+
+This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 62173087. Corresponding author: Zeyu Guo.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/d9xHa3zSc0/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/d9xHa3zSc0/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..1bd96a4ae85cdeb44cc79d919e63421063ec77ce
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/d9xHa3zSc0/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,359 @@
+# Improved Catch Fish Optimization Algorithm with Personalized Fishing Strategy for Global Optimization
+
+${1}^{\text{st }}$ Bowen Xue
+
+School of Electrical and Information
+
+Engineering
+
+Northeast Petroleum University
+
+Daqing, China
+
+xuebowen@stu.nepu.edu.cn
+
+${2}^{\text{nd }}$ Heming Jia*
+
+School of Information Engineering
+
+Sanming University
+
+Sanming, China
+
+jiaheming@fjsmu.edu.cn
+
+${3}^{\text{rd }}$ Honghua Rao
+
+School of Electrical and Information
+
+Engineering
+
+Northeast Petroleum University
+
+Daqing, China
+
+20200862235@fjsmu.edu.cn
+
+${4}^{\text{th }}$ Jinrui Zhang
+
+School of Information Engineering
+
+Sanming University
+
+Sanming, China
+
+ruiruiz2308@163.com
+
+${5}^{\text{th }}$ Yilong Du
+
+School of Information and Electrical Engineering Heilongjiang Bayi Agricultural University
+
+Daqing, China
+
+wy15093488812@163.com
+
+${6}^{\text{th}}$ Zekai Ai
+
+College of Design and Engineering
+
+National University of Singapore
+
+Singapore
+
+aizekai@u.nus.edu
+
+Abstract-Catch Fish Optimization Algorithm (CFOA) is a new meta-heuristic optimization algorithm with human behavior. In this algorithm, search agents simulate the process of rural fishermen fishing in the pond. Therefore, the CFOA generally consists of two phases of the update: the exploration phase and the exploitation phase. However, it still falls under the local optimum and has a low convergence rate. To this end, we propose an improved catch fish optimization algorithm(ICFOA) based on personalized fishing strategies. First, the adaptive Gaussian perturbation is adopted to the exploration stage process to increase the global search capability, expand the search range, and improve efficiency while avoiding falling into the local optima. Then, based on the personalized fishing strategy, the personal position of fishermen is updated by randomly selecting "freehand fishing" factors or "using fishing net" factors to accelerate the algorithm's convergence speed. Furthermore, comparative experiments were performed using the CEC2020 test suite to compare the performance of ICFOA and other excellent meta-heuristics. Finally, Wilcoxon's rank-sum test was used to verify the validity of our statistical experimental results. Moreover, the performance of ICFOA in reducer design also indicates that ICFOA can get the optimal solution in solving practical engineering optimization problems. The results show that ICFOA has a more competitive performance than the original $\mathbf{{CFOA}.}$
+
+Keywords-Catch Fish Optimization Algorithm, adaptive Gaussian perturbation, Personalized Fishing Strategy
+
+## I. INTRODUCTION
+
+In the current era of rapid technological advancement, optimization problems hold a critical position across various domains, including engineering design, economic management, and computer science. Examples include the Crayfish Optimization Algorithm (COA) [1], Whale Optimization Algorithm (WOA) [2], and Grey Wolf Optimization (GWO) [3]. COA excels in exploration but may converge slowly. WOA balances exploration and exploitation well but can get trapped in local optima. GWO is strong in convergence but requires careful parameter tuning. Human behavior-based optimization algorithms are a class of optimization techniques designed to tackle complex optimization problems by emulating human or other biological behaviors and decision-making processes. By mimicking natural phenomena such as evolution, foraging, and social interactions, these algorithms can effectively search and optimize complex solution spaces.
+
+In solving problems related to economic scheduling, functional optimization, and engineering design, human behavior-based optimization algorithms are especially adept at avoiding local optima and discovering global optima or solutions close to the global optimum. For instance, the Human Behavior-Based Optimization (HBBO) [4] algorithm models human behavior patterns, particularly focusing on how humans learn and solve problems through interaction and communication. This algorithm integrates multiple human behavioral traits, such as experiential learning, imitation, social interaction, and collaboration, to achieve efficient search and optimization in complex problems.
+
+Heming Jia et al. [5] In 2024, an innovative optimization algorithm inspired by human behavior, namely the catch fish optimization algorithm(CFOA). The main inspiration for the CFOA comes from the fishing practices of the fishermen. The CFOA contains updated rules based on different fishing practices. As intelligent humans, fishermen often use a variety of ways to find fish, such as sharing fishing experiences, using different fishing tools, etc., so their location update rules are based on individuals and teams. Furthermore, as capture rates decline, fishermen will choose whether to change their fishing strategy. Experimental results show that the proposed algorithm outperforms others in finding the optimal solution and convergence speed. However, as stated by the NFL Theorem [6], given the diversity and complexity of optimization problems, no universal algorithm can be directly applied to address all types of optimization challenges. This reality requires the exploration and adoption of more rigorous and targeted strategies to continuously improve and optimize the algorithm design.
+
+Just as the problems faced by many optimization algorithms, the original CFOA algorithm found it difficult to completely avoid the limitations of low convergence efficiency and easily fall into local optimal solutions in specific optimization tasks. Given this, the optimization and upgrading of the CFOA algorithm can not only improve the efficiency of its algorithm but also broaden its scope of application. Therefore, this paper presents an improved CFOA algorithm (ICFOA) based on a personalized fishing strategy. PFS greatly enhances the solving performance of CFOA in complex optimization problems. At the same time, the position of the fishermen is updated based on the personalized fishing strategy, which not only makes the algorithm more detailed and comprehensive when searching for the solution space but also enhances the algorithm's ability to escape local optima and find the global optimal solution. Finally, to test ICFOA, to test the improved optimization algorithm, this paper utilizes ten commonly used benchmark functions and chooses the optimization algorithm (CFOA) for evaluation. and five representative meta-heuristic algorithms for comparative experiments to validate the effectiveness and advantages of ICFOA.
+
+---
+
+* Corresponding author.
+
+This work is supported by the Natural Science Foundation of Fujian Province under Grant 2021J011128.
+
+---
+
+The remainder of this paper is structured as follows: Section II provides the concept of the original CFOA, Section III details the proposed algorithm ICFOA, Sections IV and V demonstrate the experiment analysis in comparison with several popular metaheuristics under the CEC2020 test suite and Wilcoxon's rank-sum test, and Section VI concludes.
+
+## II. CATCH FISH OPTIMIZATION ALGORITHM
+
+The CFOA simulates the fishing behavior of village fishermen. To catch fish more easily, fishermen choose different fishing methods to catch fish. Similar to other metaheuristic algorithms (MAs), CFOA consists of three distinct stages: initialization, exploration, and exploitation.
+
+## 1) Initialization phase
+
+The matrix $F$ represents the location data of $N$ search agents in a $d$ -dimensional space, and the formula is shown below:
+
+$$
+F = {\left\lbrack \begin{matrix} {F}_{1,1} & {F}_{1,2} & \cdots & {F}_{1, n} \\ {F}_{2,1} & {F}_{2,2} & \cdots & {F}_{2, n} \\ \vdots & \vdots & \ddots & \vdots \\ {F}_{n,1} & {F}_{n,2} & \cdots & {F}_{n, n} \end{matrix}\right\rbrack }_{N \times d} \tag{1}
+$$
+
+$$
+{F}_{i, j} = l{b}_{j} + \left( {u{b}_{j} - l{b}_{j}}\right) \times \text{ rand } \tag{2}
+$$
+
+The matrix $\mathrm{F}$ represents the position information of $\mathrm{N}$ search agents within a d-dimensional space. Its initialization formula is as follows: $\mathrm{{Fi}},\mathrm{j}$ denotes the position of the ith agent in the jth dimension, where ubj and lbj represent the maximum and minimum limits of the jth dimension, respectively, rand is a random number in the interval(0,1).
+
+Using the current position data of each fisherman, we apply the fitness evaluation function fobj to determine their fitness scores, yielding the following fitness matrix:
+
+$$
+f = \operatorname{fobj}\left( F\right) = \left\lbrack \begin{matrix} {f}_{1} \\ {f}_{2} \\ \vdots \\ {f}_{N} \end{matrix}\right\rbrack \tag{3}
+$$
+
+In the above formula, ${f}_{1}$ represents the fitness value of the first fisherman, ${f}_{2}$ denotes the fitness value of the second fisherman, and so on. We use a value of 0.5 to evenly distribute the balance between exploitation and exploration across iterations. In the initial part of the phase (when $\mathrm{{EFs}}/\mathrm{{MaxEFs}} < {0.5}$ ), individuals focus on global exploration, while during the latter part of the phase (when $\mathrm{{EFs}}/$ MaxEFs $\geq {0.5}$ ), they shift towards exploitation.
+
+## 2) Individual and group fishing (exploration phase)
+
+When fishermen explore, initially mainly through independent search and using group encirclement as an aid. As the exploration proceeds, the environmental advantages gradually shift from the fish side to the fishermen. In addition, continuous capture will lead to a decrease in fish population and capture rate. Fishermen will shift from independent exploration to mainly relying on collective encirclement, with personal strengths as assistance. The transformation in this mode is modeled using the capture rate parameter, expressed as $\delta$ .
+
+$$
+\delta = {\left( 1 - \frac{3 \times {EFs}}{2 \times {MaxEFs}}\right) }^{\frac{3 \times {EFs}}{2 \times {MaxEFs}}} \tag{4}
+$$
+
+where ${EFs}$ and ${MaxEFs}$ indicate the current number and maximum number of estimates, respectively.
+
+## a) Individual fishing(when ${EFs}/{MaxEFs} < {0.5}$ )
+
+Fishermen disturb the water to float the fish, determine the position of the fish and adjust the direction of exploration. The update formula is as follows:
+
+$$
+{Exp} = \frac{{f}_{i} - {f}_{\text{pos }}}{{f}_{\max } - {f}_{\min }} \tag{5}
+$$
+
+$$
+R = \text{ Dis } \times \sqrt{\left| \text{ Exp }\right| } \times \left( {1 - \frac{\text{ EFs }}{\text{ Max EFs }}}\right) \tag{6}
+$$
+
+$$
+{F}_{i, j}^{T + 1} = {F}_{i, j}^{T} + \left( {{F}_{{pos}, j}^{T} - {F}_{i, j}^{T}}\right) \times \text{ Exp } \tag{7}
+$$
+
+$$
++ \operatorname{rand} \times s \times R
+$$
+
+In the formula mentioned above, ${Exp}$ represents the empirical analysis value obtained by the $i$ -th fisherman using any other fisherman $p$ (where ${pos} = 1,2\cdots$ or $N, p \neq i$ ) as the reference object, with values ranging from -1 to $1.{f}_{\max }$ and ${f}_{\min }$ represent the lowest and highest fitness values, respectively, following the Tth complete position update. $T$ is the number of iterations fishermen’s positions. ${F}_{i, j}^{T}$ and ${F}_{i, j}^{T + 1}$ are position of the ${i}^{\text{th }}$ fisherman in $j$ -dimension after the iterations of ${T}^{\text{th }}$ and ${\left( T + I\right) }^{\text{th }}$ . Dis denotes the Euclidean distance between the $i$ -th individual and the reference point, while $s$ is a random unit vector in $d$ dimensions.
+
+## b) group fishing (when ${EFs}/{MaxEFs} \geq {0.5}$ )
+
+Fishermen utilize nets to enhance their fishing efficiency and collaborate with each other. They organize into random groups of 3-4 members to collectively encircle potential targets. By leveraging their individual mobility, they can explore the area more comprehensively and accurately. The corresponding formula are outlined below:
+
+$$
+\text{Centre} = \operatorname{mean}\left( {F}_{c}^{T}\right) \tag{8}
+$$
+
+$$
+{F}_{c, i, j}^{T + l} = {F}_{c, i, j}^{T} + {r}_{2} \times \left( {{\text{ Centre }}_{c} - {F}_{c, i, j}^{T}}\right) + {\left( 1 - \frac{2 \times {EFs}}{MaxEFs}\right) }^{2} \times {r}_{3} \tag{9}
+$$
+
+Where $c$ represents a cluster of 3 to 4 individuals whose positions remain unaltered. Centre ${}_{c}$ is the target point for group $c$ ’s encirclement. ${F}_{c, i, j}^{T + 1}$ and ${F}_{c, i, j}^{T}$ are the position of the ${i}^{\text{th }}$ fisherman in group $c$ in the $j$ -dimension after the ${\left( T + I\right) }^{\text{th }}$ and ${T}^{\text{th }}$ updates. ${r}_{2}$ represents the speed at which a fisherman moves toward the center, varying individually and falling within the range of $\left( {0,1}\right) .{r}_{3}$ is the offset of the move, ranging from(-1,1), and decreases progressively as EFs increase.
+
+3) Collective capture (exploitation phase)
+
+All fishermen searched under a uniform strategy, purposefully bringing hidden fish to the same location and around. The position of the fishermen during the trapping process is updated as follows:
+
+$$
+\sigma = \sqrt{\left( \frac{2\left( {1 - \frac{EFs}{MaxEFs}}\right) }{\left( {\left( 1 - \frac{EFs}{MaxEFs}\right) }^{2} + 1\right) }\right) } \tag{10}
+$$
+
+$$
+{F}_{i}^{T + 1} = \text{ Gbest } + {GD}\left( {0,\frac{{r}_{4} \times \sigma \times \left| {\text{ mean }\left( F\right) - \text{ Gbest }}\right| }{3}}\right) \tag{11}
+$$
+
+Within this group, GD is a Gaussian distribution function with a mean $\mu$ of 0, and its overall variance $\sigma$ decreases from 1 to 0 as the number of evaluations increases. The position of the ${i}^{\text{th }}$ fisherman after the ${\left( T + I\right) }^{\text{th }}$ update. Mean(F)signifies the matrix of mean values for each dimension at the center of the fishermen's positions, while Gbest indicates the global optimum.
+
+## III. Proposed Algorithm
+
+## A. Adaptive Gaussian Perturbation (AGP)
+
+Adaptive Gaussian Perturbation dynamically enables the optimization algorithm to flexibly balance exploration and exploitation in different iteration stages as follows:
+
+$$
+{\sigma }_{p} = \left( {1 - \frac{EFs}{MaxEFs}}\right) \cdot \operatorname{std}\left( {F}_{i}^{T}\right) \cdot \exp \left( {-\frac{EFs}{{MaxEFs}/{10}}}\right) \tag{12}
+$$
+
+$$
+{F}_{i}^{T + 1} = {F}_{i}^{T} + \mathcal{N}\left( {0,{\sigma }_{p}^{2}}\right) \tag{13}
+$$
+
+where ${\sigma }_{p}$ is the perturbation strength of the current iteration, The std is the standard deviation of the current solution, $\mathcal{N}\left( {0,{\sigma }_{p}^{2}}\right)$ is a normally distributed random variable with mean 0 and variance ${\sigma }_{p}^{2}$ .
+
+## B. Personalized Fishing Strategy (PFS)
+
+PFS is a widely adopted and effective strategy that enhances an algorithm's optimization capability within the search space. Overall, PFS enhances the exploitation capability and accelerates the convergence speed of metaheuristic algorithms (MAs). The principle of PFS is to generate random actions $\alpha$ and $\beta$ based on the original action step, update the position according to different movements, and change according to the capture parameter $\delta$ , making the whole fishing process more closely related.
+
+$$
+{F}_{i}^{T + 1} = {F}_{i}^{T} + \text{ step } \cdot {kd} \tag{14}
+$$
+
+$$
+{kd} = {0.5} + {0.5} * \text{rand} \tag{15}
+$$
+
+where step represents action choice and kd indicates the action skill proficiency of fishermen, whose value is $\left\lbrack {{0.1},1}\right\rbrack$ . The personalized fishing strategy includes two factors, "freehand capture" and "using tools", which are used to improve the global search capability of the algorithm. The updated formula is shown as follows:
+
+$$
+\begin{array}{l} \text{ step } = \left\{ \begin{array}{l} \alpha = 2 * \left( {1 - \frac{EFs}{MaxEFs}}\right) * \left( {{0.4} * \delta }\right) * \text{ rand } > {0.5} \\ \beta = 1 * \left( {1 - \frac{EFs}{MaxEFs}}\right) \text{ otherwise } \end{array}\right. \end{array} \tag{16}
+$$
+
+## C. Details of ICFOA
+
+CFOA is widely used and easy to implement for optimization tasks. However, its search capabilities (exploration and exploitation) are limited when tackling complex problems, making convergence within a finite number of iterations difficult or even unattainable. To address these challenges, this paper integrates PFS to enhance ICFOA for global optimization. The PFS improves exploitation efficiency and accelerates the convergence rate of the conventional CFOA.
+
+The pseudo-code of ICFOA is shown in Algorithm 1, and Fig. 1 illustrates the flowchart of ICFOA.
+
+Algorithm 1 the pseudo-code of ICFOA
+
+---
+
+ Initialization parameters
+
+ Initialize the population Fisher
+
+ While (EFs $\leq$ MaxEFs)
+
+ Reckon the values of fit and get the globally optimal
+
+solution(Gbest)
+
+ if EFs/MaxEFs $< {0.5}$
+
+ Reckon the values of $\delta$ by Eq. (4)
+
+ Randomly shuffle the order of each fisherman
+
+ If $\mathrm{p} < \delta$
+
+ Using Eq. (7) to update new position of fisher
+
+ Else
+
+ Randomly group the fisherman
+
+ Using Eq. (9) to update new position of fisher
+
+ End
+
+ Else
+
+ Using Eq. (13) to update new position of fisher
+
+ End
+
+ End
+
+ Output the globally optimal solution
+
+---
+
+
+
+Fig. 1. Flowchart of the proposed algorithm ICFOA
+
+## D. Computation Complexity of ICFOA
+
+Initialization and position updates are the core components of ICFOA. The computational complexity of initialization is $\mathrm{O}\left( {N \times \text{Dim }}\right)$ , where $N$ represents the population size and Dim denotes the dimensionality. The complexity of fishing with bare hands and using fishing nets varies and can reach up to $\mathrm{O}\left( {T \times N \times \text{Dim }}\right)$ , where $T$ is the maximum number of evaluations.Both the PFS and AGP methods have a complexity of $\mathrm{O}\left( {T \times N \times \text{Dim }}\right)$ . Therefore, the overall computational complexity of ICFOA is $\mathrm{O}\left( {\left( {4 \times T + 1}\right) \times N \times \text{Dim}}\right)$ .
+
+## IV. RESULTS OF GLOBAL OPTIMIZATION EXPERIMENTS
+
+This section uses 10 benchmark functions in CEC2020 [7]to evaluate the optimized performance of the proposed working ICFOA. First, the definitions of the 10 benchmark test functions are introduced. Second, the experimental setup and the comparison groups are described in detail, including other well-known MAs.
+
+## A. Definition of 10 Benchmark Functions
+
+Benchmark functions are essential for assessing the performance of various algorithms. This paper selects 10 representative benchmark functions, categorized into: (1) unimodal functions (F1-F4) and (2) multimodal functions (F5-F10). Table 1 provides a detailed description of these functions, and D denotes the dimensionality. Unimodal functions have a single global optimal solution, making them suitable for evaluating the exploitation capability of metaheuristic algorithms (MAs). Conversely, multimodal functions possess multiple local optima and a single global optimum. providing a basis for assessing the exploration capability of MAs and their ability to escape local optima.
+
+TABLE I. DEFINITION OF 10 BENCHMARK FUNCTIONS
+
+| Function | | ICFOA vs. |
| CFOA | ROA | $\mathbf{{AOA}}$ | SCA | SFO | SHO | SCA |
| F1 | ${3.34} \times {10}^{-{01}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F2 | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F3 | ${4.57} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F4 | NaN | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | $\mathbf{{NaN}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F5 | ${4.83} \times {10}^{-{01}}$ | ${1.07} \times {10}^{-{07}}$ | ${3.02} \times {10}^{-{11}}$ | ${3.02} \times {10}^{-{11}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F6 | ${4.62} \times {10}^{-{10}}$ | ${3.02} \times {10}^{-{11}}$ | ${3.02} \times {10}^{-{11}}$ | ${3.02} \times {10}^{-{11}}$ | ${1.21} \times {10}^{-{12}}$ | $\mathbf{{NaN}}$ | ${1.21} \times {10}^{-{12}}$ |
| F7 | ${6.10} \times {10}^{-{03}}$ | ${3.11} \times {10}^{-{01}}$ | ${7.98} \times {10}^{-{02}}$ | ${3.02} \times {10}^{-{11}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F8 | ${1.07} \times {10}^{-{07}}$ | ${3.02} \times {10}^{-{11}}$ | ${1.22} \times {10}^{-{02}}$ | ${3.02} \times {10}^{-{11}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F9 | ${1.21} \times {10}^{-{12}}$ | ${4.57} \times {10}^{-{12}}$ | ${1.22} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
| F10 | $\mathbf{{NaN}}$ | ${1.21} \times {10}^{-{12}}$ | $\mathbf{{NaN}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ | ${1.21} \times {10}^{-{12}}$ |
+
+
+
+
+
+Fig. 2. Convergence of ICFOA and Comparison Algorithm on Some Functions
+
+## V. REDUCER STRUCTURE MODEL DESIGN PROBLEM
+
+The model for the reducer design problem is illustrated in Figure 3. The primary objective is to minimize the mass of the reducer while satisfying the given constraints. This problem involves seven decision variables and eleven constraints. For detailed descriptions of the variables $\mathrm{x}1$ to x11), refer to reference [14]. The mathematical formulation of the problem is identical to that in reference [14].
+
+Table 5 presents the comparison results of ICFOA, CFOA, ROA, AOA, SFO, SHO, and SCA in solving the reducer design problem. It's evident that ICFOA produces strong results while effectively meeting the constraints.
+
+TABLE V. COMPARISON RESULTS OF DIFFERENT OPTIMIZATION ALGORITHMS
+
+| Instituation | Product | Functions and Characteristics |
| Ascenz Marorka | Electronic Fuel Monitoring System(EFMS) | Monitor real-time data on fuel usage. Develop energy management plan to reduce fuel consumption. |
| Germanischer Lloyd | ECO Assistant | Calculate optimum trim angle without making extra devices. Improve the operational fuel efficiency of ships. |
| Jeppesen Marine | Vessel and Voyage Optimization Services (VVOS) | Optimize route for on-time arrival. Voyage plan can be received by ECDIS and INS. |
| NAPA | NAPA Voyage Optimization (NAPA-VO) | Decrease operation cost by improved schedule adjustment procedure. Increase efficiency in altering the voyage plans. |
| ABB | ABB Dynafin | Cutting annual greenhouse emissions by at least ${50}\%$ in future. Decrease the energy consumed by propulsion systems. |
| Hangzhou Yagena Technology Co., LTD. | Intelligent Ship System and Equipment (ISSE) | Calculate the carbon intensity index. Rank the carbon emission level. |
+
+
+
+Fig. 4. (a) Structure of Radiation Pattern for S-IES, (b) Structure of Ring Pattern for S-IES, (c) Structure of Two-End Pattern for S-IES.
+
+
+
+Fig. 5. Structure for S-IES
+
+At present, with the risen on awareness of sustainable development, the economic benefit ${F}_{\mathrm{{EC}}}$ is not the only factor which should be concerned in the energy management of S-IES. For this reason, the environmental ${F}_{\mathrm{{CA}}}$ , and social ${F}_{\mathrm{{SC}}}$ benefits are considered in this paper as well. Moreover, to ensure the high-security sailing and provide high-quality energy to the ships, the utilization of energy is also another major factor. Therefore, the objective function of the energy management problem for S-IES is constructed as,
+
+$$
+\min \left\{ {{F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}},{F}_{\mathrm{{SC}}}}\right\} \text{.} \tag{1}
+$$
+
+Note that, the above objective function represents a compromise between economic benefits and carbon emissions, rather than indicating the smaller of ${F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}}$ and ${F}_{\mathrm{{SC}}}$ as the objective function for S-IES energy management. In addition, physical constraints need to be imposed on ships and energy equipment to achieve secure performance during sailing voyage. Then, the energy management model of S-IES can be established as below.
+
+$$
+\min \left\{ {{F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}},{F}_{\mathrm{{SC}}}}\right\} \text{,} \tag{2}
+$$
+
+$$
+\text{s.t.}A\left( X\right) = 0, B\left( X\right) \leq 0\text{,}
+$$
+
+where, $X$ is the decision variable of the energy management problem, which presents the energy output, $A\left( X\right)$ and $B\left( X\right)$ are equality and inequality constraints, respectively. Specifically, $A\left( X\right)$ contains supply-demand balance constraints, voltage security constraints. $B\left( X\right)$ includes energy output constraints, velocity constraints, which are shown below.
+
+1) Supply-demand balance constraints:
+
+$$
+\sum \left( {{p}_{i}^{\mathrm{{fu}}} + {p}_{i}^{\mathrm{{re}}} + {p}_{i}^{\mathrm{{chp}}} - {p}_{i}^{\mathrm{{ld}}}}\right) = 0 \tag{3}
+$$
+
+$$
+\sum \left( {{h}_{i}^{\mathrm{{fu}}} + {h}_{i}^{\mathrm{{re}}} + {h}_{i}^{\mathrm{{chp}}} - {h}_{i}^{\mathrm{{ld}}}}\right) = 0 \tag{4}
+$$
+
+where $p$ and $h$ are power and heating outputs, respectively. $i, l$ presents the node sequence of S-IES. fu, re, chp, ld are fuel-based generators, renewable-based generators, combining heating and power device, and load, respectively.
+
+2) Voltage security constraints:
+
+$$
+\sum {\alpha }_{i, l}^{\max }{p}_{i} + \Delta {V}_{l}^{\max } = {\pi }_{l}^{\max } \tag{5}
+$$
+
+$$
+\sum {\alpha }_{i, l}^{\min }{p}_{i} + \Delta {V}_{l}^{\min } = {\pi }_{l}^{\min } \tag{6}
+$$
+
+where $\alpha$ notes the voltage sensitivity coefficients, which can be calculated by a given transmission topology of S-IES. ${\Delta V}$ is the voltage excursion index, which indicates the voltage security margin. $\pi$ is a constant influencing by the reactive power. min and max are noted as the minimal and maximal values of the corresponding variables, respectively. Specifically, (5) and (6) present the upper and lower bounds of velocity security boundaries, respectively.
+
+3) Energy output constraints:
+
+$$
+{p}_{i}^{\min } \leq {p}_{i} \leq {p}^{\max },{h}^{\min } \leq {h}_{i} \leq {h}^{\max } \tag{7}
+$$
+
+4) Velocity constraints:
+
+$$
+{v}^{\min } \leq v \leq {v}^{\max } \tag{8}
+$$
+
+where $v$ describes the velocity of the ship.
+
+## B. Communication Topology for S-IES
+
+1) Centralized Energy Management Strategy: It relies on a centralized controller, which has significant advantages in terms of calculation rate and accuracy. The controller collects real-time information on the operation of all energy devices in the S-IES, and calculates the optimal energy management schemes based on the relevant data to complete the centralized control and management for the S-IES. However, the centralized strategy is prone to single point of failure, poor stability, limited scalability and huge investment costs.
+
+2) Decentralized Energy Management Strategy: According to the operating mechanism and working status of the energy equipment, the ship can be divided into multiple compartments, and the decentralized energy management strategy places the controllers in the compartments. Thus the control and management of the local energy system can be realized. Moreover, the decentralized strategy does not require real-time communication between the controllers in each cabin, and the energy equipment distributes the energy by itself, so the system response speed is fast and the scalability is strong. However, due to the lack of collaborative control between the controllers, the global optimization at the system level cannot be achieved. Therefore it is highly susceptible to external interference, which has a negative impact on secure navigation.
+
+However, the flat distributed structure of the energy network of S-IES presents that the traditional centralized and decentralized energy management strategies are no longer applicable, and a fully distributed energy management strategy continues to be utilized to complete the search for the optimal solution in order to achieve the reliable operation of S-IES.
+
+3) Distributed Energy Management Strategy: It relies on the communication network to achieve real-time information exchange between energy devices within the S-IES, so as to achieve complementary interconnection of information between neighbouring energy devices. When a local energy device fails to operate normally, the controllers of each energy device can achieve local interconnections and send signals to the controllers of neighbouring energy devices to ensure safe and stable operation of the S-IES based on the distributed communication network.
+
+## C. Main Algorithm
+
+To simplify the multi-objective functions of the energy management problems, the linear weighted sum method, the energy management model can be reconstructed as below.
+
+$$
+\min \left\{ {{\beta }_{1} \cdot {F}_{\mathrm{{EC}}} + {\beta }_{2} \cdot {F}_{\mathrm{{CA}}} + {\beta }_{3} \cdot {F}_{\mathrm{{SC}}}}\right\} , \tag{9}
+$$
+
+$$
+\text{s.t.}A\left( X\right) = 0, B\left( X\right) \leq 0\text{,}
+$$
+
+where ${\beta }_{1},{\beta }_{2}$ and ${\beta }_{3}$ are constant parameters of ${F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}}$ , and ${F}_{\mathrm{{SC}}}$ , respectively. Moreover, the related parameters satisfy ${\beta }_{1} + {\beta }_{2} + {\beta }_{3} = 1$ .
+
+Define the communication topology of the constructed S-IES as $G = \{ T, B, W\}$ , where $T, B$ , and $W$ are node set, connected edge set and connected weighted sets, respectively. Specifically, $T = \left\lbrack {{v}_{i} \mid i \in \Omega }\right\rbrack$ and $\Omega$ is noted as the energy device set. Moreover, $B \subseteq T \times T$ is related to the node set. $W = \left\lbrack {{w}_{i, l} \mid i, l \in \Omega }\right\rbrack$ , where ${w}_{i, l}$ is the connected weighted parameter between the $i$ th node and the $l$ th node. Considering the different relationships between the $i$ th node and the $l$ th node, ${w}_{i, l}$ can be calculated as below.
+
+$$
+{w}_{i, l} = \left\{ \begin{array}{l} 1/\left( {\left| {N}_{zi}\right| + \left| {N}_{l}\right| + \varepsilon }\right) , l \in {N}_{i} \\ 1 - \mathop{\sum }\limits_{{l \in {N}_{i}}}1/\left( {\left| {N}_{i}\right| + \left| {N}_{l}\right| + \varepsilon }\right) , i = l \\ 0,\text{ otherwise } \end{array}\right. \tag{10}
+$$
+
+where $\varepsilon$ is a small positive constant. ${N}_{i}$ and $\left| {N}_{i}\right|$ are the neighbor set of the $i$ th node and its cardinal number. Similarity, ${N}_{l}$ and $\left| {N}_{l}\right|$ are the neighbor set of the $l$ th node and its cardinal number. When the connected weighted value between $i$ th node and the $l$ th node equals to 0, it means that the information exchange can not occur between the mentioned nodes.
+
+Since the high calculation speed, accuracy, and reliable performed by the alternating direction method of multipliers (ADMM), a fully distributed energy management strategy for S-IES is designed in this paper. It realizes the bi-directional transmissions of energy and information, which improves the reduction of communication resource and is suitable to the flat and distributed structure of S-IES. Then, the main algorithm can be designed as below.
+
+1) Iteration of energy output:
+
+$$
+{X}_{i, k + 1} \in \arg \min \left\{ {f\left( X\right) + \psi + {\lambda }_{i, k}^{\mathrm{T}}{W}^{\mathrm{T}}A{X}_{i, k}}\right\} , \tag{11}
+$$
+
+where $\lambda$ is the incremental cost. $A$ describes the physical relationship among nodes of S-IES, which is defined by physical constraints. $k$ is the sequence of time slot.
+
+2) Iteration of output error:
+
+$$
+{d}_{i, k + 1} = W{d}_{i, k} + A\left( {{X}_{i, k + 1} - {X}_{i, k}}\right) , \tag{12}
+$$
+
+where $d$ is the output error of S-IES.
+
+3) Iteration of incremental cost:
+
+$$
+{\left\lbrack {\overset{\overleftrightarrow{} }{\lambda }}_{{\Omega }_{\mathrm{W}}^{z}, k + 1},{\lambda }_{z, k + 1}\right\rbrack }^{\mathrm{T}} = W{\left\lbrack {\overset{\overleftrightarrow{} }{\lambda }}_{{\Omega }_{\mathrm{W}}^{z}, k},{\lambda }_{z, k}\right\rbrack }^{\mathrm{T}} + \tau {d}_{z, k + 1}. \tag{13}
+$$
+
+Repeat the iterations of energy output, output error, and incremental cost until each variable converge a preset threshold.
+
+## III. Simulation Results
+
+To verify the effectiveness of the designed algorithm, a 5- node S-IES is utilized as a test system, and the detailed physical/communication topology containing connected weighted parameters is shown in Fig.6. Moreover, in the test system, there are 2-fu, 1-re, 2-ld, and the operational coefficients and carbon emission parameters are presented in (14). Assume the load demand of power and heating are $\left\lbrack {{4.5},{2.4}}\right\rbrack \left( \mathrm{{MW}}\right)$ , respectively.
+
+$$
+{C}_{1}^{\mathrm{{fu}}} = {0.040} * {h}^{2} + {25} * h + {99}
+$$
+
+$$
+{C}_{1}^{re} = {0.043} * {p}^{2} + {22} * p + {80}
+$$
+
+$$
+{C}_{2}^{\mathrm{{fu}}} = {0.035} * {p}^{2} + {18} * p + {120}
+$$
+
+$$
+{C}_{1}^{\mathrm{{ld}}} = - {0.013} * {p}^{2} + {46} * p + {30} \tag{14}
+$$
+
+$$
+{C}_{2}^{\mathrm{{ld}}} = - {0.015} * {h}^{2} + {70} * h + {40}
+$$
+
+$$
+{E}_{1}^{\mathrm{{fu}}} = {0.0648} * {h}^{2} - {2.7} * h + {41}
+$$
+
+$$
+{E}_{2}^{\mathrm{{fu}}} = {0.0520} * {p}^{2} - {2.3} * p + {50}
+$$
+
+
+
+Fig. 6. Structure for the test S-IES
+
+The trajectories of incremental costs of 5 nodes in the test S-IES are depicted in Fig.7. It can be found that the variable of incremental cost can be converged within 20 iteration steps. And the specified value of the final incremental cost is 0.3066 (p.u.). Moreover, the calculated optimized energy management solution is same as the solutions obtained by centralized strategy, which verifies the accuracy of the designed algorithm.
+
+
+
+Fig. 7. Trajectories of incremental costs
+
+## IV. CONCLUSION
+
+In this paper, the development of S-IES has been analyzed. A multi-objective energy management model for S-IES has been constructed considering economic benefits and carbon emissions. Meanwhile, the entire sailing requirements for ships are considered in the construction of energy management model. Additionally, to search for the optimization solutions in a distributed manner, an energy management algorithm has been proposed based on ADMM theory. Simulation results proves the accuracy and effectiveness of the designed energy management model and the distributed algorithm.
+
+## REFERENCES
+
+[1] F. Teng, Y. Zhang, T. Yang, T. Li, Y. Xiao and Y. Li, "Distributed Optimal Energy Management for We-Energy Considering Operation Security," IEEE Transactions on Network Science and Engineering, vol. 11, no. 1, pp. 225-235, Jan.-Feb. 2024, doi: 10.1109/TNSE.2023.3295079.
+
+[2] Y. Zhang, Y. Xiao, F. Teng and T. Li, "Distributed Energy Management Method With EEOI Limitation for the Ship-Integrated Energy System," IEEE Systems Journal, vol. 18, no. 2, pp. 1332-1343, June 2024, doi: 10.1109/JSYST.2024.3361709.
+
+[3] F. Teng, Z. Ban, T. Li, Q. Sun and Y. Li, "A Privacy-Preserving Distributed Economic Dispatch Method for Integrated Port Micro-grid and Computing Power Network," IEEE Transactions on Industrial Informatics, vol. 20, no. 8, pp. 10103-10112, Aug. 2024, doi: 10.1109/TII.2024.3393569.
+
+[4] M. Rafiei, J. Boudjadar and M. -H. Khooban, -Energy Management of a Zero-Emission Ferry Boat With a Fuel-Cell-Based Hybrid Energy System: Feasibility Assessment," IEEE Transactions on Industrial Electronics, vol. 68, no. 2, pp. 1739-1748, Feb. 2021, doi: 10.1109/TIE.2020.2992005.
+
+[5] F. D. Kanellos, G. J. Tsekouras and N. D. Hatziargyriou, -Optimal Demand-Side Management and Power Generation Scheduling in an All-Electric Ship," IEEE Transactions on Sustainable Energy, vol. 5, no. 4, pp. 1166-1175, Oct. 2014, doi: 10.1109/TSTE.2014.2336973.
+
+[6] Y. Li, T. Li, H. Zhang, X. Xie and Q. Sun, -Distributed Resilient Double-Gradient-Descent Based Energy Management Strategy for Multi-Energy System Under DoS Attacks," IEEE Transactions on Network Science and Engineering, vol. 9, no. 4, pp. 2301-2316, 1 July-Aug. 2022, doi: 10.1109/TNSE.2022.3162669.
+
+[7] F. D. Kanellos, "Optimal Power Management With GHG Emissions Limitation in All-Electric Ship Power Systems Comprising Energy Storage Systems," IEEE Transactions on Power Systems, vol. 29, no. 1, pp. 330-339, Jan. 2014, doi: 10.1109/TPWRS.2013.2280064.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/kzbTcOzvFr/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/kzbTcOzvFr/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..453795676fe04f5602d491519972f40b0faa26a4
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/kzbTcOzvFr/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,269 @@
+§ DISTRIBUTED ENERGY MANAGEMENT FOR SHIP-INTEGRATED ENERGY SYSTEM CONSIDERING ECONOMIC AND ENVIRONMENTAL BENEFITS
+
+${1}^{\text{ st }}$ Yuxin Zhang
+
+Navigation College
+
+Dalian Maritime University
+
+Dalian, China
+
+liam_zhang@dlmu@edu.cn
+
+${2}^{\text{ nd }}$ Qihe Shan
+
+Navigation College
+
+Dalian Maritime University
+
+Dalian, China
+
+tengfei@dlmu.edu.cn
+
+${3}^{\text{ rd }}$ Haoran Liu
+
+Navigation College
+
+Dalian Maritime University
+
+Dalian, China
+
+lhr6@dlmu.edu.cn
+
+${4}^{\text{ th }}$ Tieshan Li
+
+School of Automation Engineering
+
+University of Electronic Science and Technology of China Chengdu, China
+
+litieshan073@uestc.edu.cn
+
+Abstract-To decrease the dependency of fuel-based resources in the shipping industry, the energy management problem is analyzed in this paper. Firstly, the development of shipboard energy system is reviewed from radiation pattern, ring pattern, two-end pattern to the ship-integrated energy system. Moreover, for ensuring the secure sailing, a multi-objective energy management model is established with the consideration of economic and environmental benefits. Meanwhile, the requirements of supply-demand balance, velocity, voltage security and so on are considered as well in the energy management. Then, a distributed energy management strategy based on ADMM algorithm is proposed. Finally, simulation results of a 5-node test system proves the effectiveness of the constructed energy management model and the distributed algorithm.
+
+Index Terms-ship-integrated energy system, distributed optimization, energy management
+
+§ I. INTRODUCTION
+
+With the deepening of economic and manufacturing cooperation among countries, we have gradually entered the era of globalization. As a major way of exchange of capital, goods, technology, and services among different countries and territories, the export value of international trade has been on the rise since 1950. Because of its low cost per unit of cargo delivery, wide route coverage, and other characteristics, the shipping industry has undertaken most of the global bulk cargo transport and has gradually become the most important manner of transport for trade exchanges between countries and supply chain operations between enterprises [1], [2]. As shown in Figure 1, between 1980 and 2022, the overall capacity of the shipping industry continued to rise, in which cruise ships (tankers), container ships (containers), and cruise ships (cruises) capacity increased significantly, affected by the epidemic and other factors, general cargo (general cargo) capacity has slightly decreased [3]. By 2022, there are 10 ports in the world with a throughput of more than 15 million TEUs (Twenty-Foot Equivalent Unit, TEU), and the order of throughput from large to small is Shanghai (CNSHA), Singapore (SGSGP), Ningbo-Zhoushan ( CNZOS), Shenzhen (CNSZN), Qingdao (CNQIN), Guangzhou (CNGUA), Busan (CNBUS), Tianjin (CNTJN), Los Angeles/Long Beach (USLSA), Hong Kong (HKHKG).
+
+However, the booming international trade and the revival of the shipping industry depend on huge fossil energy consumption, accompanied by the emission of various greenhouse gases (GHG) such as nitrogen and sulphur, which aggravate global warming and the melting of glaciers, and run counter to the concept of ecologically sustainable development [4]. In 2022, the transport sector will account for approximately ${20}\%$ of global greenhouse gas emissions, second only to the electricity sector. As a necessary guarantee for international trade transactions, the greenhouse gas emissions of international shipping, international aviation and international rail account for ${58.8}\% ,{35.3}\%$ and ${5.9}\%$ of the total international trade transport emissions, respectively. Carbon dioxide, as the main component of greenhouse gas emissions from shipping, accounts for more than 90 per cent of the total, and its total emissions have been on a sharp upward trend overall since 1970 until 2021. In 2021 alone, the shipping industry will emit about 700 million tonnes of carbon dioxide into the atmosphere, an increase of about $5\%$ over the previous year. According to the International Maritime Organization (IMO), current ${\mathrm{{CO}}}_{2}$ emissions from shipping industry have doubled since 1990 and reached a staggering 701.9 million tonnes in 2017, as shown in Figure 2. If timely and effective improvement measures are not taken, the total amount of carbon dioxide generated by the shipping industry from fossil energy consumption will rapidly grow to 2.50-3.65 billion tonnes in 2050, accounting for about ${18}\%$ of total global carbon emissions. To address the contradiction between the development of the maritime economy and the high level of carbon pollution from shipping, IMO and its subsidiary body, the Maritime Environment Protection Committee (MEPC), have established a number of regulations and strategic targets to reduce the total carbon emissions from the global shipping industry since 1997, as shown in Figure 3. In 2023, IMO member states unanimously agreed to adopt strategic carbon reduction targets that are expected to reduce emissions from international shipping by at least ${20}\%$ and ${70}\%$ relative to 2008 levels by 2030 and 2040, respectively, and to achieve zero GHG emissions by 2050. To this end, national classification societies and shipbuilding enterprises have paid extensive attention to the application of new energy in ship energy systems and invested in the research and development of onboard new energy equipment, thus promoting the innovation of ship energy systems.
+
+This paper is supported by the National Natural Science Foundation of China (under Grants 52371360, 52201407, 51939001, 61976033) and the China Scholarship Council Program (under Grant 202406570011). Corresponding author: Fei Teng
+
+ < g r a p h i c s >
+
+Fig. 1. Trends of Export Trade and International Maritime Trade
+
+According to the type of ship and its operating characteristics, the connection of energy devices in the current ship energy system with new energy devices and traditional fossil fuel devices can be broadly classified into three types, i.e., radiation pattern for small and medium-sized ships, ring pattern for large ships, and two-end pattern, which are shown in Fig. 4. Among them, radiation pattern is the most common and relatively mature technology, which is widely used. Compared with the radiation pattern, the ring pattern and the two-end pattern are equipped with a large number of non-professional energy supply equipment, the structure is relatively complex, and are mostly used in the energy network of large ships [5].
+
+However, with the continuous progress and development of renewable technology, ship technology and information technology, more and more non-professional energy device and intelligent device integrated into the ship, the energy system presents a fully distributed flat structure coupled with a variety of heterogeneous energy sources. The cruise ship Ecoship, integrating diesel, natural gas, and photovoltaic panel, reduces carbon emissions by ${40}\%$ compared with the average 60,000-tonne-class large cruise ship. Ship integrated energy system (S-IES) is a typical power-heating coupling energy system, which takes energy management system as the core and heterogeneous energy conversion centre as the hub, and uses both traditional energy devices and renewable energy device. It can reduce the dependence on traditional fossil fuels for sailing and improve the efficiency of energy utilization [6], which provides continuous and high-quality energy for the normal operation [7]. Therefore, S-IES and its energy management problem are gradually gaining wide attention in the related fields.
+
+§ II. SHIP-INTEGRATED ENERGY SYSTEM AND ITS ENERGY MANAGEMENT
+
+§ A. STRUCTURE AND FRAMEWORK FOR S-IES
+
+As a core unit to ensure secure and reliable navigation of ships, S-IES provides continuous and high-quality energy for the energy system, communication and navigation system, and mechanical towing system. Ship energy management system carries out real-time monitoring and collection of shipboard load-side equipment, gathers communication and navigation systems as well as wind, wave, and other climate perturbation information to predict the shipload demand, and finally sends the load demand value to the energy supply side, and solves the optimal energy management scheme based on the intelligent algorithm to realize secure sailing in a long-distance voyage, and the structure of the system is shown in Fig.5.
+
+Energy devices in S-IES can be roughly divided into five categories: power-only device (PO), heating-only device (HO), combined heating and power device (CHP), energy storage device (ESD) and load device (LD), storage device (ESD) and load device (LD).
+
+ < g r a p h i c s >
+
+Fig. 3. Regulations development for GHG Emission Reduction
+
+For improving the energy supply performance of S-IES and ensuring that the energy supply side equipment provides continuous and high-quality energy for the safe navigation of the ship, it is necessary to carry out an in-depth analysis of the energy management problems of S-IES. The energy management system of S-IES is based on the theory of multi-intelligent body system, which realizes the bidirectional interaction of information and energy among the shipboard-distributed energy equipment. Then it formulates the optimal energy management strategy for the distributed energy management of S-IES. Ship energy efficiency monitoring technology, as a key role in ensuring the accuracy of the analysis of S-IES energy management issues, provides the necessary guarantee for maximizing the economic and environmental benefits of S-IES and has received extensive attention worldwide, as shown in Table I. Specifically, the EFMS designed by Ascenz Marorka, includes fuel pilferage prevention measures and reporting capabilities for tracking fuel usage and analyzing data over time, which plays a vital role in saving operational costs by analyzing the information collected by the EFMS. For improving the operational fuel efficiency of ships, Germanis-cher Lloyd proposes the ECO-Assistant. It allows mariner to sail at optimal trim during voyage by acquiring the optimal trim angle in different sailing conditions. Moreover, the ECO-Assistant depends on the existing shipborne devices without installing extra modifications to the ships. By utilization of the routing algorithms, VVOS is constructed to optimize each route for on-time arrival while minimizing fuel consumption and avoiding weather damage. Additionally, the Electronic Chart Display and Information System (ECDIS) and Integrated Navigation System (INS) can receive the voyage scheme planed by VVOS to realizing secure check and execution. NAPA-VO is a software designed by NAPA, which realizes the improvement of operational efficiency. The created route with NAPA-VO concerns different fuel types and the corresponding features under emission control areas. A novel propulsion concept for ships is proposed by ABB Dynafin, which reduces greenhouse emissions by at least half. Moreover, compared to the traditional shaftline ships, the new technology is set to decrease propulsion energy consumption by up to ${22}\%$ . ISSE can automatically calculate the carbon intensity index and related data sampling, which realizes the CII rating automatically and provides guarantees for sailing voyages.
+
+TABLE I
+
+ENERGY CONSUMPTION DETECTION TECHNOLOGY AROUND THE WORLD
+
+max width=
+
+Instituation Product Functions and Characteristics
+
+1-3
+Ascenz Marorka Electronic Fuel Monitoring System(EFMS) Monitor real-time data on fuel usage. Develop energy management plan to reduce fuel consumption.
+
+1-3
+Germanischer Lloyd ECO Assistant Calculate optimum trim angle without making extra devices. Improve the operational fuel efficiency of ships.
+
+1-3
+Jeppesen Marine Vessel and Voyage Optimization Services (VVOS) Optimize route for on-time arrival. Voyage plan can be received by ECDIS and INS.
+
+1-3
+NAPA NAPA Voyage Optimization (NAPA-VO) Decrease operation cost by improved schedule adjustment procedure. Increase efficiency in altering the voyage plans.
+
+1-3
+ABB ABB Dynafin Cutting annual greenhouse emissions by at least ${50}\%$ in future. Decrease the energy consumed by propulsion systems.
+
+1-3
+Hangzhou Yagena Technology Co., LTD. Intelligent Ship System and Equipment (ISSE) Calculate the carbon intensity index. Rank the carbon emission level.
+
+1-3
+
+ < g r a p h i c s >
+
+Fig. 4. (a) Structure of Radiation Pattern for S-IES, (b) Structure of Ring Pattern for S-IES, (c) Structure of Two-End Pattern for S-IES.
+
+ < g r a p h i c s >
+
+Fig. 5. Structure for S-IES
+
+At present, with the risen on awareness of sustainable development, the economic benefit ${F}_{\mathrm{{EC}}}$ is not the only factor which should be concerned in the energy management of S-IES. For this reason, the environmental ${F}_{\mathrm{{CA}}}$ , and social ${F}_{\mathrm{{SC}}}$ benefits are considered in this paper as well. Moreover, to ensure the high-security sailing and provide high-quality energy to the ships, the utilization of energy is also another major factor. Therefore, the objective function of the energy management problem for S-IES is constructed as,
+
+$$
+\min \left\{ {{F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}},{F}_{\mathrm{{SC}}}}\right\} \text{ . } \tag{1}
+$$
+
+Note that, the above objective function represents a compromise between economic benefits and carbon emissions, rather than indicating the smaller of ${F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}}$ and ${F}_{\mathrm{{SC}}}$ as the objective function for S-IES energy management. In addition, physical constraints need to be imposed on ships and energy equipment to achieve secure performance during sailing voyage. Then, the energy management model of S-IES can be established as below.
+
+$$
+\min \left\{ {{F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}},{F}_{\mathrm{{SC}}}}\right\} \text{ , } \tag{2}
+$$
+
+$$
+\text{ s.t. }A\left( X\right) = 0,B\left( X\right) \leq 0\text{ , }
+$$
+
+where, $X$ is the decision variable of the energy management problem, which presents the energy output, $A\left( X\right)$ and $B\left( X\right)$ are equality and inequality constraints, respectively. Specifically, $A\left( X\right)$ contains supply-demand balance constraints, voltage security constraints. $B\left( X\right)$ includes energy output constraints, velocity constraints, which are shown below.
+
+1) Supply-demand balance constraints:
+
+$$
+\sum \left( {{p}_{i}^{\mathrm{{fu}}} + {p}_{i}^{\mathrm{{re}}} + {p}_{i}^{\mathrm{{chp}}} - {p}_{i}^{\mathrm{{ld}}}}\right) = 0 \tag{3}
+$$
+
+$$
+\sum \left( {{h}_{i}^{\mathrm{{fu}}} + {h}_{i}^{\mathrm{{re}}} + {h}_{i}^{\mathrm{{chp}}} - {h}_{i}^{\mathrm{{ld}}}}\right) = 0 \tag{4}
+$$
+
+where $p$ and $h$ are power and heating outputs, respectively. $i,l$ presents the node sequence of S-IES. fu, re, chp, ld are fuel-based generators, renewable-based generators, combining heating and power device, and load, respectively.
+
+2) Voltage security constraints:
+
+$$
+\sum {\alpha }_{i,l}^{\max }{p}_{i} + \Delta {V}_{l}^{\max } = {\pi }_{l}^{\max } \tag{5}
+$$
+
+$$
+\sum {\alpha }_{i,l}^{\min }{p}_{i} + \Delta {V}_{l}^{\min } = {\pi }_{l}^{\min } \tag{6}
+$$
+
+where $\alpha$ notes the voltage sensitivity coefficients, which can be calculated by a given transmission topology of S-IES. ${\Delta V}$ is the voltage excursion index, which indicates the voltage security margin. $\pi$ is a constant influencing by the reactive power. min and max are noted as the minimal and maximal values of the corresponding variables, respectively. Specifically, (5) and (6) present the upper and lower bounds of velocity security boundaries, respectively.
+
+3) Energy output constraints:
+
+$$
+{p}_{i}^{\min } \leq {p}_{i} \leq {p}^{\max },{h}^{\min } \leq {h}_{i} \leq {h}^{\max } \tag{7}
+$$
+
+4) Velocity constraints:
+
+$$
+{v}^{\min } \leq v \leq {v}^{\max } \tag{8}
+$$
+
+where $v$ describes the velocity of the ship.
+
+§ B. COMMUNICATION TOPOLOGY FOR S-IES
+
+1) Centralized Energy Management Strategy: It relies on a centralized controller, which has significant advantages in terms of calculation rate and accuracy. The controller collects real-time information on the operation of all energy devices in the S-IES, and calculates the optimal energy management schemes based on the relevant data to complete the centralized control and management for the S-IES. However, the centralized strategy is prone to single point of failure, poor stability, limited scalability and huge investment costs.
+
+2) Decentralized Energy Management Strategy: According to the operating mechanism and working status of the energy equipment, the ship can be divided into multiple compartments, and the decentralized energy management strategy places the controllers in the compartments. Thus the control and management of the local energy system can be realized. Moreover, the decentralized strategy does not require real-time communication between the controllers in each cabin, and the energy equipment distributes the energy by itself, so the system response speed is fast and the scalability is strong. However, due to the lack of collaborative control between the controllers, the global optimization at the system level cannot be achieved. Therefore it is highly susceptible to external interference, which has a negative impact on secure navigation.
+
+However, the flat distributed structure of the energy network of S-IES presents that the traditional centralized and decentralized energy management strategies are no longer applicable, and a fully distributed energy management strategy continues to be utilized to complete the search for the optimal solution in order to achieve the reliable operation of S-IES.
+
+3) Distributed Energy Management Strategy: It relies on the communication network to achieve real-time information exchange between energy devices within the S-IES, so as to achieve complementary interconnection of information between neighbouring energy devices. When a local energy device fails to operate normally, the controllers of each energy device can achieve local interconnections and send signals to the controllers of neighbouring energy devices to ensure safe and stable operation of the S-IES based on the distributed communication network.
+
+§ C. MAIN ALGORITHM
+
+To simplify the multi-objective functions of the energy management problems, the linear weighted sum method, the energy management model can be reconstructed as below.
+
+$$
+\min \left\{ {{\beta }_{1} \cdot {F}_{\mathrm{{EC}}} + {\beta }_{2} \cdot {F}_{\mathrm{{CA}}} + {\beta }_{3} \cdot {F}_{\mathrm{{SC}}}}\right\} , \tag{9}
+$$
+
+$$
+\text{ s.t. }A\left( X\right) = 0,B\left( X\right) \leq 0\text{ , }
+$$
+
+where ${\beta }_{1},{\beta }_{2}$ and ${\beta }_{3}$ are constant parameters of ${F}_{\mathrm{{EC}}},{F}_{\mathrm{{CA}}}$ , and ${F}_{\mathrm{{SC}}}$ , respectively. Moreover, the related parameters satisfy ${\beta }_{1} + {\beta }_{2} + {\beta }_{3} = 1$ .
+
+Define the communication topology of the constructed S-IES as $G = \{ T,B,W\}$ , where $T,B$ , and $W$ are node set, connected edge set and connected weighted sets, respectively. Specifically, $T = \left\lbrack {{v}_{i} \mid i \in \Omega }\right\rbrack$ and $\Omega$ is noted as the energy device set. Moreover, $B \subseteq T \times T$ is related to the node set. $W = \left\lbrack {{w}_{i,l} \mid i,l \in \Omega }\right\rbrack$ , where ${w}_{i,l}$ is the connected weighted parameter between the $i$ th node and the $l$ th node. Considering the different relationships between the $i$ th node and the $l$ th node, ${w}_{i,l}$ can be calculated as below.
+
+$$
+{w}_{i,l} = \left\{ \begin{array}{l} 1/\left( {\left| {N}_{zi}\right| + \left| {N}_{l}\right| + \varepsilon }\right) ,l \in {N}_{i} \\ 1 - \mathop{\sum }\limits_{{l \in {N}_{i}}}1/\left( {\left| {N}_{i}\right| + \left| {N}_{l}\right| + \varepsilon }\right) ,i = l \\ 0,\text{ otherwise } \end{array}\right. \tag{10}
+$$
+
+where $\varepsilon$ is a small positive constant. ${N}_{i}$ and $\left| {N}_{i}\right|$ are the neighbor set of the $i$ th node and its cardinal number. Similarity, ${N}_{l}$ and $\left| {N}_{l}\right|$ are the neighbor set of the $l$ th node and its cardinal number. When the connected weighted value between $i$ th node and the $l$ th node equals to 0, it means that the information exchange can not occur between the mentioned nodes.
+
+Since the high calculation speed, accuracy, and reliable performed by the alternating direction method of multipliers (ADMM), a fully distributed energy management strategy for S-IES is designed in this paper. It realizes the bi-directional transmissions of energy and information, which improves the reduction of communication resource and is suitable to the flat and distributed structure of S-IES. Then, the main algorithm can be designed as below.
+
+1) Iteration of energy output:
+
+$$
+{X}_{i,k + 1} \in \arg \min \left\{ {f\left( X\right) + \psi + {\lambda }_{i,k}^{\mathrm{T}}{W}^{\mathrm{T}}A{X}_{i,k}}\right\} , \tag{11}
+$$
+
+where $\lambda$ is the incremental cost. $A$ describes the physical relationship among nodes of S-IES, which is defined by physical constraints. $k$ is the sequence of time slot.
+
+2) Iteration of output error:
+
+$$
+{d}_{i,k + 1} = W{d}_{i,k} + A\left( {{X}_{i,k + 1} - {X}_{i,k}}\right) , \tag{12}
+$$
+
+where $d$ is the output error of S-IES.
+
+3) Iteration of incremental cost:
+
+$$
+{\left\lbrack {\overset{\overleftrightarrow{} }{\lambda }}_{{\Omega }_{\mathrm{W}}^{z},k + 1},{\lambda }_{z,k + 1}\right\rbrack }^{\mathrm{T}} = W{\left\lbrack {\overset{\overleftrightarrow{} }{\lambda }}_{{\Omega }_{\mathrm{W}}^{z},k},{\lambda }_{z,k}\right\rbrack }^{\mathrm{T}} + \tau {d}_{z,k + 1}. \tag{13}
+$$
+
+Repeat the iterations of energy output, output error, and incremental cost until each variable converge a preset threshold.
+
+§ III. SIMULATION RESULTS
+
+To verify the effectiveness of the designed algorithm, a 5- node S-IES is utilized as a test system, and the detailed physical/communication topology containing connected weighted parameters is shown in Fig.6. Moreover, in the test system, there are 2-fu, 1-re, 2-ld, and the operational coefficients and carbon emission parameters are presented in (14). Assume the load demand of power and heating are $\left\lbrack {{4.5},{2.4}}\right\rbrack \left( \mathrm{{MW}}\right)$ , respectively.
+
+$$
+{C}_{1}^{\mathrm{{fu}}} = {0.040} * {h}^{2} + {25} * h + {99}
+$$
+
+$$
+{C}_{1}^{re} = {0.043} * {p}^{2} + {22} * p + {80}
+$$
+
+$$
+{C}_{2}^{\mathrm{{fu}}} = {0.035} * {p}^{2} + {18} * p + {120}
+$$
+
+$$
+{C}_{1}^{\mathrm{{ld}}} = - {0.013} * {p}^{2} + {46} * p + {30} \tag{14}
+$$
+
+$$
+{C}_{2}^{\mathrm{{ld}}} = - {0.015} * {h}^{2} + {70} * h + {40}
+$$
+
+$$
+{E}_{1}^{\mathrm{{fu}}} = {0.0648} * {h}^{2} - {2.7} * h + {41}
+$$
+
+$$
+{E}_{2}^{\mathrm{{fu}}} = {0.0520} * {p}^{2} - {2.3} * p + {50}
+$$
+
+ < g r a p h i c s >
+
+Fig. 6. Structure for the test S-IES
+
+The trajectories of incremental costs of 5 nodes in the test S-IES are depicted in Fig.7. It can be found that the variable of incremental cost can be converged within 20 iteration steps. And the specified value of the final incremental cost is 0.3066 (p.u.). Moreover, the calculated optimized energy management solution is same as the solutions obtained by centralized strategy, which verifies the accuracy of the designed algorithm.
+
+ < g r a p h i c s >
+
+Fig. 7. Trajectories of incremental costs
+
+§ IV. CONCLUSION
+
+In this paper, the development of S-IES has been analyzed. A multi-objective energy management model for S-IES has been constructed considering economic benefits and carbon emissions. Meanwhile, the entire sailing requirements for ships are considered in the construction of energy management model. Additionally, to search for the optimization solutions in a distributed manner, an energy management algorithm has been proposed based on ADMM theory. Simulation results proves the accuracy and effectiveness of the designed energy management model and the distributed algorithm.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/lAtPkUMK1M/Initial_manuscript_md/Initial_manuscript.md b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/lAtPkUMK1M/Initial_manuscript_md/Initial_manuscript.md
new file mode 100644
index 0000000000000000000000000000000000000000..a354a951ca4f317ee43eb7071ddb7a44a50c5538
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/lAtPkUMK1M/Initial_manuscript_md/Initial_manuscript.md
@@ -0,0 +1,521 @@
+# Adaptive dynamic programming-based optimal heading control for state constrained unmanned sailboat
+
+${1}^{\text{st }}$ Shitong Zhang
+
+School of Mechanical Engineering
+
+Yanshan University
+
+Qinhuangdao, China
+
+bben@stumail.ysu.edu.cn
+
+${2}^{\text{st }}$ Yifei Xu
+
+School of Mechanical Engineering Yanshan University
+
+Qinhuangdao, China
+
+xyf@ysu.edu.cn
+
+${3}^{\text{st }}$ Yingjie Deng*
+
+School of Mechanical Engineering
+
+Yanshan University
+
+Qinhuangdao, China
+
+dyj@ysu.edu.cn
+
+${4}^{\text{st }}$ Sheng Xu
+
+Shenzhen Institute of Advanced Technology
+
+Chinese Academy of Sciences
+
+Shenzhen, China
+
+${Abstract}$ -This paper proposes a new state-constrained adaptive optimal control strategy for unmanned sailboat heading angle tracking considering the motion constraints. To improve the tracking accuracy, a combination of backstepping and adaptive dynamic programming (ADP) is employed. Thus, the issue of virtual control rate derivatives in conventional backstepping control is resolved with satisfactory precision. Firstly, the motion constraints are considered by using the Barrier Lyapunov function (BLF), and the neural networks (NNs) is employed to approximate the model uncertainties and disturbances. Secondly, an adaptive backstepping feedforward controller is proposed, transforming the sailboat's affine nonlinear system tracking problem into a regulation problem. Thirdly, according to the ADP theory, critic NNs are constructed to approximate the analytical solution of the Hamilton-Jacobi-Bellman (HJB) equation, and the optimal feedback control is obtained by online learning. Finally, simulation results demonstrate the effectiveness and optimality of the proposed controller.
+
+Index Terms-Optimal control, Adaptive dynamic programming (ADP), barrier Lyapunov function (BLF), Neural networks (NNs), Unmanned sailboat
+
+## I. INTRODUCTION
+
+In the past few decades, surface and underwater intelligent vehicles have played important roles in sea patrols, resource exploration, and rescue. However, due to the consumption of fuel, electricity, and other energy sources, unmanned ships and submarines require a large amount of power support to complete long-distance tasks, which also leads to huge cost problems in [1]-[6]. Since unmanned sailboats can use sails as power, have low costs, and can transmit data in real-time through their sensors, research on unmanned sailboats has gradually entered the interest of scholars. However, due to the complex marine environment, the disturbances of wind and waves can easily generate obstructive forces on the sails and keel, making the sails control difficult. Therefore, the authors of [7] proposed a path tracking method for unmanned sailboats that combines logic virtual ship (LVS) guidance law and dynamic event-triggered control. In [8], a path tracking control scheme is proposed for unmanned sailboats that combined backstepping and dynamic surface control technology. In order to reduce the sideslip angle error during sailing, the authors of [9] proposed double finite-time observers-based line-of-sight guidance (DFLOS) and adaptive finite-time control (DFLOS-AFC) strategies. However, they did not consider the issue of state constraints. When subjected to significant interference, to return the sailboat to the reference heading, a larger rudder angle is required, which will damage the actuator as it cannot accept significant deflection in a short period. Therefore, while ensuring the accuracy of sailboat heading control, it is also necessary to ensure that the turning speed of the boat heading satisfies the prescribed constraints.
+
+In order to solve the problem of unmanned sailboat heading angle tracking accuracy, many scholars have conducted research on it. In [10], a control strategy of adaptive echo state networks and backstepping is proposed to control the steering angle of the rudder. The authors of [11] designed a nonlinear heading controller of velocity vector direction to track the reference heading angle. In [12], the ${L}_{1}$ adaptive control theory is proposed to complete heading control and ensure the stability of the required heading angle. However, most papers focused on the heading control of sailboats using backstepping or sign functions, which results in low tracking accuracy and cannot achieve optimal control effects.
+
+The key problem of the nonlinear optimal control is that the analytical solution of the Hamilton-Jacobi-Bellman (HJB) equation is difficult to solve. In order to obtain the analytical solution, the authors of [13] proposed the ADP theory, which approximates the solution of the equation online. However, the problem of slow approximation speed and multiple iterations has led to an explosion in computational complexity. Recently, with the rapid development of NNs, their approximation performance and speed for unknown nonlinear functions have become increasingly excellent. Therefore, the research combining ADP and NNs has solved the above problems. In [14], a new event-triggered optimal trajectory tracking control method based on goal representation heuristic dynamic programming (GrHDP) for underactuated ships is proposed. In [15], they proposed a model free dual heuristic dynamic programming (DHP) method for unmanned aerial vehicle attitude control. The authors of [16] applied ADP theory to the path planning problem of mobile robots. However, to the best of our knowledge, there is limited research on applying ADP theory to the heading control of unmanned sailboats.
+
+---
+
+This work is partially supported by the Natural Science Foundation of China (No.52101375), the Hebei Province Natural Science Fund (No.E2022203088, E2024203179), the Innovation Capacity Enhancement Program of Hebei Province (No.24461901D), the Joint Funds of the National Natural Science Foundation of China (No.U20A20332), and the Key Research and Development Project of Hebei Province (No.21351802D).
+
+---
+
+Taking inspiration from the above analysis, we introduce the LBF function to solve the state constraint problem. In the control design of heading tracking, while introducing the backstepping method, the ADP theory is also introduced to ensure the optimal tracking accuracy. We constructed an evaluation NN to approximate the analytical solution of the HJB equation and obtained the optimal feedback control through online learning. The main contributions are concluded below:
+
+- Compared with previous papers of [7]-[9], we proposed a LBF-based method to overcome the issues of the motion constraints caused by the turning rate limitation of unmanned sailboats.
+
+- By using the ADP optimal feedback control strategy, our designed control method achieves optimal tracking accuracy in heading control compared to [9]-[12].
+
+- According to the construction of the critic network, a training speed acceleration method is developed using online learning methods.
+
+The remaining of this paper is organized as follows. Section II elaborates on the heading angle control model of unmanned sailboats and lemma. Section III designs the backstepping feedforward controller and the optimal feedback controller. Section IV provides simulation verification of the proposed strategy superiority. Section V gives some conclusions.
+
+## II. MATHEMATICAL MODEL AND PRELIMINARIES
+
+The sailboat is divided into four parts: the sail, rudder, keel, and hull. Combined with the theory of gas fluid dynamics, force analysis is conducted on each part, ignoring the undulating and pitching motion of the sailboat to establish a 3-DOF mathematical model of sailboat motion. The sailboat model considering external interference and control input rudder angle in this paper is
+
+$$
+\left\{ \begin{array}{l} \dot{\psi } = r \\ \dot{r} = {f}_{r}\left( {\psi , r,{\delta }_{s}, u, v,{\tau }_{wr}}\right) + {g}_{r}{\tau }_{r} \end{array}\right. \tag{1}
+$$
+
+where $\psi \in {\mathbb{R}}^{M}$ and $r \in {\mathbb{R}}^{M}$ are system state variables, respectively; $u \in {\mathbb{R}}^{M}$ is the forward velocity; $v \in {\mathbb{R}}^{M}$ is the lateral velocity; ${\delta }_{s} \in {\mathbb{R}}^{M}$ is the sail angle; ${\tau }_{wr}$ is the external disturbance; ${f}_{r}\left( \cdot \right)$ is the nonlinear function of the unknown model of a sailboat; ${g}_{r}$ is the unknown control gain; ${\tau }_{r}$ is the control input of the system.
+
+Assumption 1 [17]: Due to the fact that unmanned sailboats navigate in limited space, there exists a normal number that satisfies the heading angle and heading angular speed to be less than or equal to this normal number. That is, $\psi \leq {k}_{\psi }$ and $r \leq {k}_{r}$ .
+
+Lemma 1 [10]: Due to the outstanding ability of NNs in function approximation, they are often used to approximate nonlinear functions. Therefore, NNs can be used to approximate unknown functions as follows:
+
+$$
+F\left( x\right) = {W}^{\mathrm{T}}\sigma \left( x\right) + \varepsilon \left( x\right) \tag{2}
+$$
+
+where $W = {\left( {w}_{1},{w}_{2},\ldots ,{w}_{n}\right) }^{\mathrm{T}}$ denotes the desired weight of the NNs and $\varepsilon \leq \bar{\varepsilon }$ denotes the approximating error.
+
+## III. CONTROL DESIGN
+
+## A. Feedforward controller design
+
+In this section, the backstepping method based on an adaptive NNs framework is adopted to transform system (1) into an affine nonlinear system.
+
+According to the system (1), define the error system function as
+
+$$
+\left\{ \begin{matrix} {\psi }_{e} = \psi - {\psi }_{d} \\ {r}_{e} = r - {r}_{d} \end{matrix}\right. \tag{3}
+$$
+
+where ${\psi }_{d}$ and ${r}_{d}$ are the reference heading angle and yaw speed, respectively. Differentiating ${\psi }_{e}$ along with (1), it has:
+
+$$
+{\dot{\psi }}_{e} = \dot{\psi } - {\dot{\psi }}_{d} = r - {\dot{\psi }}_{d} = \left( {{r}_{e} + {r}_{d}}\right) - {\dot{\psi }}_{d} \tag{4}
+$$
+
+where ${r}_{d} = {r}_{d}^{\alpha } + {r}_{d}^{ * }$ is the virtual control input of yaw speed. ${r}_{d}^{\alpha }$ denotes the feedforward virtual yaw speed input and ${r}_{d}^{ * }$ denotes the feedback virtual yaw speed input. Therefore, we can get
+
+$$
+{\dot{\psi }}_{e} = {r}_{e} + {r}_{d}^{\alpha } + {r}_{d}^{ * } - {\dot{\psi }}_{d} \tag{5}
+$$
+
+To construct the desired feedforward yaw speed virtual control input, consider the BLF as
+
+$$
+{V}_{1} = \frac{1}{2}\log \left( \frac{{k}_{\psi }^{2}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\right) \tag{6}
+$$
+
+where ${k}_{\psi }$ is a positive value of the state constraint. Calculate the derivate of ${V}_{1}$ , we have
+
+$$
+{\dot{V}}_{1} = \frac{{\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\left( {{r}_{e} + {r}_{d}^{\alpha } + {r}_{d}^{ * } - {\dot{\psi }}_{d}}\right) \tag{7}
+$$
+
+Therefore, the feedforward virtual yaw speed input ${r}_{d}^{\alpha }$ can be designed as
+
+$$
+{r}_{d}^{\alpha } = - \left( {{k}_{\psi }^{2} - {\psi }_{e}^{2}}\right) {k}_{1}{\psi }_{e} + {\dot{\psi }}_{d} \tag{8}
+$$
+
+where ${k}_{1} > 0$ is a tuning parameters. Substituting (8) into (7), we have
+
+$$
+{\dot{V}}_{1} = - {k}_{1}{\psi }_{e}^{2} + \frac{{\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\left( {{r}_{e} + {r}_{d}^{ * }}\right) \tag{9}
+$$
+
+Taking the derivative of the second equation of (3) yields
+
+$$
+{\dot{r}}_{e} = {f}_{r}\left( \cdot \right) + {f}_{r}\left( {e}_{d}\right) - {f}_{r}\left( {e}_{d}\right) + {g}_{r}{\tau }_{r} - {\dot{r}}_{d} \tag{10}
+$$
+
+where ${e}_{d} = {\left\lbrack {\psi }_{d},{r}_{d}\right\rbrack }^{\mathrm{T}}$ . The unknown model uncertainty function ${f}_{r}\left( \cdot \right)$ and ${\dot{r}}_{d}$ can be transferred by the following function
+
+$$
+{F}_{2}\left( {z}_{2d}\right) = {f}_{r}\left( {e}_{d}\right) - {\dot{r}}_{d} \tag{11}
+$$
+
+where ${z}_{2d} = {\left\lbrack {e}_{d}^{\mathrm{T}},{\psi }_{e},{r}_{e}\right\rbrack }^{\mathrm{T}}$ . According to the Lemma 1, the above (11) can be approximated by NNs as follows:
+
+$$
+{F}_{2}\left( {z}_{2d}\right) = \left( {{\widehat{W}}_{2}^{\mathrm{T}} + {\widetilde{W}}_{2}^{\mathrm{T}}}\right) {\sigma }_{2}\left( {z}_{2d}\right) + {\varepsilon }_{2}\left( {z}_{2d}\right) \tag{12}
+$$
+
+where ${\widetilde{W}}_{2}^{\mathrm{T}} = {W}_{2} - {\widehat{W}}_{2}$ is the NNs approximate error and ${\widehat{W}}_{2}$ is the estimation of optimal weight vector ${W}_{2}$ . Through (11) and (12), ${f}_{r}\left( \cdot \right) - {f}_{r}\left( {e}_{d}\right)$ can be approximated as follows:
+
+$$
+{f}_{r}\left( \cdot \right) - {f}_{r}\left( {e}_{d}\right) = {F}_{2}\left( {z}_{2}\right) - {F}_{2}\left( {z}_{2d}\right)
+$$
+
+$$
+= p\left( e\right) + {\widetilde{W}}_{2}^{\mathrm{T}}\left\lbrack {{\sigma }_{2}\left( {z}_{2}\right) - {\sigma }_{2}\left( {z}_{2d}\right) }\right\rbrack + {\varepsilon }_{2}\left( {z}_{2}\right)
+$$
+
+$$
+- {\varepsilon }_{2}\left( {z}_{2d}\right)
+$$
+
+(13)
+
+where $p\left( e\right) = {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) - {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) , e = {\left\lbrack {\psi }_{e},{r}_{e}\right\rbrack }^{\mathrm{T}}$ and ${F}_{2}\left( {z}_{2}\right) = {f}_{r}\left( \cdot \right) - {\dot{r}}_{d}$ is a function of ${r}_{d}$ and ${\dot{r}}_{d}$ . The input ${z}_{2}$ is chosen as ${\left( {e}_{d}^{\mathrm{T}},\psi , r,{\delta }_{s}, u, v\right) }^{\mathrm{T}}$ . According to (12) and (13), (10) can be written as
+
+$$
+{\dot{r}}_{e} = p\left( e\right) + {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) + {\widetilde{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) + {\varepsilon }_{2}\left( {z}_{2}\right) \tag{14}
+$$
+
+$$
++ {g}_{r}{\tau }_{r}^{\alpha } + {g}_{r}{\tau }_{r}^{ * }
+$$
+
+To construct the feedforward virtual control input ${\tau }_{r}^{\alpha }$ , consider the BLF as
+
+$$
+{V}_{2} = {V}_{1} + \frac{1}{2}\log \left( \frac{{k}_{r}^{2}}{{k}_{r}^{2} - {r}_{e}^{2}}\right) + \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2} \tag{15}
+$$
+
+where ${k}_{r}$ is a positive value of the motion constraint. Calculate the derivate of ${V}_{2}$ , we have
+
+$$
+{\dot{V}}_{2} = - {k}_{1}{\psi }_{e}^{2} + \frac{{\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\left( {{r}_{e} + {r}_{d}^{ * }}\right) + \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}(p\left( e\right)
+$$
+
+$$
++ {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) + {\widetilde{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) + {\varepsilon }_{2}\left( {z}_{2}\right) + {g}_{r}{\tau }_{r}^{\alpha } \tag{16}
+$$
+
+$$
+\left. {+{g}_{r}{\tau }_{r}^{ * }}\right) - {\widetilde{W}}_{2}^{\mathrm{T}}{\dot{\widehat{W}}}_{2}
+$$
+
+According to the Young's inequality, we can get that
+
+$$
+\frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}{\varepsilon }_{2}\left( {z}_{2}\right) \leq \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}{\bar{\varepsilon }}_{2} \leq \frac{1}{2}\frac{{r}_{e}^{2}}{{\left( {k}_{r}^{2} - {r}_{e}^{2}\right) }^{2}} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} \tag{17}
+$$
+
+Substituting (17) into (16), we have
+
+$$
+{\dot{V}}_{2} = - {k}_{1}{\psi }_{e}^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} - {\widetilde{W}}_{2}^{\mathrm{T}}{\dot{\widehat{W}}}_{2} + \frac{{\psi }_{e}{r}_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}} + \frac{{\psi }_{e}{r}_{d}^{ * }}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}
+$$
+
+$$
++ \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}\left( {p\left( e\right) + {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) + {\widetilde{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) }\right.
+$$
+
+$$
+\left. {+{\varepsilon }_{2}\left( {z}_{2}\right) + {g}_{r}{\tau }_{r}^{\alpha } + {g}_{r}{\tau }_{r}^{ * }}\right)
+$$
+
+(18)
+
+Therefore, the feedforward control input ${\tau }_{r}^{\alpha }$ can be designed as
+
+$$
+{\tau }_{r}^{\alpha } = - \frac{1}{{g}_{r}}\left\lbrack {\left( {{k}_{r}^{2} - {r}_{e}^{2}}\right) {k}_{2}{r}_{e} + \frac{\left( {{k}_{r}^{2} - {r}_{e}^{2}}\right) {\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}} + {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) }\right\rbrack
+$$
+
+(19)
+
+where ${k}_{2} > 0$ is a tuning parameters. The NNs weight vector adaptation law ${\widehat{W}}_{2}$ can be designed as
+
+$$
+{\dot{\widehat{W}}}_{2} = \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}{\sigma }_{2}\left( {z}_{2}\right) - {\beta }_{2}{\widehat{W}}_{2} \tag{20}
+$$
+
+where ${\beta }_{2} > 0$ is also a tuning parameters. Substituting (19) and (20) into (18), we have
+
+$$
+{\dot{V}}_{2} \leq - {k}_{1}{\psi }_{e}^{2} - {k}_{2}{r}_{e}^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} + {\beta }_{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widehat{W}}_{2} + \frac{p\left( e\right) {r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}} \tag{21}
+$$
+
+$$
++ \frac{{r}_{e}{g}_{r}}{{k}_{r}^{2} - {r}_{e}^{2}}{\tau }_{r}^{ * } + \frac{{\psi }_{e}{r}_{d}^{ * }}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}
+$$
+
+According to the Young's inequality, we can get that
+
+$$
+{\widetilde{W}}_{2}^{\mathrm{T}}{\widehat{W}}_{2} = {\widetilde{W}}_{2}^{\mathrm{T}}\left( {{W}_{2} - {\widetilde{W}}_{2}}\right) = {\widetilde{W}}_{2}^{\mathrm{T}}{W}_{2} - {\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+$$
+\leq \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2} + \frac{1}{2}{W}_{2}^{\mathrm{T}}{W}_{2} - {\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2} \tag{22}
+$$
+
+$$
+= \frac{1}{2}{W}_{2}^{\mathrm{T}}{W}_{2} - \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+Therefore, the above (22) can be written as
+
+$$
+{\dot{V}}_{2} \leq - \underline{k}\parallel E{\parallel }^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} + \frac{1}{2}{\beta }_{2}{W}_{2}^{\mathrm{T}}{W}_{2} - \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+$$
++ \frac{p\left( e\right) {r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}} + \frac{{r}_{e}{g}_{r}}{{k}_{r}^{2} - {r}_{e}^{2}}{\tau }_{r}^{ * } + \frac{{\psi }_{e}{r}_{d}^{ * }}{{k}_{\psi }^{2} - {\psi }_{e}^{2}} \tag{23}
+$$
+
+where $E = \left\lbrack {{\psi }_{e},{r}_{e}}\right\rbrack \mathrm{T},\underline{k} = \min \left( {{k}_{1},{k}_{2}}\right)$ .
+
+In previous research, the feedforward controller ${\tau }_{r}^{\alpha }$ was calculated based on the derivative of the virtual controller ${\dot{r}}_{d}^{\alpha }$ . However, in practical applications, it is not easy to get analytical solutions for ${\dot{r}}_{d}^{\alpha }$ . Our proposed control method denotes ${r}_{d} = {r}_{d}^{\alpha } + {r}_{d}^{ * }$ , which is determined by both ${r}_{d}^{\alpha }$ and ${r}_{d}^{ * }$ and they are obtained through NNs weights and the system's state. Therefore, we approximate the derivative of the virtual controller using NNs in (11). With this approach, the feedforward controller ${r}_{d}^{\alpha }$ and ${\tau }_{r}^{\alpha }$ can be obtained directly from the current system without the need for the derivative of the virtual controller used in previous studies. Consequently, compared to previous work, the method we propose is more feasible to implement in practical applications.
+
+Rewriting (23) as follow:
+
+$$
+{\dot{V}}_{2} \leq - \underline{k}\parallel E{\parallel }^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} + \frac{1}{2}{\beta }_{2}{W}_{2}^{\mathrm{T}}{W}_{2} - \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+$$
++ \frac{{E}^{\mathrm{T}}}{{\widetilde{E}}^{\mathrm{T}}}\left( {\left\lbrack \begin{matrix} 0 \\ p\left( e\right) \end{matrix}\right\rbrack + \left\lbrack \begin{matrix} 1 & 0 \\ 0 & {g}_{r} \end{matrix}\right\rbrack \left\lbrack \begin{matrix} {r}_{d}^{ * } \\ {\tau }_{r}^{ * } \end{matrix}\right\rbrack }\right) \tag{24}
+$$
+
+where $\widetilde{E} = {\left\lbrack {k}_{\psi }^{2} - {\psi }_{e}^{2},{k}_{r}^{2} - {r}_{e}^{2}\right\rbrack }^{\mathrm{T}}$ . The feedforward controller is expressed as ${U}^{\alpha } = \left\lbrack {{r}_{d}^{\alpha },{\tau }_{r}^{\alpha }}\right\rbrack$ . The feedback optimal controller ${U}^{ * } = \left\lbrack {{r}_{d}^{ * },{\tau }_{r}^{ * }}\right\rbrack$ will be designed in the subsection. Therefore, they constitute the controller of the entire system.
+
+## B. Feedback optimal controller design
+
+According to (24), the design of an individual feedforward controller for ${U}^{\alpha }$ cannot guarantee the stability of the entire closed-loop system. Therefore, to ensure the stability of the last term in (24), a feedback optimal controller is designed based on ADP theory. With this design, not only the tracking ability of the system can be optimized, but also the system's stability can be ensured.
+
+The last term in (24) can be written as:
+
+$$
+\dot{E} = \left\lbrack \begin{matrix} 0 \\ p\left( e\right) \end{matrix}\right\rbrack + \left\lbrack \begin{matrix} 1 & 0 \\ 0 & {g}_{r} \end{matrix}\right\rbrack {U}^{ * } \tag{25}
+$$
+
+Further, it can be obtained that
+
+$$
+\dot{E} = P\left( E\right) + G\widehat{U} \tag{26}
+$$
+
+where $E = {\left\lbrack {\psi }_{e},{r}_{e}\right\rbrack }^{\mathrm{T}}$ are the heading angle error and yaw speed error, $P\left( E\right) = {\left\lbrack 0, p\left( e\right) \right\rbrack }^{\mathrm{T}}, G = \operatorname{diag}{\left\lbrack 1,{g}_{r}\right\rbrack }^{\mathrm{T}}$ .
+
+According to the ADP theory, the performance index function can be define as
+
+$$
+J\left( E\right) = {\int }_{t}^{\infty }{E}^{\mathrm{T}}{QE} + {\widehat{U}}^{\mathrm{T}}R\widehat{U}\mathrm{\;d}\tau \tag{27}
+$$
+
+where $Q \in {\mathbb{R}}^{2 \times 2}$ and $R \in {\mathbb{R}}^{2 \times 2}$ are positive definite matrices. The Hamiltonian function of the performance index function can be defined as
+
+$$
+H\left( {E,\widehat{U},\nabla J\left( E\right) }\right) = {E}^{\mathrm{T}}{QE} + {\widehat{U}}^{\mathrm{T}}R\widehat{U}
+$$
+
+$$
++ \nabla J{\left( E\right) }^{\mathrm{T}}\left( {P\left( E\right) + G\widehat{U}}\right) \tag{28}
+$$
+
+where $\nabla J\left( E\right) = \frac{\partial J\left( E\right) }{\partial E}$ denotes the derivative of $J\left( E\right)$ with regard to $E$ . In order to solve the HJB equation, the feedback optimal control ${U}^{ * }$ can be designed as
+
+$$
+{U}^{ * }\left( E\right) = - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}\nabla {J}^{ * }\left( E\right) \tag{29}
+$$
+
+From (28), the optimal performance index function ${J}^{ * }\left( E\right)$ can be obtained by
+
+$$
+\mathop{\min }\limits_{{\widehat{U}\left( E\right) }}H\left( {E,\widehat{U},\nabla {J}^{ * }\left( E\right) }\right) = 0 \tag{30}
+$$
+
+Substituting the above (30) into (28), the HJB equation can be rewritten as follow:
+
+$$
+{E}^{\mathrm{T}}{QE} + {\left( \nabla {J}^{ * }\left( E\right) \right) }^{\mathrm{T}}P\left( E\right) - \frac{1}{4}\left( {\left( \nabla {J}^{ * }\left( E\right) \right) }^{\mathrm{T}}\right. \tag{31}
+$$
+
+$$
+\left. {G{R}^{-1}{G}^{\mathrm{T}}\nabla {J}^{ * }\left( E\right) }\right) = 0
+$$
+
+It is obvious that the above equation is a nonlinear partial differential equation, so it is difficult to obtain its analytical solution. Therefore, to address this issue, the ADP theory is adopted. By constructing a single-layer NN to approximate the following optimal performance index function as
+
+$$
+{J}^{ * }\left( E\right) = {W}_{c}^{\mathrm{T}}\sigma \left( E\right) + {\varepsilon }_{c}\left( E\right) \tag{32}
+$$
+
+where ${W}_{c}$ denotes the optimal weight vector of critic NNs, $\sigma \left( \cdot \right)$ is the activation function, ${\varepsilon }_{c}\left( E\right)$ is the critic NNs approximation error.
+
+The gradient of the optimal performance index function ${J}^{ * }\left( E\right)$ with regard to $E$ can be defined as
+
+$$
+\nabla {J}^{ * }\left( E\right) = {\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{W}_{c} + \nabla {\varepsilon }_{c}\left( E\right) \tag{33}
+$$
+
+From (33) and (32), (29) can be rewritten as
+
+$$
+{U}^{ * }\left( E\right) = - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{W}_{c} - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}\nabla {\varepsilon }_{c}\left( E\right)
+$$
+
+(34)
+
+Therefore, the HJB equation can be further designed as
+
+$$
+H\left( {E,{U}^{ * },{W}_{c}}\right) = {E}^{\mathrm{T}}{QE} + {W}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) P\left( E\right) + {\varepsilon }_{HJB}
+$$
+
+$$
+- \frac{1}{4}\left( {{W}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) G{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{W}_{c}}\right)
+$$
+
+$$
+= 0
+$$
+
+(35)
+
+where ${\varepsilon }_{HJB}$ is the error.
+
+By using NNs to estimate the desired weights of performance index function as follow:
+
+$$
+\widehat{J}\left( E\right) = {\widehat{W}}_{c}^{\mathrm{T}}\sigma \left( E\right) \tag{36}
+$$
+
+where ${\widehat{W}}_{c}$ and $\widehat{J}\left( E\right)$ are estimations of ${W}_{c}$ and $J\left( E\right)$ , respectively. Let weight estimation error as ${\widetilde{W}}_{c} = {W}_{c} - {\widehat{W}}_{c}$ , the estimate of optimal control ${U}^{ * }$ can be designed as
+
+$$
+\bar{U}\left( E\right) = - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{\widehat{W}}_{c} \tag{37}
+$$
+
+Then, the HJB equation can be approximated as
+
+$$
+H\left( {E,\bar{U},{W}_{c}}\right) = {E}^{\mathrm{T}}{QE} + {W}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) P\left( E\right)
+$$
+
+$$
+- \frac{1}{4}\left( {{\widehat{W}}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) G{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{\widehat{W}}_{c}}\right)
+$$
+
+$$
+= {e}_{c}
+$$
+
+(38)
+
+The objective error function of critic NNs is defined as
+
+$$
+{E}_{c} = \frac{1}{2}{e}_{c}^{2} \tag{39}
+$$
+
+From the [18], we can design a appropriate critic NNs updating law, which can guarantee that ${\widehat{W}}_{c}$ converges to ${W}_{c}$ and also minimize the objective error function (39).
+
+$$
+{\dot{\widehat{W}}}_{c} = - {k}_{c}\frac{\Gamma }{{\left( 1 + {\Gamma }^{\mathrm{T}}\Gamma \right) }^{2}}{e}_{c} + \frac{{k}_{c}}{2}\Delta \nabla \sigma \left( E\right) G\nabla V\left( E\right)
+$$
+
+$$
++ {k}_{c}\left\lbrack {\frac{1}{4}\frac{\Gamma }{{\left( 1 + {\Gamma }^{\mathrm{T}}\Gamma \right) }^{2}}{\widehat{W}}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) G{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{\widehat{W}}_{c}}\right\rbrack
+$$
+
+$$
++ {k}_{c}\left\lbrack {{K}_{1}{\zeta }^{\mathrm{T}}{\widehat{W}}_{c} - {K}_{2}{\widehat{W}}_{c}}\right\rbrack
+$$
+
+(40)
+
+where ${k}_{c} > 0$ is the tuning parameter, $\Gamma = \nabla \sigma \left( E\right) (P\left( E\right) +$ $G\bar{U}),\zeta = \frac{\Gamma }{1 + {\Gamma }^{\mathrm{T}}\Gamma },{K}_{1}$ and ${K}_{2}$ are the tuning parameter. $\Delta$ is designed as
+
+$$
+\Delta = \left\{ \begin{array}{l} 0,{\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}\left( {P\left( E\right) + G\bar{U}}\right) < 0 \\ 1,\text{ else } \end{array}\right. \tag{41}
+$$
+
+where $V\left( E\right)$ is a Lyapunov function. From this, we can obtain
+
+$$
+\dot{V}\left( E\right) = {\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}\dot{E} = {\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}\left( {P\left( E\right) + G{U}^{ * }}\right) \tag{42}
+$$
+
+$$
+= - {\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}S\nabla V\left( E\right) \leq 0
+$$
+
+where $S$ is a positive definite matrix. Specifically, $V\left( E\right)$ is a function of the state variable $E$ and can be chosen appropriately, for example, $V\left( E\right) = {E}^{\mathrm{T}}E$ .
+
+Remark 1: The weight ${\widehat{W}}_{c}$ update process consists of the following four components: The first component employs gradient descent for design. The second component ensures the boundedness of the weights. The third and fourth components guarantee the stability of the weights. Through this design, the proposed control strategy achieves a higher tracking accuracy while ensuring the rapid and stable update of the neural network weights.
+
+## IV. SIMULATION
+
+The model parameters of the unmanned sailboat are selected from [10]. In order to facilitate simulation analysis without losing generality, the reference heading is set as ${\psi }_{d} = \sin \left( \mathrm{t}\right)$ . Select control parameters as ${k}_{\psi } = {1.2},{k}_{r} = {1.5},{k}_{1} = 3,{k}_{2} =$ $6,{k}_{c} = {3.8},{\beta }_{2} = 4,{K}_{1} = {0.0001}\mathrm{I},{K}_{2} = {0.00001}\mathrm{I}, Q = \mathrm{I}$ , $R = {0.25}\mathrm{I}$ . The time step is set as 0.05 . The initial states are defined as $\psi \left( 0\right) = {0.05}$ and $r\left( 0\right) = 0$ . The activation function of the critic network is chosen as $\sigma \left( E\right) = \left\lbrack {{e}_{1},{e}_{1}^{2},{e}_{2},{e}_{2}^{2},{e}_{1}{e}_{2}}\right\rbrack$ , the network weights are selected randomly in $\left\lbrack {0,1}\right\rbrack$ . Simulate real ocean and wind disturbances by using first-order Markov perturbations.
+
+To verify the superiority of the proposed strategy, we will compare the "LSBG" strategy of [10]. Fig. 1 shows the real-time curve of heading angle tracking, and the results show that the designed optimal control method can track the reference signal with smaller errors and within state constraints. The heading angular velocity tracking and its state constraints are shown in Fig. 2. Fig. 3 and Fig. 4 illustrate the error curves of heading angle and heading angular velocity, indicating that the proposed strategy achieves better tracking performance than the "LSBG" strategy. Fig. 5 displays the control inputs for control input ${\tau }_{r}$ under "LSBG" strategy, backstepping feedforward control, optimal feedback control, and system control under the proposed strategy, respectively. Fig. 6 shows the update curve of the evaluation network weights, it is obvious that the speed of online learning has reached stability in a very short time. From the above analysis, the proposed strategy can not only ensure better tracking accuracy, but also avoid system state violations of constraints.
+
+## V. CONCLUSION
+
+In this paper, the optimal control method based on ADP is proposed for the tracking control of unmanned sailboats with heading turning constraints. The proposed LBF-based method solves the problem of state constraints. The feedforward backstepping controller and the feedback optimal controller were designed using the backstepping method and ADP theory, respectively. The learning ability of critic NNs has been accelerated through online learning strategies. Finally, the simulation verified the optimality of the proposed strategy. In the future, we will apply this method to the path-tracking task of unmanned sailboats in pratice.
+
+
+
+Fig. 1. Comparison of heading angle tracking under different strategies.
+
+
+
+Fig. 2. Comparison of heading angle speed tracking under different strategies.
+
+
+
+Fig. 3. Comparison of heading angle error under different strategies.
+
+
+
+Fig. 4. Comparison of heading angle speed error under different strategies.
+
+
+
+Fig. 5. Control input ${\tau }_{r}$ under "LSBG" strategy, feedforward control input ${\tau }_{r}^{\alpha }$ , feedback control input ${\tau }_{r}^{ * }$ and optimal control input ${\tau }_{r}$ .
+
+## REFERENCES
+
+[1] Y. Deng, S. Zhang, J. Yan, N. Im and W. Zhou, "Adaptive asymptotic tracking control of autonomous underwater vehicles based on Bernstein polynomial approximation," Ocean Engineering, vol. 288, pp. 116220, 2023.
+
+[2] J. Zhang, T. Yang and T. Chai, "Neural Network Control of Underactuat-ed Surface Vehicles With Prescribed Trajectory Tracking Performance," IEEE Transactions on Neural Networks and Learning Systems, vol. 35, no. 6, pp. 8026-8039, June 2024.
+
+[3] G. Zhu, Y. Ma, Z. Li, R. Malekian and M. Sotelo, "Event-Triggered Adaptive Neural Fault-Tolerant Control of Underactuated MSVs With Input Saturation," IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 7, pp. 7045-7057, July 2022.
+
+[4] J. Chen, X. Hu, C. Lv, Z. Zhang and R. Ma, "Adaptive event-triggered fuzzy tracking control for underactuated surface vehicles under external disturbances," Ocean Engineering, vol. 283, pp. 115026, 2023.
+
+[5] G. Wu, Y. Ding, T. Tahsin and I. Atilla, "Adaptive neural network and extended state observer-based non-singular terminal sliding modetracking control for an underactuated USV with unknown uncertainties," Applied Ocean Research, vol. 135, pp. 103560, 2023.
+
+
+
+Fig. 6. Critic ADP weight update curve.
+
+[6] Ruiting Chu and Zhiquan Liu and Zhenzhong Chu, "Improved super-twisting sliding mode control for ship heading with sideslip angle compensation," Ocean Engineering, vol. 260, pp. 111996, 2022.
+
+[7] G. Zhang, L. Wang, J. Li and W. Zhang, "Improved LVS guidance and path-following control for unmanned sailboat robot with the minimum triggered setting," Ocean Engineering, vol. 272, pp. 113860, 2023.
+
+[8] Z. Shen, Y. Liu, Y. Nie and H. Yu, "Prescribed performance LOS guidance-based dynamic surface path following control of unmanned sailboats," Ocean Engineering, vol. 284, pp. 115182, 2023.
+
+[9] K. Shao, N. Wang, H. Qin, "Sideslip angle observation-based LOS and adaptive finite-time path following control for sailboat," Ocean Engineering, vol. 281, pp. 114636, 2023.
+
+[10] Y. Deng, X. Zhang and G. Zhang, "Line-of-Sight-Based Guidance and Adaptive Neural Path-Following Control for Sailboats," IEEE Journal of Oceanic Engineering, vol. 45, no. 4, pp. 1177-1189, Oct. 2020
+
+[11] H. Saoud, M. -D. Hua, F. Plumet and F. Ben Amar, "Routing and course control of an autonomous sailboat," 2015 European Conference on Mobile Robots (ECMR), Lincoln, UK, 2015, pp. 1-6.
+
+[12] X. Xiao, T. I. Fossen and J. Jouffroy, "Nonlinear Robust Heading Control for Sailing Yachts," IFAC Proceedings Volumes, vol. 45, no. 27, pp. 404-409, 2012.
+
+[13] P. Werbos, "Advanced forecasting methods for global crisis warning and models of intelligence," General System Yearbook, pp. 25-38, 1977.
+
+[14] Y. Deng, S. Zhang, Y. Xu, X. Zhang and W. Zhou, "Event-triggered optimal trajectory tracking control of underactuated ships based on goal representation heuristic dynamic programming," Ocean Engineering, vol. 308, pp. 118251, 2024.
+
+[15] X. Huang, J. Liu, C. Jia, Z. Wang and W. Li, "Online self-learning attitude tracking control of morphing unmanned aerial vehicle based on dual heuristic dynamic programming," Aerospace Science and Technology, vol. 143, pp. 108727, 2023.
+
+[16] X. Li, L. Wang, Y. An, Q. Huang, Y. Cui and H. Hu, "Dynamic path planning of mobile robots using adaptive dynamic programming," Expert Systems with Applications, vol. 235, pp. 121112, 2024.
+
+[17] J. Wang, P. Zhang, Y. Wang and Z. Ji, "Adaptive dynamic programming-based optimal control for nonlinear state constrained systems with input delay," Nonlinear Dynamics, vol. 111, pp. 19133-19149, 2023.
+
+[18] X. Yang, D. Liu and Q. Wei, "Online approximate optimal control for affine non-linear systems with unknown internal dynamics using adaptive dynamic programming," IET Control Theory & Applications, vol. 8, pp. 1676-1688, 2014.
\ No newline at end of file
diff --git a/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/lAtPkUMK1M/Initial_manuscript_tex/Initial_manuscript.tex b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/lAtPkUMK1M/Initial_manuscript_tex/Initial_manuscript.tex
new file mode 100644
index 0000000000000000000000000000000000000000..912d05f8d65b1c296290342b79088d380a158805
--- /dev/null
+++ b/papers/IEEE/IEEE ICIST/IEEE ICIST 2024/IEEE ICIST 2024 Conference/lAtPkUMK1M/Initial_manuscript_tex/Initial_manuscript.tex
@@ -0,0 +1,475 @@
+§ ADAPTIVE DYNAMIC PROGRAMMING-BASED OPTIMAL HEADING CONTROL FOR STATE CONSTRAINED UNMANNED SAILBOAT
+
+${1}^{\text{ st }}$ Shitong Zhang
+
+School of Mechanical Engineering
+
+Yanshan University
+
+Qinhuangdao, China
+
+bben@stumail.ysu.edu.cn
+
+${2}^{\text{ st }}$ Yifei Xu
+
+School of Mechanical Engineering Yanshan University
+
+Qinhuangdao, China
+
+xyf@ysu.edu.cn
+
+${3}^{\text{ st }}$ Yingjie Deng*
+
+School of Mechanical Engineering
+
+Yanshan University
+
+Qinhuangdao, China
+
+dyj@ysu.edu.cn
+
+${4}^{\text{ st }}$ Sheng Xu
+
+Shenzhen Institute of Advanced Technology
+
+Chinese Academy of Sciences
+
+Shenzhen, China
+
+${Abstract}$ -This paper proposes a new state-constrained adaptive optimal control strategy for unmanned sailboat heading angle tracking considering the motion constraints. To improve the tracking accuracy, a combination of backstepping and adaptive dynamic programming (ADP) is employed. Thus, the issue of virtual control rate derivatives in conventional backstepping control is resolved with satisfactory precision. Firstly, the motion constraints are considered by using the Barrier Lyapunov function (BLF), and the neural networks (NNs) is employed to approximate the model uncertainties and disturbances. Secondly, an adaptive backstepping feedforward controller is proposed, transforming the sailboat's affine nonlinear system tracking problem into a regulation problem. Thirdly, according to the ADP theory, critic NNs are constructed to approximate the analytical solution of the Hamilton-Jacobi-Bellman (HJB) equation, and the optimal feedback control is obtained by online learning. Finally, simulation results demonstrate the effectiveness and optimality of the proposed controller.
+
+Index Terms-Optimal control, Adaptive dynamic programming (ADP), barrier Lyapunov function (BLF), Neural networks (NNs), Unmanned sailboat
+
+§ I. INTRODUCTION
+
+In the past few decades, surface and underwater intelligent vehicles have played important roles in sea patrols, resource exploration, and rescue. However, due to the consumption of fuel, electricity, and other energy sources, unmanned ships and submarines require a large amount of power support to complete long-distance tasks, which also leads to huge cost problems in [1]-[6]. Since unmanned sailboats can use sails as power, have low costs, and can transmit data in real-time through their sensors, research on unmanned sailboats has gradually entered the interest of scholars. However, due to the complex marine environment, the disturbances of wind and waves can easily generate obstructive forces on the sails and keel, making the sails control difficult. Therefore, the authors of [7] proposed a path tracking method for unmanned sailboats that combines logic virtual ship (LVS) guidance law and dynamic event-triggered control. In [8], a path tracking control scheme is proposed for unmanned sailboats that combined backstepping and dynamic surface control technology. In order to reduce the sideslip angle error during sailing, the authors of [9] proposed double finite-time observers-based line-of-sight guidance (DFLOS) and adaptive finite-time control (DFLOS-AFC) strategies. However, they did not consider the issue of state constraints. When subjected to significant interference, to return the sailboat to the reference heading, a larger rudder angle is required, which will damage the actuator as it cannot accept significant deflection in a short period. Therefore, while ensuring the accuracy of sailboat heading control, it is also necessary to ensure that the turning speed of the boat heading satisfies the prescribed constraints.
+
+In order to solve the problem of unmanned sailboat heading angle tracking accuracy, many scholars have conducted research on it. In [10], a control strategy of adaptive echo state networks and backstepping is proposed to control the steering angle of the rudder. The authors of [11] designed a nonlinear heading controller of velocity vector direction to track the reference heading angle. In [12], the ${L}_{1}$ adaptive control theory is proposed to complete heading control and ensure the stability of the required heading angle. However, most papers focused on the heading control of sailboats using backstepping or sign functions, which results in low tracking accuracy and cannot achieve optimal control effects.
+
+The key problem of the nonlinear optimal control is that the analytical solution of the Hamilton-Jacobi-Bellman (HJB) equation is difficult to solve. In order to obtain the analytical solution, the authors of [13] proposed the ADP theory, which approximates the solution of the equation online. However, the problem of slow approximation speed and multiple iterations has led to an explosion in computational complexity. Recently, with the rapid development of NNs, their approximation performance and speed for unknown nonlinear functions have become increasingly excellent. Therefore, the research combining ADP and NNs has solved the above problems. In [14], a new event-triggered optimal trajectory tracking control method based on goal representation heuristic dynamic programming (GrHDP) for underactuated ships is proposed. In [15], they proposed a model free dual heuristic dynamic programming (DHP) method for unmanned aerial vehicle attitude control. The authors of [16] applied ADP theory to the path planning problem of mobile robots. However, to the best of our knowledge, there is limited research on applying ADP theory to the heading control of unmanned sailboats.
+
+This work is partially supported by the Natural Science Foundation of China (No.52101375), the Hebei Province Natural Science Fund (No.E2022203088, E2024203179), the Innovation Capacity Enhancement Program of Hebei Province (No.24461901D), the Joint Funds of the National Natural Science Foundation of China (No.U20A20332), and the Key Research and Development Project of Hebei Province (No.21351802D).
+
+Taking inspiration from the above analysis, we introduce the LBF function to solve the state constraint problem. In the control design of heading tracking, while introducing the backstepping method, the ADP theory is also introduced to ensure the optimal tracking accuracy. We constructed an evaluation NN to approximate the analytical solution of the HJB equation and obtained the optimal feedback control through online learning. The main contributions are concluded below:
+
+ * Compared with previous papers of [7]-[9], we proposed a LBF-based method to overcome the issues of the motion constraints caused by the turning rate limitation of unmanned sailboats.
+
+ * By using the ADP optimal feedback control strategy, our designed control method achieves optimal tracking accuracy in heading control compared to [9]-[12].
+
+ * According to the construction of the critic network, a training speed acceleration method is developed using online learning methods.
+
+The remaining of this paper is organized as follows. Section II elaborates on the heading angle control model of unmanned sailboats and lemma. Section III designs the backstepping feedforward controller and the optimal feedback controller. Section IV provides simulation verification of the proposed strategy superiority. Section V gives some conclusions.
+
+§ II. MATHEMATICAL MODEL AND PRELIMINARIES
+
+The sailboat is divided into four parts: the sail, rudder, keel, and hull. Combined with the theory of gas fluid dynamics, force analysis is conducted on each part, ignoring the undulating and pitching motion of the sailboat to establish a 3-DOF mathematical model of sailboat motion. The sailboat model considering external interference and control input rudder angle in this paper is
+
+$$
+\left\{ \begin{array}{l} \dot{\psi } = r \\ \dot{r} = {f}_{r}\left( {\psi ,r,{\delta }_{s},u,v,{\tau }_{wr}}\right) + {g}_{r}{\tau }_{r} \end{array}\right. \tag{1}
+$$
+
+where $\psi \in {\mathbb{R}}^{M}$ and $r \in {\mathbb{R}}^{M}$ are system state variables, respectively; $u \in {\mathbb{R}}^{M}$ is the forward velocity; $v \in {\mathbb{R}}^{M}$ is the lateral velocity; ${\delta }_{s} \in {\mathbb{R}}^{M}$ is the sail angle; ${\tau }_{wr}$ is the external disturbance; ${f}_{r}\left( \cdot \right)$ is the nonlinear function of the unknown model of a sailboat; ${g}_{r}$ is the unknown control gain; ${\tau }_{r}$ is the control input of the system.
+
+Assumption 1 [17]: Due to the fact that unmanned sailboats navigate in limited space, there exists a normal number that satisfies the heading angle and heading angular speed to be less than or equal to this normal number. That is, $\psi \leq {k}_{\psi }$ and $r \leq {k}_{r}$ .
+
+Lemma 1 [10]: Due to the outstanding ability of NNs in function approximation, they are often used to approximate nonlinear functions. Therefore, NNs can be used to approximate unknown functions as follows:
+
+$$
+F\left( x\right) = {W}^{\mathrm{T}}\sigma \left( x\right) + \varepsilon \left( x\right) \tag{2}
+$$
+
+where $W = {\left( {w}_{1},{w}_{2},\ldots ,{w}_{n}\right) }^{\mathrm{T}}$ denotes the desired weight of the NNs and $\varepsilon \leq \bar{\varepsilon }$ denotes the approximating error.
+
+§ III. CONTROL DESIGN
+
+§ A. FEEDFORWARD CONTROLLER DESIGN
+
+In this section, the backstepping method based on an adaptive NNs framework is adopted to transform system (1) into an affine nonlinear system.
+
+According to the system (1), define the error system function as
+
+$$
+\left\{ \begin{matrix} {\psi }_{e} = \psi - {\psi }_{d} \\ {r}_{e} = r - {r}_{d} \end{matrix}\right. \tag{3}
+$$
+
+where ${\psi }_{d}$ and ${r}_{d}$ are the reference heading angle and yaw speed, respectively. Differentiating ${\psi }_{e}$ along with (1), it has:
+
+$$
+{\dot{\psi }}_{e} = \dot{\psi } - {\dot{\psi }}_{d} = r - {\dot{\psi }}_{d} = \left( {{r}_{e} + {r}_{d}}\right) - {\dot{\psi }}_{d} \tag{4}
+$$
+
+where ${r}_{d} = {r}_{d}^{\alpha } + {r}_{d}^{ * }$ is the virtual control input of yaw speed. ${r}_{d}^{\alpha }$ denotes the feedforward virtual yaw speed input and ${r}_{d}^{ * }$ denotes the feedback virtual yaw speed input. Therefore, we can get
+
+$$
+{\dot{\psi }}_{e} = {r}_{e} + {r}_{d}^{\alpha } + {r}_{d}^{ * } - {\dot{\psi }}_{d} \tag{5}
+$$
+
+To construct the desired feedforward yaw speed virtual control input, consider the BLF as
+
+$$
+{V}_{1} = \frac{1}{2}\log \left( \frac{{k}_{\psi }^{2}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\right) \tag{6}
+$$
+
+where ${k}_{\psi }$ is a positive value of the state constraint. Calculate the derivate of ${V}_{1}$ , we have
+
+$$
+{\dot{V}}_{1} = \frac{{\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\left( {{r}_{e} + {r}_{d}^{\alpha } + {r}_{d}^{ * } - {\dot{\psi }}_{d}}\right) \tag{7}
+$$
+
+Therefore, the feedforward virtual yaw speed input ${r}_{d}^{\alpha }$ can be designed as
+
+$$
+{r}_{d}^{\alpha } = - \left( {{k}_{\psi }^{2} - {\psi }_{e}^{2}}\right) {k}_{1}{\psi }_{e} + {\dot{\psi }}_{d} \tag{8}
+$$
+
+where ${k}_{1} > 0$ is a tuning parameters. Substituting (8) into (7), we have
+
+$$
+{\dot{V}}_{1} = - {k}_{1}{\psi }_{e}^{2} + \frac{{\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\left( {{r}_{e} + {r}_{d}^{ * }}\right) \tag{9}
+$$
+
+Taking the derivative of the second equation of (3) yields
+
+$$
+{\dot{r}}_{e} = {f}_{r}\left( \cdot \right) + {f}_{r}\left( {e}_{d}\right) - {f}_{r}\left( {e}_{d}\right) + {g}_{r}{\tau }_{r} - {\dot{r}}_{d} \tag{10}
+$$
+
+where ${e}_{d} = {\left\lbrack {\psi }_{d},{r}_{d}\right\rbrack }^{\mathrm{T}}$ . The unknown model uncertainty function ${f}_{r}\left( \cdot \right)$ and ${\dot{r}}_{d}$ can be transferred by the following function
+
+$$
+{F}_{2}\left( {z}_{2d}\right) = {f}_{r}\left( {e}_{d}\right) - {\dot{r}}_{d} \tag{11}
+$$
+
+where ${z}_{2d} = {\left\lbrack {e}_{d}^{\mathrm{T}},{\psi }_{e},{r}_{e}\right\rbrack }^{\mathrm{T}}$ . According to the Lemma 1, the above (11) can be approximated by NNs as follows:
+
+$$
+{F}_{2}\left( {z}_{2d}\right) = \left( {{\widehat{W}}_{2}^{\mathrm{T}} + {\widetilde{W}}_{2}^{\mathrm{T}}}\right) {\sigma }_{2}\left( {z}_{2d}\right) + {\varepsilon }_{2}\left( {z}_{2d}\right) \tag{12}
+$$
+
+where ${\widetilde{W}}_{2}^{\mathrm{T}} = {W}_{2} - {\widehat{W}}_{2}$ is the NNs approximate error and ${\widehat{W}}_{2}$ is the estimation of optimal weight vector ${W}_{2}$ . Through (11) and (12), ${f}_{r}\left( \cdot \right) - {f}_{r}\left( {e}_{d}\right)$ can be approximated as follows:
+
+$$
+{f}_{r}\left( \cdot \right) - {f}_{r}\left( {e}_{d}\right) = {F}_{2}\left( {z}_{2}\right) - {F}_{2}\left( {z}_{2d}\right)
+$$
+
+$$
+= p\left( e\right) + {\widetilde{W}}_{2}^{\mathrm{T}}\left\lbrack {{\sigma }_{2}\left( {z}_{2}\right) - {\sigma }_{2}\left( {z}_{2d}\right) }\right\rbrack + {\varepsilon }_{2}\left( {z}_{2}\right)
+$$
+
+$$
+- {\varepsilon }_{2}\left( {z}_{2d}\right)
+$$
+
+(13)
+
+where $p\left( e\right) = {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) - {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) ,e = {\left\lbrack {\psi }_{e},{r}_{e}\right\rbrack }^{\mathrm{T}}$ and ${F}_{2}\left( {z}_{2}\right) = {f}_{r}\left( \cdot \right) - {\dot{r}}_{d}$ is a function of ${r}_{d}$ and ${\dot{r}}_{d}$ . The input ${z}_{2}$ is chosen as ${\left( {e}_{d}^{\mathrm{T}},\psi ,r,{\delta }_{s},u,v\right) }^{\mathrm{T}}$ . According to (12) and (13), (10) can be written as
+
+$$
+{\dot{r}}_{e} = p\left( e\right) + {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) + {\widetilde{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) + {\varepsilon }_{2}\left( {z}_{2}\right) \tag{14}
+$$
+
+$$
++ {g}_{r}{\tau }_{r}^{\alpha } + {g}_{r}{\tau }_{r}^{ * }
+$$
+
+To construct the feedforward virtual control input ${\tau }_{r}^{\alpha }$ , consider the BLF as
+
+$$
+{V}_{2} = {V}_{1} + \frac{1}{2}\log \left( \frac{{k}_{r}^{2}}{{k}_{r}^{2} - {r}_{e}^{2}}\right) + \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2} \tag{15}
+$$
+
+where ${k}_{r}$ is a positive value of the motion constraint. Calculate the derivate of ${V}_{2}$ , we have
+
+$$
+{\dot{V}}_{2} = - {k}_{1}{\psi }_{e}^{2} + \frac{{\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}\left( {{r}_{e} + {r}_{d}^{ * }}\right) + \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}(p\left( e\right)
+$$
+
+$$
++ {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) + {\widetilde{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) + {\varepsilon }_{2}\left( {z}_{2}\right) + {g}_{r}{\tau }_{r}^{\alpha } \tag{16}
+$$
+
+$$
+\left. {+{g}_{r}{\tau }_{r}^{ * }}\right) - {\widetilde{W}}_{2}^{\mathrm{T}}{\dot{\widehat{W}}}_{2}
+$$
+
+According to the Young's inequality, we can get that
+
+$$
+\frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}{\varepsilon }_{2}\left( {z}_{2}\right) \leq \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}{\bar{\varepsilon }}_{2} \leq \frac{1}{2}\frac{{r}_{e}^{2}}{{\left( {k}_{r}^{2} - {r}_{e}^{2}\right) }^{2}} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} \tag{17}
+$$
+
+Substituting (17) into (16), we have
+
+$$
+{\dot{V}}_{2} = - {k}_{1}{\psi }_{e}^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} - {\widetilde{W}}_{2}^{\mathrm{T}}{\dot{\widehat{W}}}_{2} + \frac{{\psi }_{e}{r}_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}} + \frac{{\psi }_{e}{r}_{d}^{ * }}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}
+$$
+
+$$
++ \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}\left( {p\left( e\right) + {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) + {\widetilde{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2}\right) }\right.
+$$
+
+$$
+\left. {+{\varepsilon }_{2}\left( {z}_{2}\right) + {g}_{r}{\tau }_{r}^{\alpha } + {g}_{r}{\tau }_{r}^{ * }}\right)
+$$
+
+(18)
+
+Therefore, the feedforward control input ${\tau }_{r}^{\alpha }$ can be designed as
+
+$$
+{\tau }_{r}^{\alpha } = - \frac{1}{{g}_{r}}\left\lbrack {\left( {{k}_{r}^{2} - {r}_{e}^{2}}\right) {k}_{2}{r}_{e} + \frac{\left( {{k}_{r}^{2} - {r}_{e}^{2}}\right) {\psi }_{e}}{{k}_{\psi }^{2} - {\psi }_{e}^{2}} + {\widehat{W}}_{2}^{\mathrm{T}}{\sigma }_{2}\left( {z}_{2d}\right) }\right\rbrack
+$$
+
+(19)
+
+where ${k}_{2} > 0$ is a tuning parameters. The NNs weight vector adaptation law ${\widehat{W}}_{2}$ can be designed as
+
+$$
+{\dot{\widehat{W}}}_{2} = \frac{{r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}}{\sigma }_{2}\left( {z}_{2}\right) - {\beta }_{2}{\widehat{W}}_{2} \tag{20}
+$$
+
+where ${\beta }_{2} > 0$ is also a tuning parameters. Substituting (19) and (20) into (18), we have
+
+$$
+{\dot{V}}_{2} \leq - {k}_{1}{\psi }_{e}^{2} - {k}_{2}{r}_{e}^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} + {\beta }_{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widehat{W}}_{2} + \frac{p\left( e\right) {r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}} \tag{21}
+$$
+
+$$
++ \frac{{r}_{e}{g}_{r}}{{k}_{r}^{2} - {r}_{e}^{2}}{\tau }_{r}^{ * } + \frac{{\psi }_{e}{r}_{d}^{ * }}{{k}_{\psi }^{2} - {\psi }_{e}^{2}}
+$$
+
+According to the Young's inequality, we can get that
+
+$$
+{\widetilde{W}}_{2}^{\mathrm{T}}{\widehat{W}}_{2} = {\widetilde{W}}_{2}^{\mathrm{T}}\left( {{W}_{2} - {\widetilde{W}}_{2}}\right) = {\widetilde{W}}_{2}^{\mathrm{T}}{W}_{2} - {\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+$$
+\leq \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2} + \frac{1}{2}{W}_{2}^{\mathrm{T}}{W}_{2} - {\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2} \tag{22}
+$$
+
+$$
+= \frac{1}{2}{W}_{2}^{\mathrm{T}}{W}_{2} - \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+Therefore, the above (22) can be written as
+
+$$
+{\dot{V}}_{2} \leq - \underline{k}\parallel E{\parallel }^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} + \frac{1}{2}{\beta }_{2}{W}_{2}^{\mathrm{T}}{W}_{2} - \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+$$
++ \frac{p\left( e\right) {r}_{e}}{{k}_{r}^{2} - {r}_{e}^{2}} + \frac{{r}_{e}{g}_{r}}{{k}_{r}^{2} - {r}_{e}^{2}}{\tau }_{r}^{ * } + \frac{{\psi }_{e}{r}_{d}^{ * }}{{k}_{\psi }^{2} - {\psi }_{e}^{2}} \tag{23}
+$$
+
+where $E = \left\lbrack {{\psi }_{e},{r}_{e}}\right\rbrack \mathrm{T},\underline{k} = \min \left( {{k}_{1},{k}_{2}}\right)$ .
+
+In previous research, the feedforward controller ${\tau }_{r}^{\alpha }$ was calculated based on the derivative of the virtual controller ${\dot{r}}_{d}^{\alpha }$ . However, in practical applications, it is not easy to get analytical solutions for ${\dot{r}}_{d}^{\alpha }$ . Our proposed control method denotes ${r}_{d} = {r}_{d}^{\alpha } + {r}_{d}^{ * }$ , which is determined by both ${r}_{d}^{\alpha }$ and ${r}_{d}^{ * }$ and they are obtained through NNs weights and the system's state. Therefore, we approximate the derivative of the virtual controller using NNs in (11). With this approach, the feedforward controller ${r}_{d}^{\alpha }$ and ${\tau }_{r}^{\alpha }$ can be obtained directly from the current system without the need for the derivative of the virtual controller used in previous studies. Consequently, compared to previous work, the method we propose is more feasible to implement in practical applications.
+
+Rewriting (23) as follow:
+
+$$
+{\dot{V}}_{2} \leq - \underline{k}\parallel E{\parallel }^{2} + \frac{1}{2}{\bar{\varepsilon }}_{2}^{2} + \frac{1}{2}{\beta }_{2}{W}_{2}^{\mathrm{T}}{W}_{2} - \frac{1}{2}{\widetilde{W}}_{2}^{\mathrm{T}}{\widetilde{W}}_{2}
+$$
+
+$$
++ \frac{{E}^{\mathrm{T}}}{{\widetilde{E}}^{\mathrm{T}}}\left( {\left\lbrack \begin{matrix} 0 \\ p\left( e\right) \end{matrix}\right\rbrack + \left\lbrack \begin{matrix} 1 & 0 \\ 0 & {g}_{r} \end{matrix}\right\rbrack \left\lbrack \begin{matrix} {r}_{d}^{ * } \\ {\tau }_{r}^{ * } \end{matrix}\right\rbrack }\right) \tag{24}
+$$
+
+where $\widetilde{E} = {\left\lbrack {k}_{\psi }^{2} - {\psi }_{e}^{2},{k}_{r}^{2} - {r}_{e}^{2}\right\rbrack }^{\mathrm{T}}$ . The feedforward controller is expressed as ${U}^{\alpha } = \left\lbrack {{r}_{d}^{\alpha },{\tau }_{r}^{\alpha }}\right\rbrack$ . The feedback optimal controller ${U}^{ * } = \left\lbrack {{r}_{d}^{ * },{\tau }_{r}^{ * }}\right\rbrack$ will be designed in the subsection. Therefore, they constitute the controller of the entire system.
+
+§ B. FEEDBACK OPTIMAL CONTROLLER DESIGN
+
+According to (24), the design of an individual feedforward controller for ${U}^{\alpha }$ cannot guarantee the stability of the entire closed-loop system. Therefore, to ensure the stability of the last term in (24), a feedback optimal controller is designed based on ADP theory. With this design, not only the tracking ability of the system can be optimized, but also the system's stability can be ensured.
+
+The last term in (24) can be written as:
+
+$$
+\dot{E} = \left\lbrack \begin{matrix} 0 \\ p\left( e\right) \end{matrix}\right\rbrack + \left\lbrack \begin{matrix} 1 & 0 \\ 0 & {g}_{r} \end{matrix}\right\rbrack {U}^{ * } \tag{25}
+$$
+
+Further, it can be obtained that
+
+$$
+\dot{E} = P\left( E\right) + G\widehat{U} \tag{26}
+$$
+
+where $E = {\left\lbrack {\psi }_{e},{r}_{e}\right\rbrack }^{\mathrm{T}}$ are the heading angle error and yaw speed error, $P\left( E\right) = {\left\lbrack 0,p\left( e\right) \right\rbrack }^{\mathrm{T}},G = \operatorname{diag}{\left\lbrack 1,{g}_{r}\right\rbrack }^{\mathrm{T}}$ .
+
+According to the ADP theory, the performance index function can be define as
+
+$$
+J\left( E\right) = {\int }_{t}^{\infty }{E}^{\mathrm{T}}{QE} + {\widehat{U}}^{\mathrm{T}}R\widehat{U}\mathrm{\;d}\tau \tag{27}
+$$
+
+where $Q \in {\mathbb{R}}^{2 \times 2}$ and $R \in {\mathbb{R}}^{2 \times 2}$ are positive definite matrices. The Hamiltonian function of the performance index function can be defined as
+
+$$
+H\left( {E,\widehat{U},\nabla J\left( E\right) }\right) = {E}^{\mathrm{T}}{QE} + {\widehat{U}}^{\mathrm{T}}R\widehat{U}
+$$
+
+$$
++ \nabla J{\left( E\right) }^{\mathrm{T}}\left( {P\left( E\right) + G\widehat{U}}\right) \tag{28}
+$$
+
+where $\nabla J\left( E\right) = \frac{\partial J\left( E\right) }{\partial E}$ denotes the derivative of $J\left( E\right)$ with regard to $E$ . In order to solve the HJB equation, the feedback optimal control ${U}^{ * }$ can be designed as
+
+$$
+{U}^{ * }\left( E\right) = - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}\nabla {J}^{ * }\left( E\right) \tag{29}
+$$
+
+From (28), the optimal performance index function ${J}^{ * }\left( E\right)$ can be obtained by
+
+$$
+\mathop{\min }\limits_{{\widehat{U}\left( E\right) }}H\left( {E,\widehat{U},\nabla {J}^{ * }\left( E\right) }\right) = 0 \tag{30}
+$$
+
+Substituting the above (30) into (28), the HJB equation can be rewritten as follow:
+
+$$
+{E}^{\mathrm{T}}{QE} + {\left( \nabla {J}^{ * }\left( E\right) \right) }^{\mathrm{T}}P\left( E\right) - \frac{1}{4}\left( {\left( \nabla {J}^{ * }\left( E\right) \right) }^{\mathrm{T}}\right. \tag{31}
+$$
+
+$$
+\left. {G{R}^{-1}{G}^{\mathrm{T}}\nabla {J}^{ * }\left( E\right) }\right) = 0
+$$
+
+It is obvious that the above equation is a nonlinear partial differential equation, so it is difficult to obtain its analytical solution. Therefore, to address this issue, the ADP theory is adopted. By constructing a single-layer NN to approximate the following optimal performance index function as
+
+$$
+{J}^{ * }\left( E\right) = {W}_{c}^{\mathrm{T}}\sigma \left( E\right) + {\varepsilon }_{c}\left( E\right) \tag{32}
+$$
+
+where ${W}_{c}$ denotes the optimal weight vector of critic NNs, $\sigma \left( \cdot \right)$ is the activation function, ${\varepsilon }_{c}\left( E\right)$ is the critic NNs approximation error.
+
+The gradient of the optimal performance index function ${J}^{ * }\left( E\right)$ with regard to $E$ can be defined as
+
+$$
+\nabla {J}^{ * }\left( E\right) = {\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{W}_{c} + \nabla {\varepsilon }_{c}\left( E\right) \tag{33}
+$$
+
+From (33) and (32), (29) can be rewritten as
+
+$$
+{U}^{ * }\left( E\right) = - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{W}_{c} - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}\nabla {\varepsilon }_{c}\left( E\right)
+$$
+
+(34)
+
+Therefore, the HJB equation can be further designed as
+
+$$
+H\left( {E,{U}^{ * },{W}_{c}}\right) = {E}^{\mathrm{T}}{QE} + {W}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) P\left( E\right) + {\varepsilon }_{HJB}
+$$
+
+$$
+- \frac{1}{4}\left( {{W}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) G{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{W}_{c}}\right)
+$$
+
+$$
+= 0
+$$
+
+(35)
+
+where ${\varepsilon }_{HJB}$ is the error.
+
+By using NNs to estimate the desired weights of performance index function as follow:
+
+$$
+\widehat{J}\left( E\right) = {\widehat{W}}_{c}^{\mathrm{T}}\sigma \left( E\right) \tag{36}
+$$
+
+where ${\widehat{W}}_{c}$ and $\widehat{J}\left( E\right)$ are estimations of ${W}_{c}$ and $J\left( E\right)$ , respectively. Let weight estimation error as ${\widetilde{W}}_{c} = {W}_{c} - {\widehat{W}}_{c}$ , the estimate of optimal control ${U}^{ * }$ can be designed as
+
+$$
+\bar{U}\left( E\right) = - \frac{1}{2}{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{\widehat{W}}_{c} \tag{37}
+$$
+
+Then, the HJB equation can be approximated as
+
+$$
+H\left( {E,\bar{U},{W}_{c}}\right) = {E}^{\mathrm{T}}{QE} + {W}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) P\left( E\right)
+$$
+
+$$
+- \frac{1}{4}\left( {{\widehat{W}}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) G{R}^{-1}{G}^{\mathrm{T}}{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{\widehat{W}}_{c}}\right)
+$$
+
+$$
+= {e}_{c}
+$$
+
+(38)
+
+The objective error function of critic NNs is defined as
+
+$$
+{E}_{c} = \frac{1}{2}{e}_{c}^{2} \tag{39}
+$$
+
+From the [18], we can design a appropriate critic NNs updating law, which can guarantee that ${\widehat{W}}_{c}$ converges to ${W}_{c}$ and also minimize the objective error function (39).
+
+$$
+{\dot{\widehat{W}}}_{c} = - {k}_{c}\frac{\Gamma }{{\left( 1 + {\Gamma }^{\mathrm{T}}\Gamma \right) }^{2}}{e}_{c} + \frac{{k}_{c}}{2}\Delta \nabla \sigma \left( E\right) G\nabla V\left( E\right)
+$$
+
+$$
++ {k}_{c}\left\lbrack {\frac{1}{4}\frac{\Gamma }{{\left( 1 + {\Gamma }^{\mathrm{T}}\Gamma \right) }^{2}}{\widehat{W}}_{c}^{\mathrm{T}}\nabla \sigma \left( E\right) G{\left( \nabla \sigma \left( E\right) \right) }^{\mathrm{T}}{\widehat{W}}_{c}}\right\rbrack
+$$
+
+$$
++ {k}_{c}\left\lbrack {{K}_{1}{\zeta }^{\mathrm{T}}{\widehat{W}}_{c} - {K}_{2}{\widehat{W}}_{c}}\right\rbrack
+$$
+
+(40)
+
+where ${k}_{c} > 0$ is the tuning parameter, $\Gamma = \nabla \sigma \left( E\right) (P\left( E\right) +$ $G\bar{U}),\zeta = \frac{\Gamma }{1 + {\Gamma }^{\mathrm{T}}\Gamma },{K}_{1}$ and ${K}_{2}$ are the tuning parameter. $\Delta$ is designed as
+
+$$
+\Delta = \left\{ \begin{array}{l} 0,{\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}\left( {P\left( E\right) + G\bar{U}}\right) < 0 \\ 1,\text{ else } \end{array}\right. \tag{41}
+$$
+
+where $V\left( E\right)$ is a Lyapunov function. From this, we can obtain
+
+$$
+\dot{V}\left( E\right) = {\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}\dot{E} = {\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}\left( {P\left( E\right) + G{U}^{ * }}\right) \tag{42}
+$$
+
+$$
+= - {\left( \nabla V\left( E\right) \right) }^{\mathrm{T}}S\nabla V\left( E\right) \leq 0
+$$
+
+where $S$ is a positive definite matrix. Specifically, $V\left( E\right)$ is a function of the state variable $E$ and can be chosen appropriately, for example, $V\left( E\right) = {E}^{\mathrm{T}}E$ .
+
+Remark 1: The weight ${\widehat{W}}_{c}$ update process consists of the following four components: The first component employs gradient descent for design. The second component ensures the boundedness of the weights. The third and fourth components guarantee the stability of the weights. Through this design, the proposed control strategy achieves a higher tracking accuracy while ensuring the rapid and stable update of the neural network weights.
+
+§ IV. SIMULATION
+
+The model parameters of the unmanned sailboat are selected from [10]. In order to facilitate simulation analysis without losing generality, the reference heading is set as ${\psi }_{d} = \sin \left( \mathrm{t}\right)$ . Select control parameters as ${k}_{\psi } = {1.2},{k}_{r} = {1.5},{k}_{1} = 3,{k}_{2} =$ $6,{k}_{c} = {3.8},{\beta }_{2} = 4,{K}_{1} = {0.0001}\mathrm{I},{K}_{2} = {0.00001}\mathrm{I},Q = \mathrm{I}$ , $R = {0.25}\mathrm{I}$ . The time step is set as 0.05 . The initial states are defined as $\psi \left( 0\right) = {0.05}$ and $r\left( 0\right) = 0$ . The activation function of the critic network is chosen as $\sigma \left( E\right) = \left\lbrack {{e}_{1},{e}_{1}^{2},{e}_{2},{e}_{2}^{2},{e}_{1}{e}_{2}}\right\rbrack$ , the network weights are selected randomly in $\left\lbrack {0,1}\right\rbrack$ . Simulate real ocean and wind disturbances by using first-order Markov perturbations.
+
+To verify the superiority of the proposed strategy, we will compare the "LSBG" strategy of [10]. Fig. 1 shows the real-time curve of heading angle tracking, and the results show that the designed optimal control method can track the reference signal with smaller errors and within state constraints. The heading angular velocity tracking and its state constraints are shown in Fig. 2. Fig. 3 and Fig. 4 illustrate the error curves of heading angle and heading angular velocity, indicating that the proposed strategy achieves better tracking performance than the "LSBG" strategy. Fig. 5 displays the control inputs for control input ${\tau }_{r}$ under "LSBG" strategy, backstepping feedforward control, optimal feedback control, and system control under the proposed strategy, respectively. Fig. 6 shows the update curve of the evaluation network weights, it is obvious that the speed of online learning has reached stability in a very short time. From the above analysis, the proposed strategy can not only ensure better tracking accuracy, but also avoid system state violations of constraints.
+
+§ V. CONCLUSION
+
+In this paper, the optimal control method based on ADP is proposed for the tracking control of unmanned sailboats with heading turning constraints. The proposed LBF-based method solves the problem of state constraints. The feedforward backstepping controller and the feedback optimal controller were designed using the backstepping method and ADP theory, respectively. The learning ability of critic NNs has been accelerated through online learning strategies. Finally, the simulation verified the optimality of the proposed strategy. In the future, we will apply this method to the path-tracking task of unmanned sailboats in pratice.
+
+ < g r a p h i c s >
+
+Fig. 1. Comparison of heading angle tracking under different strategies.
+
+ < g r a p h i c s >
+
+Fig. 2. Comparison of heading angle speed tracking under different strategies.
+
+ < g r a p h i c s >
+
+Fig. 3. Comparison of heading angle error under different strategies.
+
+ < g r a p h i c s >
+
+Fig. 4. Comparison of heading angle speed error under different strategies.
+
+ < g r a p h i c s >
+
+Fig. 5. Control input ${\tau }_{r}$ under "LSBG" strategy, feedforward control input ${\tau }_{r}^{\alpha }$ , feedback control input ${\tau }_{r}^{ * }$ and optimal control input ${\tau }_{r}$ .
\ No newline at end of file