WM and then the effectiveness of such an algorithm is validated in Section 4. Finally, in Section 5, the conclusion is summarized and the future work is discussed.
Notations. $\mathbb{R}$ and $\mathbb{N}^+$ represent the space of all real numbers and positive integers, respectively. $\mathbb{R}^n$ and $\mathbb{R}^{m \times n}$ stand for the $n$-dimensional Euclidean space and the set of all $m \times n$ real matrices, respectively. For any vector $x \in \mathbb{R}^n$, $|x|$ denotes the Euclidean norm of $x$. Additionally, for any nonnegative matrix $P \in \mathbb{R}^{n \times n}$, $|x|_P$ means the weighted norm of $x$ (i.e., $|x|_P = \sqrt{x^T P x}$).
2. PROBLEM FORMULATION
Consider the following nonlinear system with IFs
where $x(k) \in \mathbb{R}^{n_x}$, $y(k) \in \mathbb{R}^{n_y}$ and $f(k) \in \mathbb{R}^{n_f}$ are the state vector, measurement output and IF signal, respectively. $w(k) \in \mathbb{R}^{n_x}$ and $v(k) \in \mathbb{R}^{n_y}$ are mutually uncorrelated zero-mean Gaussian white noises with respective covariance matrices $R_w$ and $R_v$. $F$ and $G$ are known matrices with appropriate dimensions. $g(\cdot)$ and $h(\cdot)$ are known nonlinear functions.
The intermittent fault $f(k)$ is assumed to satisfy the following form
where $k_{s,1}$ and $k_{s,2}$ are the sth unknown appearing time and disappearing time of IF $f(k)$, respectively. $\Theta(\cdot)$ is a function satisfying $\Theta(i) = 1$ ($i \ge 0$) and $\Theta(i) = 0$ ($i < 0$). $m_s(k)$ is the sth unknown fault magnitude. Define $d_{s,1} = k_{s,2} - k_{s,1}$ and $d_{s,2} = k_{s+1,1} - k_{s,2}$ as the sth active duration time and inactive duration time of $f(k)$. In this paper, we suppose that there exist two known constants $\bar{d}_1 > 0$ and $\bar{d}2 > 0$ satisfying $d{s,1} \le \bar{d}1$ and $d{s,2} \le \bar{d}_2$ ($s \in \mathbb{N}^+$), where $\bar{d}_1$ and $\bar{d}_2$ are respectively called the lower bounds of fault active duration and fault inactive duration.
If a residual $r(k)$ satisfies the following two conditions:
(1) there exists a constant $0 \le \tau_1 < \bar{d}1$ such that $r(k) \ge J{\text{th},1}$ holds for all $k \in [k_{s,1} + \tau_1, k_{s,2})$ ($s \in \mathbb{N}^+$), where $J_{\text{th},1}$ is the detection threshold for the appearing time and $k_{s,1} + \tau_1$ is the sth appearing time detected by the residual $r(k)$;
(2) there exists a constant $0 \le \tau_2 < \bar{d}2$ such that $r(k) < J{\text{th},2}$ holds for all $k \in [k_{s,2} + \tau_2, k_{s+1,1})$ ($s \in \mathbb{N}^+$), where $J_{\text{th},2}$ is the detection threshold for the disappearing time and $k_{s,2} + \tau_2$ is the sth disappearing time detected by the residual $r(k)$,
it is said that IF $f(k)$ is detectable by the residual $r(k)$.
Remark 1. The core task for IF detection is to detect all appearing and disappearing times of IFs. If condition (1) is fulfilled, the designed residual $r(k)$ must be larger than the threshold $J_{\text{th},1}$ before fault $f(k)$ disappears, which means that there must exist a period of alarm time during $[k_{s,1}, k_{s,2})$ ($s \in \mathbb{N}^+$). Condition (2) shows that $r(k)$ can decrease below the threshold $J_{\text{th},2}$ before the next fault
Fig. 1. IF and the residual of EKF in the case of $f_a = 1$
Fig. 2. IF and the residual of EKF in the case of $f_a = 2$
$f(k)$ appears, which ensures that the sth disappearing time and the s+1th appearing time of IF $f(k)$ can be clearly distinguished. Combining the two conditions, it is easy to deduce that all appearing and disappearing times of IF $f(k)$ can be detected by the residual $r(k)$.
Example 1: Consider the nonlinear system with the following parameters
The IF $f(k)$ is chosen as
By means of EKF, the estimate $\hat{x}^*(k)$ can be derived. Then the residual is defined as $r(k) = y(k) - h(\hat{x}^*(k))$. The trajectories of $r(k)$ in the case of $f_a = 1$ and $f_a = 2$ are respectively depicted in Figs. 1 and 2.