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samples/texts/1072043/page_10.md

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+Fig. 19. AC voltage and current waveforms under the step down transient condition
+
+<table><thead><tr><th>Electrical parameters</th><th colspan="2">Voltage, current and power for transient step down conditions</th><th colspan="2">Voltage, current and power for transient step up conditions</th></tr></thead><tbody><tr><td>AC output voltage (V)</td><td>220</td><td>220</td><td>220</td><td>220</td></tr><tr><td>AC output current (A)</td><td>7</td><td>2</td><td>2</td><td>7</td></tr><tr><td>AC output power (W)</td><td>1540</td><td>440</td><td>440</td><td>1540</td></tr></tbody></table>
+
+Table 2. Inverter operations under step up/down conditions
+
+# 5. Results and discussion
+
+In the next step, data waveforms are divided into the “estimate data set” and the “validate data set”. Examples are shown in Fig. 20, whereby the first part of the AC and DC voltage waveforms are used as the estimate data set and the second part the validate data set. The system identification process is executed according to mentioned descriptions on the Hammerstein-Wiener modeling.
+
+The validation of models is taken by considering (i) model order by adjusting the number of poles plus zeros. The system must have the lowest-order model that adequately captures the system dynamics.(ii) the best fit, comparing between modeling and experimental outputs, (iii) FPE and AIC, both of these values need be lowest for high accuracy of modeling (iv)

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+Nonlinear behavior characteristics. For example, linear interval of saturation, zero interval of dead-zone, wavenet, sigmoid network requiring the simplest and less complex function to explain the system. Model properties, estimators, percentage of accuracy, final Prediction Error-FPE and Akaikae Information Criterion-AIC are as follows [58]:
+
+Fig. 20. Data divided into Estimated and validated data
+
+Criteria for Model selection
+
+The percentage of the best fit accuracy in equation (13) is obtained from comparison between experimental waveform and simulation modeling waveform.
+
+$$
+\text{Best fit} = 100 * \left( \frac{1 - \operatorname{norm}(y^* - y)}{\operatorname{norm}(\bar{y} - \bar{y}^*)} \right) \quad (13)
+$$
+
+where $y^*$ is the simulated output, $y$ is the measured output and $\bar{y}$ is the mean of output.
+FPE is the Akaike Final Prediction Error for the estimated model, of which the error
+calculation is defined as equation (14)
+
+$$
+FPE = V \left( \frac{1 + d/N}{1 - d/N} \right) \qquad (14)
+$$
+
+where V is the loss function, d is the number of estimated parameters, N is the number of estimation data. The loss function V is defined in Equation (15) where θ_N represents the estimated parameters.
+
+$$
+V = \det \left( \frac{1}{N} \sum_{1}^{N} \varepsilon(t, \theta_{N}) (\varepsilon(t, \theta_{N}))^{T} \right) \quad (15)
+$$
+
+The Final Prediction Error (FPE) provides a measure of a model quality by simulating
+situations where the model is tested on a different data set. The Akaike Information

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+# Modeling of Photovoltaic Grid Connected Inverters Based on Nonlinear System Identification for Power Quality Analysis
+
+Nopporn Patcharaprakiti¹,², Krissanapong Kirtikara¹,²,
+Khanchai Tunlasakun¹, Juttrit Thongpron¹,², Dheerayut Chenvidhya¹,
+Anawach Sangswang¹, Veerapol Monyakul¹ and Ballang Muenpinij¹
+
+¹King Mongkut's University of Technology Thonburi, Bangkok,
+²Rajamangala University of Technology Lanna, Chiang Mai
+Thailand
+
+## 1. Introduction
+
+Photovoltaic systems are attractive renewable energy sources for Thailand because of high daily solar irradiation, about 18 MJ/m²/day. Furthermore, renewable energy is boosted by the government incentive on adders on electricity from renewable energy like solar PV, wind and biomass, introduced in the second half of 2000s. For PV systems, domestic rooftop PV units, commercial rooftop PV units and ground-based PV plants are appealing. Applications of electricity supply from PV plants that have been filed total more than 1000 MW. With the adder incentive, more households will be attracted to produce electricity with a small generating capacity of less than 10 kW, termed a very small power producer (VSPP). A possibility of expanding domestic roof-top grid-connected units draw our attention to study single phase PV-grid connected systems. Increased PV penetration can have significant [1-2] and detrimental impacts on the power quality (PQ) of the distribution networks [3-5]. Fluctuation of weather condition, variations of loads and grids, connecting PV-based inverters to the power system, requires power quality control to meet standards of electrical utilities. PV can reduce or improve power quality levels [6-9]. Different aspects should be taken into account. In particular, large current variations during PV connection or disconnection can lead to significant voltage transients [10]. Cyclic variations of PV power output can cause voltage fluctuations [11]. Changes of PV active and reactive power and the presence of large numbers of single phase domestic generators can lead to long-duration voltage variations and unbalances [12]. The increasing values of fault currents modify the voltage sag characteristics. Finally, the waveform distortion levels are influenced in different ways according to types of PV connections to the grid, i.e. direct connection or by power electronic interfaces. PV can improve power quality levels, mainly as a consequence of increase of short circuit power and of advanced controls of PWM converters and custom devices. [13]
+
+Grid-connected inverter technology is one of the key technologies for reliable and safety grid interconnection operation of PV systems [14-15]. An inverter being a power

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+Fig. 21. Comparison of AC voltage and current output waveforms of a steady state FVMC MIMO model
+
+In Table 3 $n_{bi}$, $n_{fi}$ and $n_{ki}$ are poles, zeros and delays of a linear model. The subscript (1, 2, 3 and 4) stands for relations between DC voltage-AC voltage, DC current-AC voltage, DC voltage-AC current and DC current-AC current respectively. Therefore, the linear parameters of the model are [$n_{b1}, n_{b2}, n_{b3}, n_{b4}$], [$n_{f1}, n_{f2}, n_{f3}, n_{f4}$], [$n_{k1}, n_{k2}, n_{k3}, n_{k4}$]. The first value of percentages of fit in each type, shown in the Table 3, is the accuracy of the voltage output, the second the current output from the model. From the results, nonlinear estimators can describe the photovoltaic grid connected system. The estimators are good in terms of accuracy, with a low order model or a low FPE and AIC. Under most of testing conditions, high accuracy of more than 85% is achieved, except the case of FVLC. This is because of under such an operating condition, the inverter has very small current, and it is operating under highly nonlinear behavior. Then complex of nonlinear function and parameter adjusted is need for achieve the high accuracy and low order of model. After obtaining the appropriate model, the PVGCS system can be analyzed by nonlinear and linear analyses. Nonlinear parts are analyzed from the properties of nonlinear function such as dead-zone interval, saturation interval, piecewise range, Sigmoid and Wavelet properties. Nonlinear properties are also considered, e.g. stability and irreversibility In order to use linear analysis, Linearization of a nonlinear model is required for linear control design and analysis, with acceptable representation of the input/output behaviors. After linearizing the model, we can use control system theory to design a controller and perform linear analysis. The linearized command for computing a first-order Taylor series approximation for a system requires specification of an operating point. Subsequently, mathematical representation can be obtained, for example, a discrete time invariant state space model, a transfer function and graphical tools.

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+We next demonstrate accuracy and precision of power quality prediction from modeling.
+Table 4 shows the comparisons. Two representative cases mentioned above are given, i.e.
+the steady state Fix Voltage High Current (FVHC) condition, and the transient step down
+condition. Comparison of THDs is shown in Fig. 23. Agreements between experiments and
+modeling results are good.
+
+Fig. 24. Comparison of measured and modeled THD of AC current of the transient step down condition
+
+**6.3 Power quality problem analysis**
+
+The power quality phenomena are classified in terms of typical duration, typical voltage magnitude and typical spectral content. They can be broken down into 7 groups on transient, short duration voltage, long duration voltage, voltage unbalance, waveform distortion, voltage fluctuation or flicker, frequency variation. Comparisons of the Standard values and modeled outputs of the FVHC and the transient step down conditions are shown in Table 5. The results show that under both the steady state and the transient cases, good power quality is achieved from the PVGCS.

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+<table><thead><tr><th>Type</th><th>Typical Duration</th><th>Typical Voltage Magnitude</th><th>Typical Spectral Content</th><th>Steady State FVHC</th><th>Transient Step down</th><th>Result</th></tr></thead><tbody><tr><td>1.Transient</td><td>5 ns - 0.1ms</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- Impulsive</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- Oscillation</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- low frequency</td><td>0.3-50 ms</td><td>0.4 pu.</td><td>&lt; 5 kHz</td><td>0.99 pu.</td><td>0.99 pu.</td><td>Pass</td></tr><tr><td>- medium frequency</td><td>5-20 ms</td><td>0-8 pu.</td><td>5-500 kHz</td><td></td><td></td><td></td></tr><tr><td>- high frequency</td><td>0-5 ms</td><td>0.4 pu.</td><td>0.5-5 MHz</td><td></td><td></td><td></td></tr><tr><td>2.Short Duration</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- voltage sag</td><td>10 ms-1 min</td><td>0.1-0.9 pu.</td><td>-</td><td>0.99 pu.</td><td>0.99 pu.</td><td>Pass</td></tr><tr><td>- voltage swell</td><td>10 ms-1 min</td><td>1.1-1.8 pu.</td><td>-</td><td></td><td></td><td></td></tr><tr><td>3. Long Duration</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- overvoltage (OV)</td><td>&gt; 1 min</td><td>&gt; 1.1 pu.</td><td>-</td><td>0.99 pu.</td><td>0.99 pu.</td><td>Pass</td></tr><tr><td>- undervoltage (UV)</td><td>&gt; 1 min</td><td>&lt; 0.9 pu.</td><td>-</td><td></td><td></td><td></td></tr><tr><td>- voltage Interruption</td><td>&gt; 1 min</td><td>0 pu.</td><td>-</td><td></td><td></td><td></td></tr><tr><td>4. Voltage Unbalance</td><td>Steady state</td><td>0.5-2%</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>5.Waveform distortion</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- Harmonic voltage</td><td>Steady state</td><td>&lt; 5% THD</td><td>0-100<sup>th</sup></td><td>1.20 %</td><td>1.24 %</td><td>Pass</td></tr><tr><td>- Harmonic Current</td><td>Steady state</td><td>&lt; 20% THD</td><td>0-100<sup>th</sup></td><td>3.25 %</td><td>3.68 %</td><td>Pass</td></tr><tr><td>- Interharmonic</td><td>Steady state</td><td>0-2%</td><td>0-6 kHz</td><td>-</td><td>-</td><td>-</td></tr><tr><td>- DC offset</td><td>Steady state</td><td>0-0.1%</td><td>&lt; 200 kHz</td><td>-</td><td>-</td><td>-</td></tr><tr><td>- Notching</td><td>Steady state</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>- Noise</td><td>Steady state</td><td>0-1%</td><td>Broad band</td><td>-</td><td>-</td><td>-</td></tr><tr><td>6.Voltage fluctuation</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>- Flicker</td><td>Intermittent</td><td>0.1-7%</th<br/> <span style="color:red;">&lt; 25 Hz &plusmn; 3 Hz &plusmn; 5 Hz &plusmn; 7 Hz &plusmn; 10 Hz &plusmn; 15 Hz &plusmn; 20 Hz &plusmn; 25 Hz &plusmn; 30 Hz &plusmn; 35 Hz &plusmn; 40 Hz &plusmn; 45 Hz &plusmn; 50 Hz &plusmn; 55 Hz &plusmn; 60 Hz &plusmn; 65 Hz &plusmn; 70 Hz &plusmn; 75 Hz &plusmn; 80 Hz &plusmn; 85 Hz &plusmn; 90 Hz &plusmn; 95 Hz &plusmn; 100 Hz &plusmn; 105 Hz &plusmn; 110 Hz &plusmn; 115 Hz &plusmn; 120 Hz &plusmn; 125 Hz &plusmn; 130 Hz &plusmn; 135 Hz &plusmn; 140 Hz &plusmn; 145 Hz &plusmn; 150 Hz &plusmn; 155 Hz &plusmn; 160 Hz &plusmn; 165 Hz &plusmn; 170 Hz &plusmn; 175 Hz &plusmn; 180 Hz &plusmn; 185 Hz &plusmn; 190 Hz &plusmn; 195 Hz &plusmn; 200 Hz &plusmn; 205 Hz &plusmn; 210 Hz &plusmn; 215 Hz &plusmn; 220 Hz &plusmn; 225 Hz &plusmn; 230 Hz &plusmn; 235 Hz &plusmn; 240 Hz &plusmn; 245 Hz &plusmn; 250 Hz &plusmn; 255 Hz &plusmn; 260 Hz &plusmn; 265 Hz &plusmn; 270 Hz &plusmn; 275 Hz &plusmn; 280 Hz &plusmn; 285 Hz &plusmn; 290 Hz &plusmn; 295 Hz &plusmn; 300 Hz &plusmn; 305 Hz &plusmn; 310 Hz &plusmn; 315 Hz &plusmn; 320 Hz &plusmn; 325 Hz &plusmn; 330 Hz &plusmn; 335 Hz &plusmn; 340 Hz &plusmn; 345 Hz &plusmn; 350 Hz &plusmn; 355 Hz &plusmn; 360 Hz &plusmn; 365 Hz &plusmn; 370 Hz &plusmn; 375 Hz &plusmn; 380 Hz &plusmn; 385 Hz &plusmn; 390 Hz &plusmn; 395 Hz &plusmn; 400 Hz &plusmn; 405 Hz &plusmn; 410 Hz &plusmn; 415 Hz &plusmn; 420 Hz &plusmn; 425 Hz &plusmn; 430 Hz &plusmn; 435 Hz &plusmn; 440 Hz &plusmn; 445 Hz &plusmn; 450 Hz &plusmn; 455 Hz &plusmn; 460 Hz &plusmn; 465 Hz &plusmn; 470 Hz &plusmn; 475 Hz &plusmn; 480 Hz &plusmn; 485 Hz &plusmn; 490 Hz &plusmn; 495 Hz &plusmn; 500 Hz&mdash;</span></th></tr><tr><th colspan="7" style="text-align:center;">Table 5. Comparison modeling output with Categories and Typical Characteristics of power system electromagnetic phenomena</th></tr></tbody></table>
+
+## **7. Conclusions**
+
+In this paper, a PVGCS system is modeled by the Hammerstein-Wiener nonlinear system identification method. Two main steps to obtain models from a system identification process are implemented. The first step is to set up experiments to obtain waveforms of DC inverter voltage/current, AC inverter voltage/current, point of common coupling (PCC) voltage, and grid and load current. Experiments are conducted under steady state and transient conditions for commercial rooftop inverters with rating of few kW, covering resistive and complex loads. In the steady state experiment, six conditions are carried out. In the transient case, two conditions of operating conditions are conducted. The second stage is to derive system models from system identification software. Collected waveforms are transmitted

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+into a computer for data processing. Waveforms data are divided in two groups. One group is used to estimate models whereas the other group to validate models. The developed programming determines various model waveforms and search for model waveforms of maximum accuracy compared with actual waveforms. This is achieved through selecting model structures and adjusting the model order of the linear terms and nonlinear estimators of nonlinear terms. The criteria for selection of a suitable model are the “Best Fits” as defined by the software, and a model order which should be minimum.
+
+After obtaining appropriate models, analysis and prediction of power quality are carried out. Modeled output waveforms relating to power quality analysis are determined from different scenarios. For example, irradiances and ambient temperature affecting DC PV outputs and nature of complex local load can be varied. From the model output waveforms, determination is made on power quality aspects such as voltage level, total harmonic distortion, complex power, power factor, power penetration and frequency deviation. Finally, power quality problems are classified.
+
+Such modelling techniques can be used for system planning, prevention of system failures and improvement of power quality of roof-top grid connected systems. Furthermore, they are not limited to PVGCS but also applicable to other distributed energy generators connected to grids.
+
+## 8. Acknowledgements
+
+The authors would like to express their appreciations to the technical staff of the CES Solar Cells Standards and Testing Center (CSSC) of King Mongkut's University of Technology Thonburi (KMUTT) for their assistance and valuable discussions. One of the authors, N. Patcharaprakiti receives a scholarship from Rajamangala University of Technology Lanna (RMUTL), a research grant from the Energy Policy and Planning Office (EPPO) and Office of Higher education commission, Ministry of Education, Thailand for enabling him to pursue his research of interests. He is appreciative of the scholarship and research grant supports.
+
+## 9. References
+
+[1] Bollen M. H. J. and Hager M., Power quality: integrations between distributed energy resources, the grid, and other customers. Electrical Power Quality and Utilization Magazine, vol. 1, no. 1, pp. 51-61, 2005.
+
+[2] Vu Van T., Belmans R., Distributed generation overview: current status and challenges. Inter-national Review of Electrical Engineering (IREE), vol. 1, no. 1, pp. 178-189, 2006.
+
+[3] Pedro González, "Impact of Grid connected Photovoltaic System in the Power Quality of a Network", Power Electrical and Electronic Systems (PE&ES), School of Industrial Engineering, University of Extremadura.
+
+[4] Barker P. P., De Mello R. W., Determining the impact of distributed generation on power systems: Part 1 - Radial distribution systems. PES Summer Meeting, IEEE, Vol. 3, pp. 1645-1656, 2000.
+
+[5] Vu Van T., Impact of distributed generation on power system operation and control. PhD Thesis, Katholieke Universiteit Leuven, 2006.

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+[21] Zheng Shi-cheng, Liu Xiao-li, Ge Lu-sheng; "Study on Photovoltaic Generation System and Its Islanding Effect," Industrial Electronics and Applications, 2007. ICIEA 2007. 2nd IEEE Conference on, 23-25 May 2007, pp.2328-2332
+
+[22] Pu kar Mahat, et al, "Review of Islanding Detection Methods for Distributed Generation", DRPT 6-9 April 2008, Nanjing, China.
+
+[23] Yamashita, H. et al. "A novel simulation technique of the PV generation system using real weather conditions Proceedings of the Power Conversion Conference, 2002 PCC Osaka 2002.
+
+[24] Golovanov, N, "Steady state disturbance analysis in PV systems", IEEE Power Engineering Society General Meeting, 2004.
+
+[25] D. Maksimovic, et al, "Modeling and simulation of power electronic converters," Proc. IEEE, vol. 89, pp. 898, Jun. 2001
+
+[26] R. D. Middlebrook and S. Cuk "A general unified approach to modeling switching converter power stages," Int.J. Electron., vol. 42, pp. 521, Jun. 1977.
+
+[27] Marisol Delgado and Hebertt Sira-Ramírez A bond graph approach to the modeling and simulation of switch regulated DC-to-DC power supplies, Simulation Practice and Theory Volume 6, Issue 7, 15 November 1998, Pages 631-646.
+
+[28] R.E. Araujo, et al, "Modelling and simulation of power electronic systems using a bond formalism", Proceeding of the 10th Mediterranean Conference on control and Automation - MED2002, Lisbon, Portugal, July 9-12 2002.
+
+[29] H. I. Cho, "A Steady-State Model of the Photovoltaic System in EMTP", The International Conference on Power Systems Transients (IPST2009), Kyoto, Japan June 3-6, 2009
+
+[30] Y. Jung, J. Sol, G. Yu, J. Choj, "Modelling and analysis of active islanding detection methods for photovoltaic power conditioning systems," Electrical and Computer Engineering, 2004. Canadian Conference on, vol.2, 2-5 May 2004, pp. 979-982.
+
+[31] Seul-Ki Kim, "Modeling and simulation of a grid-connected PV generation system for electromagnetic transient analysis", Solar Energy, Volume 83, Issue 5, May 2009, Pages 664-678.
+
+[32] Onbilgin, Guven, et al, "Modeling of power electronics circuits using wavelet theory", Sampling Theory in Signal and Image Processing, September, 2007.
+
+[33] George A. Bekey, "System identification- an introduction and a survey, Simulation, Transaction of the society for modeling and simulation international", October 1970 vol. 15 no. 4, 151-166.
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+[34] J.sjoberg, et al., "Nonlinear black-box Modeling in system identification: a unified overview, Automatica Journal of IFAC, Vol 31, Issue 12, December 1995.
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+[35] K. T. Chau and C. C. Chan "Nonlinear identification of power electronics systems," Proc. Int. Conf. Power Electronics and D20ve Systems, vol. 1, 1995, p. 329.
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+[36] J. Y. Choi , et al. "System identification of power converters based on a black-box approach," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, pp. 1148, Nov. 1998.
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+[37] F.O. Resende and J.A.Pecas Lopes, "Development of Dynamic Equivalents for Microgrids using system identification theory", IEEE of Power Technology, Lausanne, pp. 1033-1038, 1-5 July 2007.

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+conditioner of a PV system consists of power electronic switching components, complex control systems [16]. In addition, their operations depends on several factors such as input weather condition, switching algorithm and maximum power point tracking (MPPT) algorithm implemented in grid-connected inverters, giving rise to a variety of nonlinear behaviors and uncertainties [17]. Operating conditions of PV based inverters can be considered as steady state condition [18], transient condition [19-20], and fault condition such islanding [21-22]. In practical operations, inverters constantly change their operating conditions due to variation of irradiances, temperatures, load or grid impedance variations. In most cases, behavior of inverters is mainly considered in a steady state condition with slowly changing grid, load and weather conditions. However, in many instances conditions suddenly change, e.g. sudden changes of input weather, cloud or shading effects, loads and grid changes from faults occurring in near PV sites [23]. In these conditions, PV based inverters operate in transient conditions. Their average power increases or decreases upon the disturbances to PV systems [24]. In order to understand the behavior of PV based inverters, modeling and simulation of PV based inverter systems is the one of essential tools for analysis, operation and impacts of inverters on the power systems [25].
+
+There are two major approaches for modeling power electronics based systems, i.e. analytical and experimental approaches. The analytical methods to study steady state, transient models and islanding conditions of PV based inverter systems, such as state space averaging method [26], graphical techniques [27-28] and computation programming [29]. In using these analytic methods, one needs to know information of system. However, PV based inverters are usually commercial products having proprietary information; system operators do not know the necessary information of products to parameterize the models. In order to build models for nonlinear devices without prior information, system identification methods are exposed [33-34]. In the method reported in this paper, specific information of inverter is not required in modeling. Instead, it uses only measured input and output waveforms.
+
+Many recent research focuses on identification modeling and control for nonlinear systems [35-37]. One of the effective identification methods is block oriented nonlinear system identification. In the block oriented models, a system consists of numbers of linear and nonlinear blocks. The blocks are connected in various cascading and parallel combinations representing the systems. Many identification methods of well known nonlinear block oriented models have been reported in the literature [38-39]. They are, for example, a NARX model [40], a Hammerstein model [41], a Wiener model [42], a Wiener-Hammerstein model and a Hammerstein-Wiener model [43]. Advantages of a Hammerstein model and a Wiener model enables combination of both models to represent a system, sensors and actuators in to one model. The Hammerstein-Wiener model is recognized as being the most effective for modeling complex nonlinear systems such PV based inverters [44].
+
+In this paper, real operating conditions weather input variation, i.e. load variations and grid variations, of PV- based inverters are considered. Then two different experiments, steady state and transient condition, are designed and carried out. Input-output data such as currents and voltages on both dc and ac sides of a PV grid-connected systems are recorded. The measured data are used to determine the model parameters by a Hammerstein-Wiener nonlinear model system identification process. In the Section II, PV system characteristics are introduced. The I-V characteristic, an equivalent model, effects of radiation and temperature on voltage and current of PV are described. In the Section III, system identification methods, particularly a Hammerstein-Wiener Model is explained. In the

samples/texts/1072043/page_24.md

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+[38] N.Patcharaprakiti, et al, "Modeling of Single Phase Inverter of Photovoltaic System Using System Identification", Computer and Network Technology (ICCNT), 2010, April 2010, Page(s): 462 – 466.
+
+[39] Hatanaka, et al., "Block oriented nonlinear model identification by evolutionary computation approach", Proceedings of IEEE Conference on Control Applications, 2003, Vol 1, pp 43-48, June 2003.
+
+[40] Li, G. Identification of a Class of Nonlinear Autoregressive Models With Exogenous Inputs Based on Kernel Machines, IEEE Transactions on Signal Processing, Vol 59, Issue 5, May 2011.
+
+[41] T. Wigren, "User choices and model validation in system identification using nonlinear Wiener models", Proc13:th IFAC Symposium on System Identification, Rotterdam, The Netherlands, pp. 863-868, August 27-29, 2003.
+
+[42] Alonge, et al, "Nonlinear Modeling of DC/DC Converters Using the Hammerstein's Approach" IEEE Transactions on Power Electronics, July 2007, Volume: 22, Issue: 4,pp. 1210-1221.
+
+[43] Guo. F. and Bretthauer, G. "Identification of cascade Wiener and Hammerstein systems", In Proceeding. of ISATED Conference on Applied Simulation and Modeling, Marbella, Spain, September, 2003.
+
+[44] N.Patcharaprakiti and et al., "Modeling of single phase inverter of photovoltaic system using Hammerstein-Wiener nonlinear system identification", Current Applied Physics 10 (2010) S532-S536,
+
+[45] A Guide To Photovoltaic Panels, PV Panels and Manufactures' Data, January 2009.
+
+[46] Tomas Markvart, "Solar electricity", Wiley, 2000
+
+[47] D.R.Myers, "Solar Radiation Modeling and Measurements for Renewable Energy Applications: Data and Model Quality", International Expert Conference on Mathematical Modeling of Solar Radiation and Daylight- Challenges for the 21st, Century, Edinburgh, Scotland
+
+[48] R.H.B. Exell, "The fluctuation of solar radiation in Thailand", Solar Energy, Volume 18, Issue 6, 1976, Pages 549-554.
+
+[49] M. Calais, J. Myrzik, T. Spooner, V.G. Agelidis, "Inverters for Single-phase Grid Connected Photovoltaic Systems - An Overview," Proc. IEEE PESC'02, vol. 2. 2002. Pp 1995-2000.
+
+[50] Photong, C. and et al, "Evaluation of single-stage power converter topologies for grid-connected Photovoltaic", IEEE International Conference on Industrial Technology (ICIT), 2010.
+
+[51] Y. Xue, L.Chang, S. B. Kj r, J. Bordonau, and T. Shimizu, "Topologies of Single-Phase Inverters for Small Distributed Power Generators: An Overview," IEEE Trans. on Power Electronics, vol. 19, no. 5, pp. 1305-1314, 2004.
+
+[52] S.H. Ko, S.R. Lee, and H. Dehbonei, "Application of Voltage- and Current- Controlled Voltage Source Inverters for Distributed Generation System," IEEE Trans. Energy Conversion, vol. 21, no.3, pp. 782-792, 2006.
+
+[53] Yunus H.I. and et al., Comparison of VSI and CSI topologies for single-phase active power filters, Power Electronics Specialists Conference, 1996. PESC '96 Record., 27th Annual IEEE

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+[54] N. Chayawatto, K. Kirtikara, V. Monyakul, C Jivacate, D Chenvidhya, "DC-AC switching converter mode lings of a PV grid-connected system under islanding phenomena", Renewable Energy 34 (2009) 2536-2544.
+
+[55] Lennart Ljung, "System identification : theory for the user", Upper Saddle River, NJ, Prentice Hall PTR, 1999.
+
+[56] Nelles, Oliver, Nonlinear system identification : from classical approaches to neural networks and fuzzy models, 2001
+
+[57] N.Patcharaprakiti and et al., "Nonlinear System Identification of power inverter for grid connected Photovoltaic System based on MIMO black box modeling", GMSTECH 2010 : International Conference for a Sustainable Greater Mekong Subregion, 26-27 August 2010, Bangkok, Thailand.
+
+[58] Lennart Ljung, System Identification Toolbox User Guide, 2009
+
+[59] IEEE 1159, 2009 Recommended practice for monitoring electric power quality.

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+**Electrical Generation and Distribution Systems and Power Quality**
+**Disturbances**
+
+*Edited by Prof. Gregorio Romero*
+
+ISBN 978-953-307-329-3
+
+Hard cover, 304 pages
+
+Publisher InTech
+
+Published online 21, November, 2011
+
+Published in print edition November, 2011
+
+The utilization of renewable energy sources such as wind energy, or solar energy, among others, is currently of greater interest. Nevertheless, since their availability is arbitrary and unstable this can lead to frequency variation, to grid instability and to a total or partial loss of load power supply, being not appropriate sources to be directly connected to the main utility grid. Additionally, the presence of a static converter as output interface of the generating plants introduces voltage and current harmonics into the electrical system that negatively affect system power quality. By integrating distributed power generation systems closed to the loads in the electric grid, we can eliminate the need to transfer energy over long distances through the electric grid. In this book the reader will be introduced to different power generation and distribution systems with an analysis of some types of existing disturbances and a study of different industrial applications such as battery charges.
+
+## How to reference
+
+In order to correctly reference this scholarly work, feel free to copy and paste the following:
+
+Nopporn Patcharaprakiti, Krissanapong Kirtikara, Khanchai Tunlasakun, Juttrit Thongpron, Dheerayut Chenvidhya, Anawach Sangswang, Veerapol Monyakul and Ballang Muenpinij (2011). Modeling of Photovoltaic Grid Connected Inverters Based on Nonlinear System Identification for Power Quality Analysis, Electrical Generation and Distribution Systems and Power Quality Disturbances, Prof. Gregorio Romero (Ed.), ISBN: 978-953-307-329-3, InTech, Available from: http://www.intechopen.com/books/electrical-generation-and-distribution-systems-and-power-quality-disturbances/modeling-of-photovoltaic-grid-connected-inverters-based-on-nonlinear-system-identification-for-power
+
+INTECH
+
+open science | open minds
+
+**InTech Europe**
+
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+
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+
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+Fax: +86-21-62489821

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+© 2011 The Author(s). Licensee IntechOpen. This is an open access article distributed under the terms of the [Creative Commons Attribution 3.0 License](http://creativecommons.org/licenses/by/3.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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+following section, the experimental design and implementation to model the system is illustrated. After that, the obtained model from prior sections is analyzed in terms of control theories. In the last section, the power quality analysis is discussed. The output prediction is performed to obtain electrical outputs of the model and its electrical power. The power quality nature is analyzed for comparison with outputs of model. Subsequently, voltage and current outputs from model are analyzed by mathematical tools such as Fast Fourier Transform-FFT, the Wavelet method in order to investigate the power quality in any operating situations.
+
+## 2. PV grid connected system (PVGCS) operation
+
+In this section, PV grid connected structures and components are introduced. Structures of PBGCS consist of solar array, power conditioners, control systems, filtering, synchronization, protection units, and loads, shown in Fig. 1.
+
+Fig. 1. Block diagram of a PV grid connected system
+
+### 2.1 Solar array
+
+Environmental inputs affecting solar array/cell outputs are temperature (T) and irradiance (I), fluctuating with weather conditions. When the ambient temperature increases, the array short circuit current slight increases with a significant voltage decrease. Temperature and I-V characteristics are related, characterized by array/cell temperature coefficients. Effects of irradiance, radiant solar energy flux density in W/m², apart from solar radiation at sea level, are determined by incident angles and array/cell envelope. Typical characteristics of relationship between environmental inputs (irradiance and temperature) and electrical parameters (current and voltage of array/cells) are shown in Fig. 2 [45]. In our experimental designs, operating conditions of PV systems under test is designed and based on typical operating conditions.
+
+### 2.2 Operating conditions of a PV grid connected system
+
+A PV system, generating power and transmitting it into the utility, can be categorized in three cases, i.e. a steady state condition, a transient condition and a fault condition like islanding. Three factors affecting the operation of inverters are input weather conditions, local loads and utility grid variations.

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+Fig. 2. Temperature and irradiance effects on I-V characteristics of PV arrays/cells [46]
+
+Fig. 3. Variations of solar irradiance and temperature throughout a day conditioning PVGCS operation
+
+Firstly, under a steady state condition, input, load and utility under consideration are treated as being constant with slightly change weather condition. Installed capacities of PV systems in a steady state are low, medium and high capacity. According to the weather conditions throughout a day as shown in Fig. 3 [47-48], a low radiation about 0-400 W/m² is common in an early morning (6:00 AM-9:00 AM) and early evening (16:00 PM-19:00 PM), medium radiation of 400-800 W/m² in late morning (9:00 AM-11:00 PM) and early afternoon (14:00 PM-16:00 PM) and high radiation of 800-1000 W/m² around noon (11.00

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+AM - 14:00 PM). Loads fluctuate upon activities of customer groups, for example, a peak load for industrial zones occurs in afternoon (13:00 - 17:00 PM) and a peak load for residential zones occurs in evening(18:00 - 21:00 PM). Variations from steady state conditions impact power quality such as overvoltage, over-current, harmonics, and so on. In case of transients, there are variations in inputs, loads and utility. Weather variations such as solar irradiance and temperature exhibit significant changes. Unexpected accidents happen. Local loads may sudden change due to activities of customers in each time. A utility has some faults in nearby locations which impact utility parameters such grid impedance. These conditions lead to short duration power quality problems with such spikes, voltage sag, voltage swell. In some extreme cases, abnormal conditions, such as very low solar irradiance or abnormal conditions such islanding, the grid-connected PV systems may collapse. The PV systems are black out and cut out of the utility grid. Such can affect power quality, stability and reliability of power systems.
+
+## 2.3 Power converter
+
+There are several topologies for converting a DC to DC voltage with desired values, for example, Push-Pull, Flyback, Forward, Half Bridge and Full Bridge [49]. The choice for a specific application is often based on many considerations such as size, weight of switching converter, generation interference and economic evaluation [50-51]. Inverters can be classified into two types, i.e. voltage source inverter (VSI) if an input voltage remains constant and a current source inverter (CSI) if input current remains constant [52-53]. The CSI is mostly used in large motor applications, whereas the VSI is adopted for and alone systems. The CSI is a dual of a VSI. A control technique for voltage source inverters consists of two types, a voltage control inverter, shown in Fig. 4(a) and a current control inverter, Fig. 4(b) [54].
+
+Fig. 4. Control techniques for an inverter
+
+# 3. System Identification
+
+System identification is the process for modeling dynamical systems by measuring the input/output from system. In this section, the principle of system identification is described. The classification is introduced and particularly a Hammerstein-Wiener model is explained. Finally, a MIMO (multi input multi output model with equation and characteristic is illustrated.

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+## 3.1 Principle of system identification
+
+A dynamical system can be classified in terms of known structures and parameters of the system, shown in Fig.5, and classified as a "White Box" if all structures and parameters are known, a "Grey Box", if some structures and parameters known and a "Black Box" if none are known [55].
+
+Fig. 5. Dynamical system classifications by structures and parameters
+
+Steps in system identification can be described as the following process, shown in Fig. 6.
+
+Fig. 6. System identification processes
+
+Each step can be described as follows
+
+### 3.1.1 Goal of modeling
+
+The primary goal of modeling is to predict behaviors of inverters for PV systems or to simulate their outputs and related values. The other important goal is to acquire

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+mathematical and physical characteristics and details of systems for the purposes of controlling, maintenance and trouble shooting of systems, and planning of managing the power system.
+
+### 3.1.2 Physical modeling
+
+Photovoltaic inverters, particularly commercial products, compose of two parts, i.e. a power circuit and a control circuit. Power electronic components convert, transfer and control power from input to output. The control system, switching topologies of power electronics are done by complex digital controls.
+
+### 3.1.3 Model structure selection
+
+Model structure selection is the stage to classify the system and choose the method of system identification. The system identification can be classified to yield a nonparametric model and a parametric model, shown in Fig 7. A nonparametric model can be obtained from various methods, e.g. Covariance function, Correlation analysis. Empirical Transfer Function Estimate and Periodogram, Impulse response, Spectral analysis, and Step response.
+
+Fig. 7. Classification of system identification
+
+Parametric models can be divided to two groups: linear parametric models and nonlinear parametric models. Examples of linear parametric models are Auto Regressive (AR), Auto Regressive Moving Average (ARMA), and Auto Regressive with Exogenous (ARX), Box-Jenkins, Output Error, Finite Impulse Response (FIR), Finite Step Response (FSR), Laplace Transfer Function (LTF) and Linear State Space (LSS). Examples of nonlinear parametric models are Nonlinear Finite Impulse Response (NFIR), Nonlinear Auto-Regressive with Exogenous (NARX), Nonlinear Output Error (NOE), and Nonlinear Auto-Regressive with

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+Moving Average Exogenous (NARMAX), Nonlinear Box-Jenkins (NBJ), Nonlinear State Space, Hammerstein model, Wiener Model, Hammerstein-Wiener model and Wiener-Hammerstein model [56]. In practice, all systems are nonlinear; their outputs are a nonlinear function of the input variables. A linear model is often sufficient to accurately describe the system dynamics as long as it operates in linear range Otherwise a nonlinear is more appropriate. A nonlinear model is often associated with phenomena such as chaos, bifurcation and irreversibility. A common approach to nonlinear problems solution is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity. Inverters of PV systems can be identified based on nonlinear parametric models using various system identification methods.
+
+### 3.1.4 Experimental design
+
+The experimental design is an important stage in achieving goals of modeling. Number parameters such as sampling rates, types and amount of data should be concerned. Grid connected inverters have four important input/output parameters, i.e. DC voltage (Vdc), DC current voltage (Idc), AC voltage (Vac) and AC current (Iac). In experiments, these data are measured, collected and send to a system identification process. Finally, a model of a PV inverter is obtained, shown in Fig. 8.
+
+Fig. 8. Experimental design of a photovoltaic inverter modeling using system identification
+
+### 3.1.5 Model estimation
+
+Data from the system are divided into two groups, i.e., data for estimation (estimate data) and data for validation (validate data). Estimate data are used in the system identification and validate data are used to check and improve the modeling to yield higher accuracy.
+
+### 3.1.6 Model validation
+
+Model validation is done by comparing experimental data or validates data and modeling data. Errors can then be calculated. In this paper, parameters of system identification are optimized to yield a high accuracy modeling by programming softwares.

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+**iii. Piecewise linear (PW) function** is defined as a nonlinear function, y=f(x) where f is a piecewise-linear (affine) function of x and there are n breakpoints (x_k,y_k) which k=1,...,n. y_k = f(x_k). f is linearly interpolated between the breakpoints. y and x are scalars.
+
+**vi. Sigmoid network (SN) activation function** Both sigmoid and wavelet network estimators which use the neural networks composing an input layer, an output layer and a hidden layer using wavelet and sigmoid activation functions as shown in Fig.12
+
+Fig. 12. Structure of nonlinear estimators
+
+A sigmoid network nonlinear estimator combines the radial basis neural network function using a sigmoid as the activation function. This estimator is based on the following expansion:
+
+$$
+y(u) = (u-r)PL + \sum_{i}^{n} a_i f((u-r)Qb_i - c_i) + d \quad (6)
+$$
+
+when u is input and y is output. r is the regressor. Q is a nonlinear subspace and P a
+linear subspace. L is a linear coefficient. d is an output offset. b is a dilation coefficient.,
+c a translation coefficient and a an output coefficient. f is the sigmoid function, given
+by the following equation (7)
+
+$$
+f(z) = \frac{1}{e^{-z} + 1} \tag{7}
+$$
+
+v. **Wavelet Network (WN) activation function.** The term wavenet is used to describe wavelet networks. A wavenet estimator is a nonlinear function by combination of a wavelet theory and neural networks. Wavelet networks are feed-forward neural networks using wavelet as an activation function, based on the following expansion in the equation (8)
+
+$$
+y = (u - r)PL + \sum_{i}^{n} as_{i} * f bs(u - r)Q + cs + \sum_{i}^{n} aw_{i} * g bw_{i}(u - r)Q + cw_{i} + d \quad (8)
+$$
+
+Which u and y are input and output functions. Q and P are a nonlinear subspace and a
+linear subspace. L is a linear coefficient. d is output offset. as and aw are a scaling
+coefficient and a wavelet coefficient. bs and bw are a scaling dilation coefficient and a

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+schematically in Fig 14. The system is connected directly to the domestic electrical system (low voltage). As we consider only domestic loads, we need not isolate our test system from the utility (high voltage) by any transformer. For system identification processes, waveforms are collected by an oscilloscope and transmitted to a computer for batch processing of voltage and current waveforms.
+
+Fig. 14. Experimental setup
+
+Fig. 15. An inverter modeling using system identification process
+
+Major steps in experimentation, analysis and system identifications are composed of Testing scenarios of six steady state conditions and two transient conditions are carried out on the inverter, from collected data from experiments, voltage and current waveform data are divided in two groups to estimate models and to validate models previously mentioned.

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+*Symposium on Cluster Computing and the Grid*, 2007; 541–548, doi:10.1109/CCGRID.2007.85.
+
+23. Yang L, Man L. On-Line and Off-Line DVS for Fixed Priority with Preemption Threshold Scheduling. *Proceedings of ICESS'09, the International Conference on Embedded Software and Systems*, 2009; 273–280, doi:10.1109/ICESS.2009.50.
+
+24. AMD processors. http://www.amd.com.
+
+25. Intel XScale technology. http://www.intel.com/design/intelxscale.
+
+26. Langen P, Juurlink B. Leakage-aware multiprocessor scheduling. *J. Signal Process. Syst.* 2009; **57**(1):73–88, doi:http://dx.doi.org/10.1007/s11265-008-0176-8.
+
+27. Chen G, Malkowski K, Kandemir M, Raghavan P. Reducing power with performance constraints for parallel sparse applications. *Proceedings of IPDPS 2005, the 19th IEEE International Parallel and Distributed Processing Symposium*, 2005; 8 pp., doi:10.1109/IPDPS.2005.378.
+
+28. Xu R, Mossé D, Melhem R. Minimizing expected energy consumption in real-time systems through dynamic voltage scaling. *ACM Trans. Comput. Syst.* 2007; **25**(4):9, doi:http://doi.acm.org/10.1145/1314299.1314300.
+
+29. Gonzalez R, Horowitz M. Energy dissipation in general purpose microprocessors. *IEEE Journal of Solid-State Circuits* Sep 1996; **31**(9):1277–1284, doi:10.1109/4.535411.
+
+30. Yao F, Demers A, Shenker S. A scheduling model for reduced CPU energy. *Proceedings of FOCS '95, the 36th Annual Symposium on Foundations of Computer Science*, IEEE Computer Society: Washington, DC, USA, 1995; 374.
+
+31. Zhang Y, Hu XS, Chen DZ. Task scheduling and voltage selection for energy minimization. *Proceedings of DAC'02, the 39th annual Design Automation Conference*, ACM: New York, NY, USA, 2002; 183–188, doi:http://doi.acm.org/10.1145/513918.513966.
+
+32. Beigne E, Clermidy F, Durupt J, Lhermet H, Miermont S, Thonnart Y, Xuan T, Valentian A, Varreau D, Vivet P. An asynchronous power aware and adaptive NoC based circuit. *Proceedings of the 2008 IEEE Symposium on VLSI Circuits*, 2008; 190–191, doi:10.1109/VLSIC.2008.4586002.
+
+33. Beigne E, Clermidy F, Miermont S, Thonnart Y, Valentian A, Vivet P. A Localized Power Control mixing hopping and Super Cut-Off techniques within a GALS NoC. *Proceedings of ICICDT 2008, the IEEE International Conference on Integrated Circuit Design and Technology and Tutorial*, 2008; 37–42, doi: 10.1109/ICICDT.2008.4567241.
+
+34. Kawaguchi H, Zhang G, Lee S, Sakurai T. An LSI for VDD-Hopping and MPEG4 System Based on the Chip. *Proceedings of ISCAS'2001, the International Symposium on Circuits and Systems*, 2001.
+
+35. Nocedal J, Wright SJ. *Numerical Optimization*. Springer, 2006.
+
+36. Schrijver A. *Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics*, vol. 24. Springer-Verlag, 2003.
+
+37. Garey MR, Johnson DS. *Computers and Intractability: A Guide to the Theory of NP-Completeness*. W. H. Freeman & Co.: New York, NY, USA, 1990.

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+**Corollary 1.** A linear chain with $n$ tasks is equivalent to a single task of cost $W = \sum_{i=1}^{n} w_i$.
+
+Indeed, in the optimal solution, the $n$ tasks are executed at the same speed, and they can be replaced by a single task of cost $W$, which is executed at the same speed and consumes the same amount of energy.
+
+### 4.2.3. Fork and join graphs
+
+Let $V = \{T_1, \dots, T_n\}$. We consider either a fork graph $G = (V \cup \{T_0\}, E)$, with $E = \{(T_0, T_i), T_i \in V\}$, or a join graph $G = (V \cup \{T_0\}, E)$, with $E = \{(T_i, T_0), T_i \in V\}$. $T_0$ is either the source of the fork or the sink of the join.
+
+**Theorem 1 (fork and join graphs)** When $G$ is a fork (resp. join) execution graph with $n+1$ tasks $T_0, T_1, \dots, T_n$, the optimal solution to MINENERGY$(G, D)$ is the following:
+
+* the execution speed of the source (resp. sink) $T_0$ is $s_0 = \frac{(\sum_{i=1}^n w_i^3)^{\frac{1}{3}} + w_0}{D}$;
+
+* for the other tasks $T_i$, $1 \le i \le n$, we have $s_i = s_0 \times \frac{w_i}{(\sum_{i=1}^n w_i^3)^{\frac{1}{3}}} \quad \text{if } s_0 \le s_{max}$.
+
+Otherwise, $T_0$ should be executed at speed $s_0 = s_{max}$, and the other speeds are $s_i = \frac{w_i}{D}$, with $D' = D - \frac{w_0}{s_{max}}$, if they do not exceed $s_{max}$ (Proposition 1 for independent tasks). Otherwise there is no solution.
+
+If no speed exceeds $s_{max}$, the corresponding energy consumption is
+
+$$ \min E(G, D) = \frac{\left( \left( \sum_{i=1}^{n} w_i^3 \right)^{\frac{1}{3}} + w_0 \right)^3}{D^2}. $$
+
+**Proof.** Let $t_0 = \frac{w_0}{s_0}$. Then, the source or the sink requires a time $t_0$ for execution. For $1 \le i \le n$, task $T_i$ must be executed within a time $D - t_0$ so that the deadline is respected. Given $t_0$, we can compute the speed $s_i$ for task $T_i$ using Theorem 1, since the tasks are independent: $s_i = \frac{w_i}{D-t_0} = w_i \cdot \frac{s_0}{s_0 D - w_0}$. The objective is therefore to minimize $\sum_{i=0}^n w_i s_i^2$, which is a function of $s_0$:
+
+$$ \sum_{i=0}^{n} w_i s_i^2 = w_0 s_0^2 + \sum_{i=1}^{n} w_i^3 \cdot \frac{s_0^2}{(s_0 D - w_0)^2} = s_0^2 \left( w_0 + \frac{\sum_{i=1}^{n} w_i^3}{(s_0 D - w_0)^2} \right) = f(s_0). $$
+
+Let $W_3 = \sum_{i=1}^{n} w_i^3$. In order to find the value of $s_0$ that minimizes this function, we study the function $f(x)$, for $x > 0$. $f'(x) = 2x(w_0 + \frac{W_3}{(xD-w_0)^2}) - 2D \cdot x^2 \cdot \frac{W_3}{(xD-w_0)^3}$, and therefore $f'(x) = 0$ for $x = (W_3^{1/3} + w_0)/D$. We conclude that the optimal speed for task $T_0$ is $s_0 = (\sum_{i=1}^{n} w_i^3)^{1/3} + w_0/D$, if $s_0 \le s_{max}$. Otherwise, $T_0$ should be executed at the maximum speed $s_0 = s_{max}$, since it is the bottleneck task. In any case, for $1 \le i \le n$, the optimal speed for task $T_i$ is $s_i = w_i s_0/(s_0 D - w_0)$.

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+Using the matrix formula we obtain
+
+$$
+\operatorname{Max}(a(m+1)+b) \geq R^{(m)}(n) = (1 \ 1)\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}^{a}\begin{pmatrix} 0 \\ 1 \end{pmatrix}=2^{a}, \quad \text{for } b \in \{0,1,\ldots,m-2\}.
+$$
+
+For $k = a(m + 1) + m - 1$ the value of $R^{(m)}(n)$ at $n$ with the greedy expansion $\langle n \rangle_m = 10^{2m-1}(10^m)^{a-1}$ is equal to
+
+$$
+\operatorname{Max}(a(m+1)+m-1) \geq R^{(m)}(n) = (1\ 1)\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}^{a-1}\begin{pmatrix} 0 \\ 1 \end{pmatrix} = 2^a + 2^{a-2}.
+$$
+
+For $k = a(m + 1) + m$ the value of $R^{(m)}(n)$ at $n$ with the greedy expansion $\langle n \rangle_m = 10^{m-1}(10^m)^a$ is equal to
+
+$$
+\operatorname{Max}(a(m+1)+m) \geq R^{(m)}(n) = (1\ 1)\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}^a\begin{pmatrix} 0 \\ 1 \end{pmatrix} = 2^a + 2^{a-1}.
+$$
+
+In the remaining part of this section we show that the above values are equal to Max(*k*).
+
+Let us describe what form of the greedy expansion an argument of the maxima may have. For
+that we introduce the following notions.
+
+**Definition 5.1** Let $\mathcal{X} = \binom{a}{c} \binom{b}{d}$ and $\tilde{\mathcal{X}} = \binom{\tilde{a}}{\tilde{c}} \binom{\tilde{b}}{\tilde{d}}$ be integer matrices with non-negative components.
+We say that $\tilde{\mathcal{X}}$ majores $\mathcal{X}$ (written $\tilde{\mathcal{X}} \succ \mathcal{X}$) if
+
+$$
+\tilde{a} \ge a, \quad \tilde{b} \ge b, \quad \tilde{a} + \tilde{c} \ge a + c \quad \text{and} \quad \tilde{b} + \tilde{d} > b + d. \tag{14}
+$$
+
+**Definition 5.2** We say that the string $10^{t_i} 10^{t_{i-1}} \cdots 10^{t_1}$ is forbidden for maximality, if there exists a word $10^{u_j} 10^{u_{j-1}} \cdots 10^{u_1}$ such that
+
+$$
+\begin{equation}
+\begin{gathered}
+t_1 + t_2 + \dots + t_i + i = u_1 + u_2 + \dots + u_j + j, \quad u_1, u_j > 0, \\
+M(t_i) M(t_{i-1}) \dots M(t_1) < M(u_j) M(u_{j-1}) \dots M(u_1).
+\end{gathered}
+\tag{15}
+\end{equation}
+$$
+
+**Proposition 5.3** Let *n* be an integer such that $\langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}$ and
+
+$$
+\text{Max}(k) = R^{(m)}(n) \quad \text{and} \quad F_k^{(m)} \le n < F_{k+1}^{(m)}.
+$$
+
+Then $10^{r_1} 10^{r_{l-1}} \cdots 10^{r_{l-i+1}}$ is not a string forbidden for maximality for any integers $l, i$, such that
+$1 \le l - i + 1 \le l \le s$.
+
+**Proof:** We prove the proposition by contradiction. Let $\langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}$ contain a string
+$10^{t_i} 10^{t_{i-1}} \cdots 10^{t_1}$ forbidden for maximality, i.e. there exists $l \le s$ such that $r_l = t_i$, $r_{l-1} = t_{i-1}$,
+$\dots$, $r_{l-i+1} = t_1$, then the word
+
+$$
+\begin{align*}
+& 10^{r_s} \cdots 10^{r_{l+1}} 10^{u_j} 10^{u_{j-1}} \cdots 10^{u_1} 10^{r_{l-i}} \cdots 10^{r_1}
+\end{align*}
+$$

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+has the same length as the greedy expansion $\langle n \rangle_m$. The condition $u_1, u_j > 0$ ensures that the new word is a greedy expansion of some integer $\tilde{n}$. Put
+
+$$
+\mathbb{A} = \left\{
+\begin{array}{ll}
+\mathbb{I}_2, & \text{if } l=s, \\
+M(r_s) \cdots M(r_{l+1}), & \text{if } l<s,
+\end{array}
+\right.
+\quad
+\mathbb{B} = \left\{
+\begin{array}{ll}
+\mathbb{I}_2, & \text{if } l-i=0, \\
+M(r_{l-i}) \cdots M(r_1), & \text{if } l-i>0.
+\end{array}
+\right.
+$$
+
+and
+
+$$
+X = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = M(t_i) \cdots M(t_1) \quad \text{and} \quad \tilde{X} = \begin{pmatrix} \tilde{a} & \tilde{b} \\ \tilde{c} & \tilde{d} \end{pmatrix} = M(u_j) \cdots M(u_1).
+$$
+
+In this notation $R^{(m)}(n) = (1 \ 1)A \times B_{\binom{0}{1}}$ and $R^{(m)}(\tilde{n}) = (1 \ 1)A \tilde{x} \tilde{B}_{\binom{0}{1}}}$. Denote $(x y) = (1 \ 1)A$ and $\binom{z}{u} = B_{\binom{0}{1}}}$. From the form of the matrices $A, B$ it can be easily seen that $x \ge y \ge 1$ and $z \ge 0, u \ge 1$. Since $X < \tilde{X}$, their components satisfy (14). We have
+
+$$
+\begin{align*}
+R^{(m)}(\tilde{n}) - R^{(m)}(n) &= (x y) \binom{\tilde{a} \, \tilde{b}}{\tilde{c} \, \tilde{d}} \binom{z}{u} - (x y) \binom{a \, b}{c \, d} \binom{z}{u} \\
+&= (\tilde{a} - a)x + (\tilde{c} - c)y, \ (\tilde{b} - b)x + (\tilde{d} - d)y) \binom{z}{u} \ge \\
+&\ge (\tilde{a} + \tilde{c} - a - c)y, \ (\tilde{b} + \tilde{d} - b - d)y) \binom{z}{u} \ge (0 \ 1) \binom{0}{1} = 1.
+\end{align*}
+$$
+
+Therefore $R^{(m)}(\tilde{n}) > R^{(m)}(n)$, and $n$ was not the argument of the maxima. $\square$
+
+The above proposition enables us to restrict the set of candidates for the arguments of the
+maxima. Let us recall that we consider $m \ge 3$, if not stated otherwise.
+
+**Claim 5.4** Let $n$ have the greedy expansion $\langle n \rangle_m = 10^{r_2} \cdots 10^{r_1}$ of length $k$ and let $\mathrm{Max}(k) = R^{(m)}(n)$. Then for every $i$ it holds that $r_i \le 2m$ or $r_i = 3m-1$.
+
+**Proof:** It is sufficient to show that the string $10^t$ where $t > 2m$ and $t \neq 3m-1$ is forbidden for maximality. Consider the string $10^{u_2}10^{u_1} = 10^{t-m-1}10^m$. Both strings $10^t$ and $10^{u_2}10^{u_1}$ have the same length, and $u_1, u_2 > 0$. In order to verify
+
+$$
+M(t) = \left( \frac{\lfloor t+1 \rfloor}{m}, \frac{\lfloor t \rfloor}{m} \right) < M(u_2)M(u_1) = \left( \frac{\lfloor t \rfloor}{m} + \frac{\lfloor t-1 \rfloor}{2} - 2, \frac{\lfloor t \rfloor}{m} + \frac{\lfloor t-1 \rfloor}{2} - 2 \right)
+$$
+
+it suffices to show that the inequality
+
+$$
+\left[ \frac{t+1}{m} \right] \leq \left[ \frac{t}{m} \right] + \left[ \frac{t-1}{m} \right] - 2
+$$
+
+is satisfied for $t > 2m$ and $t \neq 3m-1$. $\square$
+
+The above claim shows which strings of the type $10^t$ are forbidden for maximality. The following claims studies this question for some other types of strings.
+
+**Claim 5.5** The string $10^{m-1}10^{3m-1}$ is forbidden for maximality.

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+for $k = 0, \dots, N-1$.
+
+### C. Algorithm Construction
+
+In order to construct an optimization algorithm we have to define the discrete cost function $C_d$, the discrete Lagrangian $L_d$, the constraints $\omega_d$ and the discrete force $f_d$ consistently. We choose the *midpoint rule* according to which
+
+$$ 
+\begin{aligned}
+C_d(q_k, q_{k+1}, f_k, f_{k+1}) &= hC(q_{k+\frac{1}{2}}, \Delta q_k, f_{k+\frac{1}{2}}), \\
+L_d(q_k, q_{k+1}) &= hL(q_{k+\frac{1}{2}}, \Delta q_k), \\
+\omega_d^a(q_k, q_{k+1}) &= \omega^a(q_{k+\frac{1}{2}}, \Delta q_k), \\
+\int_{kh}^{(k+1)h} f(t) \cdot \delta q(t) dt &\approx h f_{k+\frac{1}{2}} \cdot \delta q_{k+\frac{1}{2}} \\
+&= \frac{h}{4}(f_k + f_{k+1}) \cdot \delta q_k + \frac{h}{4}(f_k + f_{k+1}) \cdot \delta q_{k+1},
+\end{aligned}
+ $$
+
+using the notation $q_{k+\frac{1}{2}} := \frac{q_k+q_{k+1}}{2}$ and $\Delta q_k := \frac{q_{k+1}-q_k}{h}$. The left and right forces then become $f_k^- = f_k^+ = \frac{h}{4}(f_k + f_{k+1})$. The midpoint rule is second order accurate. Higher order integrators using composition methods or symplectic partitioned Runge-Kutta methods can also be constructed [24].
+
+## III. REDUCED DISCRETIZATION
+
+The discrete equations derived in the previous section might become singular or the choice of coordinates might not be globally valid. When symmetries are present such problems can be avoided by applying reduction. In this section we use the reduced integrators derived in [25] to formulate a reduced optimization framework for Chaplygin systems that are relevant to car-like vehicles.
+
+### A. Reduced Lagrange-d'Alembert Equations
+
+Assume that we are given a Lie group $G$ acting to $Q$. We can pick local coordinates $q = (r,g)$, $q \in Q$, $r \in M$, $g \in G$, where $M = Q/G$ is the shape space. Assume that the Lagrangian $L$ and constraint distribution $\mathcal{D}$ are invariant under the induced action of $G$ on $TQ$. Then we can define the reduced Lagrangian $\ell: TQ/G \to \mathbb{R}$ satisfying $L(r,\dot{r},\dot{g},g) = \ell(r,\dot{r},g^{-1}\dot{g})$, and the constrained reduced Lagrangian $\ell_c: \mathcal{D}/G \to \mathbb{R}$, such that $\ell(r,\dot{r},g^{-1}\dot{g}) = \ell_c(r,\dot{r})$. The main point is that the Lagrange-D'Alembert equations on $TQ$ induce well-defined reduced Lagrange-D'Alembert equations on $\mathcal{D}/G$, vector fields in $\mathcal{D}$ are also $G$-invariant and define reduced vector fields on $\mathcal{D}/G$ [28].
+
+Whenever the group directions (the set of vector fields obtained by differentiating the group flow) complement the constraints we have the *principle kinematic case* or the *Chaplygin case*. In this case there is a principal connection one-form
+
+$$ A(r, g) \cdot (\dot{r}, \dot{g}) = Ad_g(g^{-1} \dot{g} + A_{loc}(r) \dot{r}), \quad (10) $$
+
+where $A_{loc}$ is the local form of the connection. This connection defines the evolution of the group variables in terms of the shape variables. It can be derived directly from the
+
+constraints. Since $A(q) \cdot \dot{q} = 0$ the constrained Lagrangian is given by
+
+$$ \ell_c(r, \dot{r}) = \ell(r, \dot{r}, -A_{loc}(r)\dot{r}) $$
+
+Assuming that the control forces are restricted to the shape, i.e. $f: TM \to T^*M$ the continuous equations of motion are
+
+$$ \frac{\partial \ell_c}{\partial r} - \frac{d}{dt} \frac{\partial \ell_c}{\partial \dot{r}} + f = \hat{f} \quad (11) $$
+
+$$ \dot{g} = -g A_{loc}(r) \dot{r}, \quad (12) $$
+
+where the forces $\hat{f}: TM \to T^*M$ arise from the curvature of $A_{loc}$ and are defined by
+
+$$ \hat{f}_{\beta} = \frac{\partial l}{\partial \xi_{b}} \left( \frac{\partial A_{\beta}^{b}}{\partial r^{\alpha}} - \frac{\partial A_{\alpha}^{b}}{\partial r^{\beta}} - C_{ac}^{b} A_{\alpha}^{a} A_{\beta}^{c} \right) \dot{r}^{\alpha}, \quad (13) $$
+
+with $\xi = -A_{loc}(r) \cdot \dot{r}$, $A_\alpha^b$ the components of $A_{loc}$, and $C_{ac}^b$ are the structure constants of the Lie algebra defined by $[e_a, e_c] = C_{ac}^b e_b$ (see [1] for details and an example). Equations (11) are independent of $g \in G$ and determine the unconstrained evolution of the system in the shape space $M$. Curves in $M$ can be *lifted* to $Q$ using (12) to produce a unique curve in $Q$ [29]. This fact allows us to reduce the optimal control problem from $Q$ to $M$ and after finding optimal trajectories in $M$ to lift them back to $Q$.
+
+Next we apply the discrete Lagrange-d'Alembert principle in the reduced space $M$. The integral of $l_c$ is approximated by the discrete constrained reduced Lagrangian $L_d^*: M \times M \to \mathbb{R}$ [25]. Then we obtain the discrete reduced equations of motion and discretized constraints:
+
+$$ 
+\begin{aligned}
+& D_2 L_d^*(r_{k-1}, r_k) + D_1 L_d^*(r_k, r_{k+1}) + f_{k-1}^+ + f_k^- \\
+&= \hat{f}_{k-1}^+ + \hat{f}_k^- \\
+& w_d(r_k, g_k, r_{k+1}, g_{k+1}) = 0,
+\end{aligned}
+ \quad (14) $$
+
+and velocity boundary conditions (corresponding to (9)):
+
+$$ 
+\begin{aligned}
+& D_2 \ell_c(r_0, \dot{r}_0) + D_1 L_d^*(r_0, r_1) + f_0^- = \hat{f}_0^- \\
+& D_2 \ell_c(r_N, \dot{r}_N) - D_2 L_d^*(r_{N-1}, r_N) - f_{N-1}^+ = \hat{f}_{N-1}^+ \\
+& w_d(r_k, g_k, r_{k+1}, g_{k+1}) = 0,
+\end{aligned}
+ \quad (15)
+ $$
+
+which determine the complete evolution of the system.
+
+When constructing a reduced algorithm with the midpoint rule we set
+
+$$ L_d^*(r_k, r_{k+1}) = h l_c(r_{k+\frac{1}{2}}, \Delta r_k) \quad (16) $$
+
+$$ \hat{f}_k^\pm = \frac{h}{2} \hat{f} (r_{k+\frac{1}{2}}, \Delta r_k) \quad (17) $$
+
+The equation for $\hat{f}_k^\pm$ was derived assuming linear dependence of the connection on the base point [25]. While a more general formulation exists, for the purpose of this paper we assume that it is a valid approximation. For the car-like examples that we consider the connection is linear in the base point and the linearity assumption is satisfied.
+
+Using the exponential map to define the midpoint (along the flow) between two configurations in $G$, the constraint equation in (14) becomes
+
+$$ g_k^{-1} g_{k+1} - \exp(h\xi_k) = 0, $$

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+Fig. 1. Differential drive optimized trajectory with $N = 32$
+
+## B. Experiments
+
+The goal is to find a trajectory between an initial and final state (fig. 1) with minimum control effort (i.e. the objective function is the sum of the squares of controls). The task constraints include bounds on the accelerations, bounds on the velocities, and obstacle avoidance. These conditions are expressed as inequality constraints. We only consider obstacles represented by arbitrary polygons and ellipses, although the method can be extended to any obstacles. The experimental results described below are averaged over runs in 10 different environments created by randomly perturbing the obstacles parameters (e.g. vertices, centers, size).
+
+The optimization is performed using an SQP solver which requires an initial guess. An $A^*$ algorithm is used to plan a shortest distance path in $(x, y)$ space and generate approximate values for the remaining coordinates. The solver is then able to transform this path into an executable trajectory and optimize it further. The resulting solution is not guaranteed to be globally optimal but there is strong evidence that it is a good optimum within its homotopy class.
+
+The method is compared against standard direct collocation with Hermite-Simpson discretization [23] of the continuous reduced equations. We are interested in finding how well the method performs as a function of the discretization resolution, i.e. the number of time steps $N$ used. The four criteria tested are: runtime (fig. 2), ratio of convergence to the optimal objective value (fig. 3), goal error after executing the trajectory (fig. 4), and average error between computed and executed trajectories (fig. 5). We use a fourth order Runge-Kutta solver with 10000 time steps as a ground truth. The resulting errors provide insight into how well the system dynamics is preserved by each method.
+
+The runtime of the two algorithms is comparable; DMOC slightly more efficient at smaller time steps $^2$. DMOC converges to its optimal objective value faster. DMOC exhibits less error at bigger time steps and hence the executed paths
+
+(by a fine integration) reach the goal closer. Therefore, computational advantage can be gained by using DMOC at reduced resolution.
+
+## V. CONCLUSIONS AND FUTURE WORK
+
+The paper proposes a new method for solving the motion planning problem for nonholonomic mechanical systems. It is based on discretizing the Lagrange-d'Alembert principle to derive discrete equations of motion serving as equality constraints in a numerical optimization scheme. Reduction by symmetry in the principal kinematic case is employed to simplify the optimal control formulation. Our experiments for car-like robots suggest that the method is a good alternative to standard collocation. It would be useful to further optimize the approach based on ideas from, e.g. [9], [27], [4].
+
+A major limitation of the approach is that it only provides a locally optimal solution. Nevertheless, it could be employed as an efficient local optimization method since it has to ability to converge to a good approximation with relatively few time steps. Another obvious application is the refinement of suboptimal trajectories computed from discrete or sampling-based motion planners.
+
+## ACKNOWLEDGMENT
+
+The authors would like to thank Eva Kanso and Mathieu Desbrun for valuable direction and advice in this project.
+
+$^2$At $N = 128$ the runtime for DMOC consistently jumps. This condition was related to sudden large memory allocation by the SQP solver only at this specific value.

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+[Bro06] Kenneth S. Brown. The homology of Richard Thompson's group $F$. In *Topological and asymptotic aspects of group theory*, volume 394 of *Contemp. Math.*, pages 47–59. Amer. Math. Soc., Providence, RI, 2006.
+
+[BT12] Carl-Friedrich Bödigheimer and Ulrike Tillmann. Embeddings of braid groups into mapping class groups and their homology. In *Configuration spaces*, volume 14 of CRM Series, pages 173–191. Ed. Norm., Pisa, 2012.
+
+[Deh06] Patrick Dehornoy. The group of parenthesized braids. *Adv. Math.*, 205(2):354–409, 2006.
+
+[FK04] L. Funar and C. Kapoudjian. On a universal mapping class group of genus zero. *Geom. Funct. Anal.*, 14(5):965–1012, 2004.
+
+[FM11] Benson Farb and Dan Margalit. *A primer on mapping class groups*. Princeton, NJ: Princeton University Press, 2011.
+
+[FN18] Louis Funar and Yurii Neretin. Diffeomorphism groups of tame Cantor sets and Thompson-like groups. *Compos. Math.*, 154(5):1066–1110, 2018.
+
+[GLU20] Anthony Genevois, Anne Lonjou, and Christian Urech. Asymptotically rigid mapping class groups I: Finiteness properties of braided thompson's and houghton's groups. *Geom. Topol.*, 2020. To appear. arXiv:2010.07225.
+
+[GRW18] Søren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. I. *J. Amer. Math. Soc.*, 31(1):215–264, 2018.
+
+[GS87] Étienne Ghys and Vlad Sergiescu. Sur un groupe remarquable de difféomorphismes du cercle. *Comment. Math. Helv.*, 62(2):185–239, 1987.
+
+[Har85] John L. Harer. Stability of the homology of the mapping class groups of orientable surfaces. *Ann. of Math. (2)*, 121(2):215–249, 1985.
+
+[Hat91] Allen Hatcher. On triangulations of surfaces. *Topology Appl.*, 40(2):189–194, 1991.
+
+[Hig74] Graham Higman. *Finitely presented infinite simple groups*. Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974. Notes on Pure Mathematics, No. 8 (1974).
+
+[HV17] Allen Hatcher and Karen Vogtmann. Tethers and homology stability for surfaces. *Algebr. Geom. Topol.*, 17(3):1871–1916, 2017.
+
+[HW10] Allen Hatcher and Nathalie Wahl. Stabilization for mapping class groups of 3-manifolds. *Duke Math. J.*, 155(2):205–269, 2010.
+
+[Nak61] Minoru Nakaoka. Homology of the infinite symmetric group. *Ann. of Math. (2)*, 73:229–257, 1961.
+
+[RWW17] Oscar Randal-Williams and Nathalie Wahl. Homological stability for automorphism groups. *Adv. Math.*, 318:534–626, 2017.
+
+[SW] Rachel Skipper and Xiaolei Wu. Finiteness properties for relatives of braided higman-thompson groups. preprint.
+
+[SW19] Markus Szymik and Nathalie Wahl. The homology of the Higman-Thompson groups. *Invent. Math.*, 216(2):445–518, 2019.
+
+[Thu17] Werner Thumann. Operad groups and their finiteness properties. *Adv. Math.*, 307:417–487, 2017.
+
+[vdK80] Wilberd van der Kallen. Homology stability for linear groups. *Invent. Math.*, 60(3):269–295, 1980.
+
+THE OHIO STATE UNIVERSITY, MATH TOWER, 231 W 18TH AVE, COLUMBUS, OH 43210, USA
+*Email address: skipper.26@osu.edu*
+
+FAKULTÄT FÜR MATHEMATIK, UNIVERSITÄT BIELEFELD, POSTFACH 100131, D-33501 BIELEFELD, GERMANY
+*Email address: xwu@math.uni-bielefeld.de*

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+Part of this work was done when the second author was a member of the Hausdorff Center of Mathematics. At the time, he was supported by Wolfgang Lück's ERC Advanced Grant "KL2MG-interactions" (no. 662400) and the DFG Grant under Germany's Excellence Strategy - GZ 2047/1, Projekt-ID 390685813.
+
+Part of this work was also done when both authors were visiting IMPAN at Warsaw during the Simons Semester "Geometric and Analytic Group Theory" which was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. We would also like to thank Kai-Uwe Bux for inviting us for a research visit at Bielefeld in May 2019 and many stimulating discussions. Special thanks go to Jonas Flechsig for his comments on preliminary versions of the paper. Furthermore, we want to thank Javier Aramayona and Stefan Witzel for discussions, Andrea Bianchi for comments and Matthew C. B. Zaremsky for some helpful communications and comments.
+
+# 1. CONNECTIVITY TOOLS
+
+In this section, we review some of the connectivity tools that we need for calculating the connectivity of our spaces. A good reference is [HV17, Section 2] although not all the tools we use can be found there.
+
+## 1.1. Complete join.
+The complete join is useful tool introduced by Hatcher and Wahl in [HW10, Section 3] for proving connectivity results. We review the basics here.
+
+**Definition 1.1.** A surjective simplicial map $\pi : Y \to X$ is called a *complete join* if it satisfies the following properties:
+
+(1) $\pi$ is injective on individual simplices.
+
+(2) For each $p$-simplex $\sigma = \langle v_0, \dots, v_p \rangle$ of $X$, $\pi^{-1}(\sigma)$ is the join $\pi^{-1}(v_0) * \pi^{-1}(v_1) * \dots * \pi^{-1}(v_p)$.
+
+**Definition 1.2.** A simplicial complex $X$ is called weakly Cohen-Macaulay of dimension $n$ if $X$ is $(n-1)$-connected and the link of each $p$-simplex of $X$ is $(n-p-2)$-connected. We sometimes shorten weakly Cohen-Macaulay to *wCM*.
+
+The main result regarding complete join that we will use is the following.
+
+**Proposition 1.3.** [HW10, Proposition 3.5] If $Y$ is a complete join complex over a wCM complex $X$ of dimension $n$, then $Y$ is also wCM of dimension $n$.
+
+**Remark 1.4.** If $\pi: Y \to X$ is a complete join, then $X$ is a retract of $Y$. In fact, we can define a simplicial map $s: X \to Y$ such that $\pi \circ s = \text{id}_X$ by sending a vertex $v \in X$ to any vertex in $\pi^{-1}(v)$ and then extending it to simplices. The fact that $s$ can be extended to simplices is granted by the condition that $\pi$ is a complete join. In particular we can also conclude that if $Y$ is $n$-connected, so is $X$.
+
+## 1.2. Bad simplices argument.
+Let $(X, Y)$ be a pair of simplicial complexes. We want to relate the $n$-connectedness of $Y$ to the $n$-connectedness of $X$ via a so called bad simplices argument, see [HV17, Section 2.1] for more information. One identifies a set of simplices in $X \setminus Y$ as bad simplices, satisfying the following two conditions:
+
+(1) Any simplex with no bad faces is in $Y$, where by a “face” of a simplex we mean a subcomplex spanned by any nonempty subset of its vertices, proper or not.
+
+(2) If two faces of a simplex are both bad, then their join is also bad.
+
+We call simplices with no bad faces good simplices. Bad simplices may have good faces or faces which are neither good nor bad. If $\sigma$ is a bad simplex, we say a simplex $\tau$ in Lk($\sigma$) is good for $\sigma$ if any bad face of $\tau * \sigma$ is contained in $\sigma$. The simplices which are good for $\sigma$ form a subcomplex of Lk($\sigma$) which we denote by $G_{\sigma}$ and call the good link of $\sigma$.
+
+**Proposition 1.5.** [HV17, Proposition 2.1] Let $X, Y$ and $G_{\sigma}$ be as above. Suppose that for some integer $n \ge 0$ the subcomplex $G_{\sigma}$ of $X$ is $(n - \dim(\sigma) - 1)$-connected for all bad simplices $\sigma$. Then the pair $(X, Y)$ is $n$-connected, i.e. $\pi_i(X, Y) = 0$ for all $i \le n$.

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+Our method of dealing with hereditary rings will be based on ideas developed by Simson in [35, 37, 40] that allow one to reduce the question to certain triangular matrix rings of the form $(B^F G^0)$ induced by division rings $F$ and $G$ and a bimodule $GB_F$, and the use of reflection functors (see, e.g., [30, 35, 37]) will be essential. We will also apply Auslander’s theory of Grassmannians [5] and Herzog’s ideas [21], to reduce our study of an arbitrary ring to the case of a hereditary (triangular matrix) ring.
+
+The paper is organized as follows. In Section 2, we discuss general properties of finendo and cofinendo modules, and characterize rings with all left modules cofinendo. In Section 3, we prove our main result on hereditary rings. Finally, Section 4 is devoted to the same questions for general rings.
+
+**2. Finendo and cofinendo modules.** Throughout this paper, $R$ is an associative ring with identity. We denote by $R$-mod the category of finitely presented left $R$-modules, and by $R$-Mod the category of all left $R$-modules. The corresponding categories of right $R$-modules are denoted by mod-$R$ and Mod-$R$. Homomorphisms between $R$-modules are assumed to operate on the side opposite to the scalars. A right $R$-module $M_R$ can be regarded in a natural way as a left End$(M_R)$-module, and similarly a left $R$-module $RN$ is a right End$(RN)$-module. We refer the reader to [1, 8, 28, 36, 42, 43] for general properties of rings, modules, and categories, and for all undefined notions used in the text.
+
+Recall that a right (or left) $R$-module $M$ is *finendo* if $M$ is finitely generated as a module over its endomorphism ring (see [16, 17]). Dually, $M$ is said to be *cofinendo* if it is finitely cogenerated over $\text{End}(M_R)$. (Note that the term “cofinendo module” was also used in [3], but with a meaning different from ours.) A module $M$ is *endofinite* (*endoartinian*, *endonoetherian*) if $M$ is of finite length (artinian, noetherian, respectively) as a module over its endomorphism ring. For a right $R$-module $M$, a subgroup $L$ of the Abelian group $M$ is called a *matrix subgroup* of $M$ if it is of the form
+
+$$ L = \operatorname{Hom}_R(Y, M)(x) = \{f(x) \mid f \in \operatorname{Hom}_R(Y, M)\} $$
+
+where $Y$ is a right $R$-module and $x \in Y$. If the module $Y$ is finitely presented, then $L$ is called a *finite matrix subgroup* of $M$.
+
+Following [27], a module $M$ is *product-complete* provided every product of copies of $M$ is a direct summand of a direct sum of copies of $M$. It is well-known that every product-complete module is finendo (see [27, Proposition 4.2]). The following characterization reflects a similar behavior of finendo modules with regard to direct products.
+
+**PROPOSITION 2.1.** Let $R$ be any ring and $M$ a right $R$-module. Then $M$ is finendo if and only if $M$ generates any product of copies of $M$.

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+*Proof.* Suppose first that $M$ is finendo with endomorphism ring $S$, and let $I$ be any index set. There is a finite generating set $\{u_1, \dots, u_n\}$ of the left $S$-module $M$. To show that $M$ generates $M^I$, it suffices to see that the trace of $M^n$ on $M^I$ is $M^I$. If we take any element $x = (x_i)_{i \in I}$ of the direct product $M^I$, then for each $i \in I$ we have $x_i = f_{1i}u_1 + \cdots + f_{ni}u_n$ for some endomorphisms $f_{1i}, \dots, f_{ni}$ of $M$. Then we define the homomorphism $g : M^n \to M^I$ as follows: $g = (g_i)_{i \in I}$, where $g_i$ is a homomorphism from $M^n$ to the $i$th component $M$ of $M^I$ such that if $y = (y_1, \dots, y_n) \in M^n$, then $g_i(y) = f_{1i}y_1 + \cdots + f_{ni}y_n$. It is clear that $g_i(u_1, \dots, u_n) = x_i$, and hence $g(u_1, \dots, u_n) = x$, proving our claim.
+
+Conversely, suppose that $M$ generates any product of copies of $M$. Consider the module $M^M$ and the element of this product whose $x$-component is $x$. By hypothesis, there exists some homomorphism $g : M^n \to M^M$ and an element $(u_1, \dots, u_n)$ of $M^n$ whose image under $g$ is the given element $(x_x)_x \in M^M$. This clearly shows that elements $u_1, \dots, u_n$ generate $M$ as a module over its endomorphism ring. $\blacksquare$
+
+We deduce the following immediate consequence.
+
+**COROLLARY 2.2.** Let $R$ be any ring. The property of being finendo as a right $R$-module is preserved under taking finite direct sums, and arbitrary direct products of copies of a single module. Moreover, if $M$ is a generator in Mod-$R$, then $M$ is finendo.
+
+We now turn to the question of when a direct sum of cofinendo modules is again cofinendo. The next result shows that this holds for finite direct sums of indecomposable modules of finite length. For any $R$-module $M$ with endomorphism ring $S$, the $S$-socle of $M$ will be referred to as the *endosocle* of $M$, and denoted as $\text{Esoc}(M)$.
+
+**LEMMA 2.3.** Let $R$ be any ring and $M = \bigoplus_{k=1}^n M_k$ be a finite direct sum of left $R$-modules with local endomorphism rings. If each $M_k$ has a finitely generated endosocle, then $M$ has a finitely generated endosocle. In particular, if each $M_k$ is cofinendo indecomposable of finite length, then $M = \bigoplus_{k=1}^n M_k$ is cofinendo.
+
+*Proof.* Let $B$ be the endosocle of $M = \bigoplus_{k=1}^n M_k$. Because each $M_k$ has a local endomorphism ring, it follows from [25, Lemma B] that there is a decomposition $B = \bigoplus_{k=1}^n B_k$ where $B_k = B \cap M_k$, and each $B_k$ is the intersection of all kernels of non-isomorphisms from $M_k$ to $M_j$ with $j = 1, \dots, n$. In particular, $B_k$ is an endosubmodule of $M_k$. Note that clearly each $B_k$ is contained in the endosocle of $M_k$ which is finitely generated by hypothesis. Thus, $B_k$ is finitely generated over $\text{End}(M_k)$. But each element of $\text{End}(M_k)$ can also be viewed as an element of $S = \text{End}(M)$. It follows that each $B_k$ has a finite generating set as a left $S$-module. Therefore the endosocle $B$ of $M$ has a finite generating set as left $S$-module, as required.

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+For the second part of the lemma, assume that each $M_k$ is cofinendo indecomposable of finite length. The left $R$-module $M = \bigoplus_{k=1}^n M_k$ is of finite length, hence it has a semiprime endomorphism ring. Thus $M$ has an essential endosocle. By the above, we see that $M$ has a finitely generated endosocle. Therefore $M$ is finitely cogenerated over its endomorphism ring. ■
+
+Following Auslander [7], for a left $R$-module $N$ with $S = \text{End}_R(N)$, the local dual of $N$ is defined as the right $R$-module $D(N) = \text{Hom}_S(N_S, C_S)$, where $C_S$ is a minimal injective cogenerator of $\text{Mod-}S$. The following results give useful connections between the finendo and cofinendo conditions for certain left and right $R$-modules through the local duality.
+
+**PROPOSITION 2.4.** Let $R$ be a ring, and $N = \bigoplus_{i \in I} N_i$ be a direct sum of finitely presented left $R$-modules. Let $M_i = D(N_i)$ be the local dual of $N_i$ and $M = \bigoplus_{i \in I} M_i$. Then $M$ is cofinendo if and only if $N$ is finendo and $N/\text{Rad}(N_S)$ is a semisimple right $S$-module, where $S$ is the endomorphism ring of $N$.
+
+*Proof.* See [15, Proposition 4.8]. ■
+
+**PROPOSITION 2.5.** Let $R$ be a ring, and $N = \bigoplus_{i \in I} N_i$ be a direct sum of finitely presented left $R$-modules each with a local endomorphism ring. Let $M_i = D(N_i)$ be the local dual of $N_i$ and $M = \bigoplus_{i \in I} M_i$. If $M$ is finendo, then $N$ is cofinendo.
+
+*Proof.* See [15, Proposition 4.9]. ■
+
+The second part of the result below was proved in [15, Proposition 4.10] under the additional hypothesis that the left functor ring $A$ of $R$ is right semiartinian. We now give an alternative proof for arbitrary rings. Note that when $RN = R$, the result says that if the ring $R$ has a finitely generated essential right socle, then the minimal injective cogenerator $E$ of $\text{Mod-}R$ is finendo, a fact that also follows from Beachy [9].
+
+**PROPOSITION 2.6.** Let $R$ be any ring, $N$ a left $R$-module, $M$ a right $R$-module and suppose that the lattice of endosubmodules of $N$ is anti-isomorphic to the lattice of matrix subgroups of $M$. If $N$ is cofinendo, then $M$ is finendo. In particular, if $N = \bigoplus_{i \in I} N_i$ is a direct sum of finitely presented left $R$-modules, set $M_i = D(N_i)$, the local dual of $N_i$, and $M = \bigoplus_{i \in I} M_i$. If $N$ is cofinendo, then $M$ is finendo.
+
+*Proof.* Since $N$ is cofinendo, any family of endosubmodules of $N$ which has zero intersection contains a finite subfamily having zero intersection (see, e.g., [43, 14.7]). It follows that the lattice of matrix subgroups of $M$ is compact, i.e. whenever $M$ is the join of a family of matrix subgroups, it is the join of a finite subfamily.

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+right artinian. Because every pure-projective right $R$-module is finendo, by Lemma 2.10 it follows that $R$ is left pure semisimple. Now Theorem 2.11 above shows that every left $R$-module is cofinendo. ■
+
+**3. The case of hereditary rings.** In this section, we study left pure semisimple hereditary rings $R$ with the property that every finitely generated indecomposable left $R$-module is cofinendo, or the property that every finitely generated indecomposable left $R$-module is finendo. A related version of these properties was also studied by Simson [35, Corollary 3.2]. We show that any left pure semisimple hereditary ring satisfying either of the above properties is of finite representation type, and we give positive answers to [13, Questions 1 and 2, pp. 122–123] in the hereditary case.
+
+The proof of our result is based on some lemmas. The first lemma below is valid without the hereditary hypothesis.
+
+**LEMMA 3.1.** Let $A$ and $B$ be Morita equivalent rings. Then every finitely generated left $A$-module is finendo (cofinendo) if and only if every finitely generated left $B$-module is finendo (respectively, cofinendo).
+
+*Proof.* Let $G: B\text{-Mod} \to A\text{-Mod}$ be a category equivalence, let $M$ be any finitely generated left $B$-module, and set $N = G(M)$. If $_AP$ is any finitely generated projective left $A$-module, then $\operatorname{Hom}_A(P, N)$ is a direct summand of a finite direct sum of copies of $\operatorname{Hom}_A(A, N) \cong N$, as a right $\operatorname{End}(N)$-module. Since $G(B)$ is finitely generated projective, it follows that if $N$ is finendo (respectively, cofinendo), then $\operatorname{Hom}_A(G(B), N)$ is finitely generated (respectively, finitely cogenerated) as a right $\operatorname{End}(N)$-module.
+
+Since $G$ is full and faithful, $\operatorname{Hom}_A(G(B), N) \cong \operatorname{Hom}_B(B, M)$, and this is a semilinear isomorphism relative to the ring isomorphism $\operatorname{End}(M) \cong \operatorname{End}(N)$. It is clear that a semilinear isomorphism preserves the lattices of submodules, and thus if $N$ is finendo (respectively, cofinendo), then $\operatorname{Hom}_B(B, M) \cong M$ is finitely generated (respectively, finitely cogenerated) as a right $\operatorname{End}(M)$-module. ■
+
+The following characterization of hereditary rings of finite representation type, due to Simson [35, 40], will give an important induction step in our study of left pure semisimple hereditary rings.
+
+**LEMMA 3.2.** Let $R$ be a basic indecomposable left pure semisimple hereditary ring. Then the following conditions are equivalent:
+
+(a) $R$ is of finite representation type.
+
+(b) For any pair of indecomposable projective direct summands $P_i \neq P_j$ of $RR$, there is a ring isomorphism
+
+$$ R_B = \begin{pmatrix} F & 0 \\ B & G \end{pmatrix} \cong \operatorname{End}(P_i \oplus P_j) $$

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+where $F = \text{End}_R P_i$ and $G = \text{End}_R P_j$ are division rings, $B = \text{Hom}_R(P_i, P_j)$, and $R_B$ is a ring of finite representation type.
+
+*Proof*. See [40, Theorem 3.4]. $\blacksquare$
+
+We also need the following simple but useful fact.
+
+**LEMMA 3.3.** *Let $R$ be a left artinian hereditary ring. Suppose that every finitely generated indecomposable projective left $R$-module is finendo. Then $R$ is right artinian.*
+
+*Proof.* Let $P$ be any finitely generated indecomposable projective left $R$-module, and let $f : P \to P$ be any non-zero homomorphism. Since $R$ is left hereditary, $\text{Im}(f)$ is a projective left $R$-module, hence $f$ splits, implying that $\text{Ker}(f)$ is a direct summand of $P$. Since $P$ is indecomposable and $f$ is non-zero, we see that $\text{Ker}(f) = 0$, so $f$ is a monomorphism. As $P$ is of finite length, $f$ must be an isomorphism. This shows that the endomorphism ring $S$ of $P$ is a division ring, and because $P$ is finendo by hypothesis, it follows that $P$ is endofinite. Since endofinite modules are preserved under taking finite direct sums [11, Proposition 4.3], the ring $R$ is endofinite as a left module over itself, i.e. $R$ is right artinian. $\blacksquare$
+
+The next proposition, which might be of independent interest, is a key
+step in the proof of our main result.
+
+**PROPOSITION 3.4.** Let $F, G$ be division rings, and let $GB_F$ be a non-zero bimodule. Suppose that the ring $R_B = (\begin{smallmatrix} F & 0 \\ B & G \end{smallmatrix})$ is left and right artinian. Set $FM_G = \text{Hom}_F(B, F)$ and consider the triangular matrix ring $R_M = (\begin{smallmatrix} G & 0 \\ M & F \end{smallmatrix})$.
+
+(a) If every finitely generated indecomposable left $R_B$-module is cofin-
+endo, then every finitely generated indecomposable left $R_M$-module
+is cofinendo.
+
+(b) If every finitely generated indecomposable left $R_B$-module is finendo, then every finitely generated indecomposable left $R_M$-module is finendo.
+
+*Proof.* (a) We assume that $R_B$ is left and right artinian, and every finitely generated indecomposable left $R_B$-module is cofinendo.
+
+First note that if $A$ is any ring and $L$ is any left $A$-module, then $L$ is cofin-
+endo if and only if for any finitely generated left $A$-module $X$, $\text{Hom}_A(X, L)$
+is finitely cogenerated over $\text{End}(L)$. Indeed, $L$ being cofinendo means that
+$\text{Hom}_A(A, L)$ is finitely cogenerated over $\text{End}(L)$, hence $\text{Hom}_A(A^m, L)$ is
+finitely cogenerated over $\text{End}(L)$ for any positive integer $m$. There is an ex-
+act sequence $A^n \to X \to 0$ in $A$-Mod which induces the exact sequence $0 \to$
+$\text{Hom}_A(X, L) \to \text{Hom}_A(A^n, L)$ in Mod-End$(L)$, implying that $\text{Hom}_A(X, L)$
+is finitely cogenerated over $\text{End}(L)$. The “if” part of the claim is trivial.

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+We recall (see, e.g., [21, p. 174]) that there is an equivalence between the category $R_B$-Mod of left $R_B$-modules and the category whose objects are the triples $(X, Y, \lambda)$, where $X, Y$ are left $F$- or $G$-modules, respectively, and $\lambda : B \otimes_F X \to Y$ is a $G$-homomorphism (equivalently, we could take instead the $F$-homomorphism $\bar{\lambda} : X \to \operatorname{Hom}_G(B, Y)$). The equivalence associates to such an object $(X, Y, \lambda)$ the left $R_B$-module whose elements are column vectors $\begin{pmatrix} x \\ y \end{pmatrix}$ with $x \in X, y \in Y$ with the usual matrix operations and using $\lambda$ to define the product $B \times_X X \to Y$. Moreover, the morphisms in that category between two objects $(X, Y, \lambda)$ and $(X', Y', \mu)$ are given by the pairs $(f, g)$ of linear maps $f : _F X \to _F X'$ and $g : _G Y \to _G Y'$ such that $g \circ \lambda = \mu \circ (1 \otimes f)$. From now on, we identify each left $R_B$-module with the corresponding triple.
+
+The construction of the Bernstein–Gelfand–Ponomarev *reflection func-tors* $S^+ : R_B$-mod $\to$ $R_M$-mod and $S^- : R_M$-mod $\to$ $R_B$-mod is given, for example, in Ringel [30], Simson [35, 37], or Herzog [21]. These func-tors can be defined as follows. Given a finitely generated left $R_B$-module $(X, Y, \lambda)$, we use the canonical isomorphism $B \otimes_F X \cong \operatorname{Hom}_F(M, X)$ to write $\tilde{\lambda} : \operatorname{Hom}_F(M, X) \to Y$, and take its kernel $K = \operatorname{Ker}(\tilde{\lambda})$, giving the left $R_M$-module $S^+(X, Y, \lambda) = (K, X, u)$, with $u$ corresponding to the in-clusion $\bar{\mu} : K \to \operatorname{Hom}_F(M, X)$. Similarly, given a finitely generated left $R_M$-module $(U, V, \mu)$ with $U \stackrel{\bar{\mu}}{\rightarrow} \operatorname{Hom}_F(M, V)$, where $U$ and $V$ are finite-dimensional $G$- (respectively, $F$-) vector spaces, we use the same isomor-phism above to obtain $\tilde{\mu} : U \to B \otimes_F V$. Then we take its cokernel $C$, giving $S^-(U, V, \mu) = (V, C, p)$, where $p : B \otimes_F V \to C$ is the projection. In view of the equivalence of categories shown in [21, Proposition 6.8] (see also [37, Lemma 3.1]), we have the following properties:
+
+(i) If $X$ is an indecomposable module in $R_M$-mod, then $S^-(X) = 0$ if and only if $X$ is isomorphic to $(G, 0, 0)$, a simple injective left $R_M$-module which we shall denote as $Q$. Moreover, if $S^-(X)$ is non-zero, then $S^+S^-(X)$ is isomorphic to $X$.
+
+(ii) If $X$ and $Y$ are indecomposable modules in $R_M$-mod such that $S^-(X)$ and $S^-(Y)$ are non-zero, then there is an isomorphism of abelian groups
+
+$$\operatorname{Hom}_{R_M}(X, Y) \cong \operatorname{Hom}_{R_B}(S^-(X), S^-(Y)).$$
+
+In particular, if $X \cong Y \not\cong Q$, then (ii) implies that $X$ and $S^-(X)$ have isomorphic endomorphism rings. Moreover, $\operatorname{Hom}_{R_M}(X, Y)$ is a right End$(Y)$-module, $\operatorname{Hom}_{R_B}(S^-(X), S^-(Y))$ is a right End$(S^-(Y))$-module, and the above abelian group isomorphism is also a semilinear isomorphism relative to this ring isomorphism.
+
+Now, let $C$ be any finitely generated indecomposable left $R_M$-module, and we want to show that $C$ is cofinendo. If $C$ is isomorphic to the simple left $R_M$-module $Q$ above, then $C$ is the module of column vectors $\begin{pmatrix} x \\ 0 \end{pmatrix}$ with

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+Let $M = \operatorname{Hom}_F(B, F)$. Then $M$ is a $F$-$G$-bimodule, and we can consider
+the triangular matrix ring
+
+$$R_M = \begin{pmatrix} G & 0 \\ M & F \end{pmatrix}.$$
+
+Then $R_M$ is a left pure semisimple ring of infinite representation type (see [21, 37]), and by Proposition 3.4(a) every finitely generated indecomposable left $R_M$-module is cofinendo. Applying Corollary 2.9, we find that $R_M$ is right artinian, and hence the vector space $M_G$ is finite-dimensional. We can take $M_1 = \operatorname{Hom}_G(M, G)$ which is a $G$-$F$-bimodule, and consider the triangular matrix ring
+
+$$R_{M_1} = \begin{pmatrix} F & 0 \\ M_1 & G \end{pmatrix}.$$
+
+Again, using arguments similar to the above, we deduce that $R_{M_1}$ is left pure semisimple representation-infinite, and every finitely generated indecomposable left $R_{M_1}$-module is cofinendo, so $R_{M_1}$ is right artinian, again by Corollary 2.9.
+
+We may extend this process so that the iterated dual bimodules are always finite-dimensional on both sides. Then, by applying [35, Theorem 3.1] or [37, Theorem 3.4], we conclude that the matrix ring $R_B$ must be of finite representation type, which is a contradiction.
+
+(d) $\Rightarrow$ (e). Suppose that (d) holds. Then Corollary 2.2 implies that every finitely generated left $R$-module is finendo. As in the proof of (c)$\Rightarrow$(e), we can use Lemma 3.1 to assume that $R$ is basic and indecomposable, and then Lemma 3.2 allows us to start with a left pure semisimple representation-infinite ring of the form $R_B = \begin{pmatrix} F & 0 \\ B & G \end{pmatrix}$ that satisfies (d). Note that, by Lemma 3.3, it follows that any hereditary ring satisfying (d) must be right artinian. Now using Proposition 3.4(b) repeatedly, as in the proof of (c)$\Rightarrow$(e), we will get a contradiction by applying again [35, Theorem 3.1] or [37, Theorem 3.4].
+
+(e) $\Rightarrow$ (a). If $R$ is of finite representation type, then every right $R$-module is endofinite (see [10, 26, 29]), hence (a) follows. $\blacksquare$
+
+**REMARK 3.6.** As pointed out to us by Professor Daniel Simson, the implications (c)⇒(e) and (d)⇒(e) of Theorem 3.5 can also be proved using [40, Corollary 2.11] (cf. [35, 37]), from which it follows that a left pure semisimple ring of the form $R_B = \begin{pmatrix} F & 0 \\ B & G \end{pmatrix}$ is of finite representation type if every indecomposable preprojective left $R$-module is endofinite.
+
+**4. The case of non-hereditary rings.** Given an arbitrary ring $R$, we have seen in Section 2 that if all right $R$-modules are finendo, then $R$ is left pure semisimple and all finitely generated indecomposable left $R$-modules are cofinendo (Theorem 2.11 and Corollary 2.12). So, the questions

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+**Table I:** The gap of expected losses.
+
+<table><thead><tr><th></th><th>q<sub>2</sub> > q̃<sub>2</sub></th><th>q<sub>2</sub> ≤ q̃<sub>2</sub></th></tr></thead><tbody><tr><td>θe > k</td><td>ETC<sub>3</sub> > ETC<sub>2</sub></td><td>ETC<sub>3</sub> ≤ ETC<sub>2</sub></td></tr><tr><td>θe ≤ k</td><td>ETC<sub>3</sub> ≤ ETC<sub>2</sub></td><td>ETC<sub>3</sub> > ETC<sub>2</sub></td></tr></tbody></table>
+
+**Lemma 4:** There exists a turning point $\bar{q}_2$, making $I_3^* \le I_2^*$ when $q_2 \le \bar{q}_2$, and $I_3^* > I_2^*$ when $q_2 > \bar{q}_2$.
+
+Lemma 4 can be illustrated by the closed-form solution of equation $I_3^* - I_2^* = 0$ and $\frac{\partial(I_3^* - I_2^*)}{\partial q_2} \ge 0$. Thus, the value of $q_2$ can change the relationship between $I_2^*$ and $I_3^*$.
+
+**Theorem 2:** The effective replacement strategy based on remaining quantity satisfies:
+
+(1) If $\bar{q}_2 > \tilde{q}_2$, when $\theta e > k$, then $q_2$ should satisfy $\tilde{q}_2 < q_2 \le \bar{q}_2$, and when $\theta e \le k$, then $q_2$ should satisfy $q_2 \le \tilde{q}_2$.
+
+(2) If $\bar{q}_2 \le \tilde{q}_2$, when $\theta e \le k$, then $q_2$ should satisfy $q_2 \le \bar{q}_2$, and when $\theta e > k$, there is no complete effective range.
+
+Theorem 2 can be demonstrated by Lemma 3 and Lemma 4, different from scenario 1, we are not sure the replacement-based strategy must be effective for airport. In order to reduce losses and enhance service level simultaneously, airports should lower the unit replacement cost by coordination or technologies, and control the size of $q_2$ at the same time.
+
+## **4.5 Further discussions about replacement and occurrence time uncertainty**
+
+In classical newsboy model, a stochastic storage model is proposed with stochastic demand and deterministic occurrence time and unlimited warehousing time (lifetime). Thus, the optimal solutions to the classical newsboy models with replacement based on remaining quantity and with non-replacement are calculated as follows.
+
+$$ \bar{I}_2 = \max(\frac{sd + kc}{2kq_2 - kq_2^2 + s + \theta e - 2\theta eq_2 + \theta eq_2^2}, c), \quad \bar{I}_3 = F^{-1}(\frac{s}{s + \theta e}). $$
+
+### * Effects of the value of ratios on inventory level
+
+**Theorem 3:** When $\theta e > k$, $\frac{\partial I_1^*}{\partial q_1} \le 0$, $\frac{\partial I_2^*}{\partial q_2} > 0$. When $\theta e \le k$, $\frac{\partial I_1^*}{\partial q_1} > 0$, $\frac{\partial I_2^*}{\partial q_2} \le 0$.
+
+The results of Theorem 3 can be illustrated well by the characteristics of replacement strategy based on remaining lifetime and replacement strategy based on remaining quantity.
+
+### * Effects of occurrence time uncertainty on inventory level
+
+**Theorem 4:** $\bar{I}_2 > I_2^*$, $\bar{I}_3 > I_3^*$.
+
+Theorem 4 can be obtained by calculating $\bar{I}_2 - I_2^*$ and $\bar{I}_3 - I_3^*$, and the results show that occurrence time uncertainty leads to lower inventory level. In theory, the optimal solution to the decision variable with stochastic occurrence time should consist with that with deterministic occurrence time under risk-neutral criteria. However, the ending time of a single emergency preparation is assumed min($t, T$), meaning that the cycle time is shortened and the order quantity is reduced accordingly.
+
+## **5. SIMULATION RESULTS**
+
+A simulation case is proposed and analysed firstly assuming stochastic occurrence time conforms to uniform distribution from *a* to *b* and stochastic demand conforms to uniform distribution from *c* to *d*. Then we simulate, analyse and compare the optimal inventory level

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+Inventory level represents service level, and the effectiveness of the replacement-based strategy can be measured by losses and inventory level through building the model that is non-replacement. The effects of occurrence time uncertainty on inventory level are analysed by building the model with deterministic occurrence time and unlimited warehousing time. Subscript 1, 2, 3 represents the replacement model based on remaining lifetime, replacement model based on remaining quantity, and general model without replacement respectively.
+
+# 3. MODELS WHEN REPLACEMENT IS ALLOWED
+
+## 3.1 Replacement model based on remaining lifetime
+
+The expected losses include shortage losses, possible expired losses, replacement costs, and expired losses for total emergency supplies.
+
+$$ 
+\begin{aligned}
+ETC_1 = & \int_a^T \int_I^d s(x-I)f(x)g(t)dxdt + \int_a^{q_1 T} \int_c^I \theta e(I-x)f(x)g(t)dxdt \\
+& + \int_{q_1 T}^T \int_c^I k(I-x)f(x)g(t)dxdt + \int_T^b \int_c^d eIf(x)g(t)dxdt
+\end{aligned}
+\quad (1) $$
+
+$$ 
+\begin{aligned}
+d ETC_1 / dI = & -\int_a^T \int_I^d sf(x)g(t)dxdt + \int_a^{q_1 T} \int_c^I \theta ef(x)g(t)dxdt \\
+& + \int_{q_1 T}^T \int_c^I kf(x)g(t)dxdt + \int_T^b \int_c^d ef(x)g(t)dxdt
+\end{aligned}
+\quad (2) $$
+
+$$ d^2 ETC_1 / dI^2 = sf(I)G(T) + \theta ef(I)G(T) + kf(I)(G(T) - G(q_1 T)) \quad (3) $$
+
+Eq. (3) is greater than zero, and the unique solution of Eq. (1) can be reached by letting Eq. (2) being equal to zero.
+
+$$ I_1^* = F^{-1}((\frac{(s+e)G(T)-e}{(s+k)G(T)+(\theta e-k)G(q_1 T)})^+) $$
+
+**Proposition 1:** $\frac{\partial I_1^*}{\partial s} \ge 0$, $\frac{\partial I_1^*}{\partial e} \le 0$, $\frac{\partial I_1^*}{\partial \theta} \le 0$, $\frac{\partial I_1^*}{\partial k} \ge 0$.
+
+Proposition 1 shows that the optimal inventory level increases with the increase of unit shortage cost, and decreases with the increase of unit expired cost, expired rate and unit replacement cost. Furthermore, the optimal inventory level may increase or decrease with the increase of lifetime, and it depends on the distribution function of stochastic occurrence time.
+
+## 3.2 Replacement model based on remaining quantity
+
+$$ 
+\begin{aligned}
+ETC_2 = & \int_a^T \int_I^d s(x-I)f(x)g(t)dxdt + \int_{q_2 I}^T \int_c^I \theta e(I-x)f(x)g(t)dxdt \\
+& + \int_a^T \int_c^{q_2 I} k(I-x)f(x)g(t)dxdt + \int_T^b \int_c^d eIf(x)g(t)dxdt
+\end{aligned}
+\quad (4) $$
+
+**Proposition 2:** The existence of solution of Eq. (4) is dependent on the distributed functions in stochastic demand. If the solution $I_2^*$ can be reached, then the following holds:
+
+$$ 
+\begin{aligned}
+& sF(I_2^*)G(T) + \theta eF(I_2^*)G(T) - \theta eF(q_2 I_2^*)G(T) - q_2\theta e(I_2^* - q_2 I_2^*)f(q_2 I_2^*)G(T) \\
+& + kF(q_2 I_2^*)G(T) + q_2k(I_2^* - q_2 I_2^*)f(q_2 I_2^*)G(T) + e - eG(T) - sG(T) = 0
+\end{aligned}
+$$
+
+**Proposition 3:** The effects of parameters (shortage cost, expired cost, replacement cost, replacement quantity ratio, expected expiration rate etc.) on optimal inventory level depend on the distribution function of stochastic demand.
+
+From Proposition 2 and Proposition 3, the existence of optimal solution and the effects of parameters is correlated to the distributed function in stochastic demand, and to but much less to the distributed function in stochastic occurrence time. The possible explanation is that the stochastic occurrence time cannot affect the integrand in this construction function.

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+**Proposition 7.** Let $G = (V, E)$ be a graph with the weights $\Omega = \{\omega_e \in [-\pi, \pi]\}_{e \in E}$ assigned to its edges and the weights $\Upsilon = \{v_v \in [-\pi, \pi]\}_{v \in V}$ assigned to its vertices, then,
+
+$$\psi_{\mathcal{X}_G}(0^{|V|}) = \frac{1}{2^{|V|}} Z_{\text{Ising}}(G; i\Omega, i\Upsilon).$$
+
+Proposition 7 is well known [20, 21]; we provide a proof in Appendix B. It will be convenient to consider graph-induced X-programs $\mathcal{X}_{G(\theta)}$ with weights that are all positive integer multiples of a real angle $\theta$. As in Section III, this model can be implemented on the augmented graph $G' = (V', E')$ with all weights equal to $\theta$ by replacing each edge with the appropriate number of parallel edges. Let us denote the graph-induced X-program of this model by $\mathcal{X}_{G'(\theta)}$. Then, we have the following proposition.
+
+**Proposition 8.**
+
+$$\psi_{\mathcal{X}_{G(\theta)}}(0^{|V|}) = \psi_{\mathcal{X}_{G'(\theta)}}(0^{|V'|}).$$
+
+We prove Proposition 8 in Appendix C. We also have the following proposition relating the principal probability amplitude to the Tutte polynomial of the augmented graph.
+
+**Proposition 9.**
+
+$$\psi_{\mathcal{X}_{G'}(\theta)}(0^{|V'|}) = e^{i\theta(r(G') - |E'|)} (i \sin(\theta))^{r(G')} T(G'; x, y),$$
+
+where $x = -i \cot(\theta)$ and $y = e^{2i\theta}$.
+
+We prove Proposition 9 in Appendix D. Notice that if we let $M = (S, I)$ be the binary matroid whose ground set $S$ is the column space of the orientated incidence matrix $A_{D(G')}$ of $G'$ with an arbitrary orientation $D(G')$ assigned to it, then we can use Proposition 6 to obtain Proposition 9.
+
+# VI. QUANTUM COMPUTATION AND THE TUTTE POLYNOMIAL
+
+In this section we show that quantum probability amplitudes may be expressed in terms of the evaluation of a Tutte polynomial. We achieve this by showing that output probability amplitudes of a class of universal quantum circuits are proportional to the principal probability amplitude of some IQP circuit.
+
+It will be convenient to define the following gate set.
+
+**Definition 13** ($\mathcal{G}_{\theta}$). For a real angle $\theta \in [-\pi, \pi]$, we define $\mathcal{G}(\theta)$ to be the gate set
+
+$$\mathcal{G}_{\theta} := \{H, e^{i\theta X}, e^{i\theta XX}\},$$
+
+where *H* denotes the Hadamard gate.
+
+It is easy to see that the gate set $\mathcal{G}_{\frac{\pi}{4}}$ generates the Clifford group and the gate set $\mathcal{G}_{\frac{\pi}{8}}$ is universal for quantum computation.
+
+In the IQP model it is easy to implement the gates $e^{i\theta X}$ and $e^{i\theta XX}$. So in order to implement the entire gate set $\mathcal{G}_{\theta}$, it remains to show that we can implement the Hadamard gate. This can be achieved by the use of postselection when $\theta = \frac{\pi}{4k}$ for $k \in \mathbb{Z}^{+}$ [6]. To apply a Hadamard gate to the target state $|\alpha\rangle_t$, consider the following Hadamard gadget. Firstly introduce an ancilla qubit in the state $|0\rangle_a$ and apply the gate $e^{i\frac{\pi}{4}(I-X)_t(I-X)_a}$ to $|\alpha\rangle_t |0\rangle_a$. Then measure qubit $t$ in the computational basis and postselect on an outcome of 0. The output state of this gadget is then $H|\alpha\rangle_a$.
+
+We shall consider quantum circuits that comprise gates from the set $\mathcal{G}_{\frac{\pi}{4k}}$ for an integer $k \in \mathbb{Z}^{+}$. Let $C_{k,n,m}$ denote such a circuit that acts on $n$ qubits and comprises $m$ Hadamard gates. Further let $\mathcal{X}_G(C_{k,n,m})$ denote the graph-induced X-program that implements the circuit $C_{k,n,m}$ by replacing each of the $m$ Hadamard gates with the Hadamard gadget. Then we have the following proposition.
+
+**Proposition 10.**
+
+$$\langle 0^n | C_{k,n,m} | 0^n \rangle = \sqrt{2}^m \psi_{\mathcal{X}_G(C_{k,n,m})} (0^{n+m}).$$
+
+*Proof.* The proof follows immediately from the application of the Hadamard gadgets. ■
+
+Any quantum amplitude may therefore be expressed as the evaluation of a Tutte polynomial by Proposition 8, Proposition 9, and Proposition 10.
+
+# VII. EFFICIENT CLASSICAL SIMULATION OF CLIFFORD CIRCUITS
+
+In this section we show how the correspondence between quantum computation and evaluations of the Tutte polynomial provides an explicit form for Clifford circuit amplitudes in terms of matroid invariants; namely, the *bicycle dimension* and *Brown's invariant*. This gives rise to an efficient classical algorithm for computing the output probability amplitudes of Clifford circuits. We note that it was first observed by Shepherd [2] that to compute the probability amplitude of a Clifford circuit, it is sufficient to evaluate the Tutte polynomial of a binary matroid at the point $(x, y)$ equals $(-i, i)$, which can be efficiently computed by Vertigan's algorithm [17]. We proceed with some definitions.
+
+Let $V$ be a linear subspace of $\mathbb{F}_2^n$. The bicycle dimension and Brown's invariant are defined as follows.
+
+**Definition 14** (Bicycle dimension). The bicycle dimension of $V$ is defined by
+
+$$d(V) := \dim(V \cap V^{\perp}).$$

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+Research Article
+
+A Real Representation Method for Solving
+Yakubovich- *j*-Conjugate Quaternion Matrix Equation
+
+Caiqin Song,<sup>1,2</sup> Jun-e Feng,<sup>1</sup> Xiaodong Wang,<sup>3</sup> and Jianli Zhao<sup>4</sup>
+
+¹ School of Mathematics, Shandong University, Jinan 250100, China
+
+² College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China
+
+³ School of Astronautics, Harbin Institute of Technology, Harbin 150001, China
+
+⁴ College of Mathematics Science, Liaocheng University, Liaocheng 252059, China
+
+Correspondence should be addressed to Caiqin Song; songcaiqin1983@163.com and Jun-e Feng; fengjune@sdu.edu.cn
+
+Received 19 October 2013; Revised 12 December 2013; Accepted 14 December 2013; Published 12 January 2014
+
+Academic Editor: Ngai-Ching Wong
+
+Copyright © 2014 Caiqin Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
+
+A new approach is presented for obtaining the solutions to Yakubovich-*j*-conjugate quaternion matrix equation $X - A\bar{X}B = CY$ based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix $A$. The closed form solution is established and the equivalent form of solution is given for this Yakubovich-*j*-conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation $X - A\bar{X}B = CY$ is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich-*j*-conjugate quaternion matrix equation $X - A\bar{X}B = CY$. Numerical example shows the effectiveness of the proposed results.
+
+# 1. Introduction
+
+The linear matrix equation $X - AXB = C$, which is called the Kalman-Yakubovich matrix equation in [1], is closely related to many problems in conventional linear control systems theory, such as pole assignment design [2], Luenberger-type observer design [3, 4], and robust fault detection [5, 6]. In recent years, many studies have been reported on the solutions to many algebraic equations including quaternion matrix equations and nonlinear matrix equations. Yuan and Liao [7] investigated the least squares solution of the quaternion *j*-conjugate matrix equation $X - A\bar{X}B = C$ (where $\bar{X}$ denotes the *j*-conjugate of quaternion matrix $X$) with the least norm using the complex representation of quaternion matrix, the Kronecker product of matrices, and the Moore-Penrose generalized inverse. The authors in [8] considered the matrix nearness problem associated with the quaternion matrix equation $AXA^H + BYB^H = C$ by means of the CCD-Q, GSVD-Q, and the projection theorem in the finite dimensional inner product space. In addition, Song et al.
+
+[9, 10] established the explicit solutions to the quaternion *j*-conjugate matrix equation $X - A\bar{X}B = C$, $XF - A\bar{X} = CY$, but here the known quaternion matrix $A$ is a block diagonal form. Wang et al. in [11, 12] investigated Hermitian tridiagonal solutions and the minimal-norm solution with the least norm of quaternionic least squares problem in quaternionic quantum theory. Besides, in [13, 14], some solutions for the Kalman-Yakubovich equation are presented in terms of the coefficients of characteristic polynomial of matrix $A$ or the Leverrier algorithm. The existence of solution to the matrix equation $X - A\bar{X}B = C$, which, for convenience, is called the Kalman-Yakubovich-conjugate matrix equation, is established, and the explicit solution is derived. Several necessary and sufficient conditions for the existence of a unique solution to the matrix equation $\sum_{i=0}^{k} A^i X B_i = E$ over quaternion field are obtained [15]. The authors in [16–18] have provided the consistency of the matrix equation $AX - \bar{X}B = C$ via the consimilarity of two matrices. In [19], Wu et al. construct some explicit expressions of the solution of the matrix equation $AX - \bar{X}B = C$ by means of a real

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+representation of a complex matrix. It is shown that there exists a unique solution if and only if $A\bar{A}$ and $B\bar{B}$ have no common eigenvalues.
+
+In this paper, we study quaternion $j$-conjugate matrix equation $X - A\tilde{X}B = CY$ by means of real representation of a quaternion matrix. Compared to the complex representation method [9, 10], the real representation method does not require any special case of the known matrix $A$. We propose the explicit solutions to the above Yakubovich-$j$-conjugate quaternion matrix equation. As the special case of quaternion $j$-conjugate matrix equation $X - A\tilde{X}B = CY$, complex conjugate matrix equation $X - A\bar{X}B = C$ and Kalman-Yakubovich quaternion matrix equation are also investigated. The explicit solutions to the complex conjugate matrix equation have been established.
+
+Throughout this paper, we use the following notations. Let $R$ denote the real number field, $C$ the complex number field, and $Q = R \oplus Ri \oplus Rj \oplus Rk$ the quaternion field, where $i^2 = j^2 = k^2 = -1$, $ij = -ji = k$. $R^{m \times n}$ ($C^{m \times n}$ or $Q^{m \times n}$) denotes the set of all $m \times n$ matrices on $R$ (or $Q$). For any matrix $A \in C^{m \times n}$, $A^T$, $\bar{A}$, $A^H$, det $A$, and $A^*$ represent the transpose, conjugate, conjugate transpose, determinant, and adjoint of $A$, respectively. In addition, symbol $A_\sigma$ is the real representation of quaternion matrix $A$. $A \otimes B = (a_{ij}B)$ denotes the Kronecker product of two matrices $A$ and $B$. If $A \in Q^{m \times n}$, let $A = A_1 + A_2 i + A_3 j + A_4 k$, where $A_t \in R^{m \times n}$, $t = 1, \dots, 4$, and define $\tilde{A} = A_1 - A_2 i + A_3 j - A_4 k$ to be the $j$-conjugate of $A$. For $A \in C^{m \times n}$, vec($A$) is defined as $\text{vec}(A) = [a_1^T \ a_2^T \ \dots \ a_n^T]^T$. Furthermore, letting $A \in Q^{n \times n}$, $B \in Q^{n \times r}$, and $C \in Q^{m \times n}$, we have the following notations associated with these matrices:
+
+$$Q_c(A, B, n) = [B \ AB \ \dots \ A^{n-1}B],$$
+
+$$Q_o(A,C,k) = \begin{bmatrix} C \\ CA \\ \vdots \\ CA^{k-1} \end{bmatrix},$$
+
+$$f_{A_\sigma}(s) = \det(sI - A_\sigma) = s^{2n} + \alpha_{2n-1}s^{2n-1} + \cdots + \alpha_1 s + \alpha_0,$$
+
+$$S_r(I, A_\sigma) = \begin{bmatrix} I_r & \alpha_2 I_r & \alpha_4 I_r & \cdots & \alpha_{2(n-1)} I_r \\ I_r & \alpha_2 I_r & \cdots & \alpha_{2(n-2)} I_r & \\ \cdots & \cdots & \cdots & & \cdots \\ I_r & \alpha_2 I_r & & & I_r \end{bmatrix}. \quad (1)$$
+
+Obviously, $Q_c(A, B, n)$ is the controllability matrix of the matrix pair $(A, B)$, $Q_o(A, C, k)$ is the observability matrix of the matrix pair $(A, C)$, and $S_r(I, A_\sigma)$ is a symmetric matrix.
+
+## 2. Quaternion-$j$-Conjugate Matrix Equation
+
+$$X - A\tilde{X}B = CY$$
+
+### 2.1. Real Matrix Equation $X - AXB = CY$
+In this subsection, we investigate the Yakubovich matrix equation over real field
+
+$$X - AXB = CY. \quad (2)$$
+
+**Theorem 1.** Suppose the real matrices $A \in R^{n \times n}$, $B \in R^{p \times p}$, $C \in R^{n \times r}$, {$s$ | det$(I - sA) = 0$} $\cap$ $\lambda(B) = \emptyset$; let
+
+$$f_{(I,A)}(s) = \det(I - sA) = \alpha_n s^n + \cdots + \alpha_1 s + \alpha_0, \quad \alpha_0 = 1,$$
+
+$$\operatorname{adj}(I - sA) = R_{n-1}s^{n-1} + \cdots + R_1s + R_0. \quad (3)$$
+
+Then, all the solutions to the Yakubovich matrix equation (2) can be established as
+
+$$X = \sum_{i=0}^{n-1} R_i CZB^i, \\
+Y = Z f_{(I,A)}(B),$$
+
+where the matrix $Z \in R^{r \times p}$ is an arbitrary matrix.
+
+*Proof.* We first show that the matrices X and Y given in (4) are solutions of the matrix equation (2). By the direct calculation we have
+
+$$
+\begin{align}
+X - AXB &= \sum_{i=0}^{n-1} R_i CZB^i - A \sum_{i=0}^{n-1} R_i CZB^i B \\
+&= \sum_{i=0}^{n-1} R_i CZB^i - \sum_{i=0}^{n-1} AR_i CZB^{i+1} \tag{5} \\
+&= R_0 CZ + \sum_{i=1}^{n-1} (R_i - AR_{i-1}) CZB^i \\
+&\qquad - AR_{n-1} CZB^n.
+\end{align}
+$$
+
+Due to the relation $(I - sA)\det(I - sA) = I\det(I - sA)$, it is easily derived that
+
+$$
+\begin{align}
+R_0 &= \alpha_0 I, \\
+R_i - AR_{i-1} &= \alpha_i I, && i = 1:n-1, \tag{6} \\
+-AR_{n-1} &= \alpha_n I.
+\end{align}
+$$
+
+So one has
+
+$$
+\begin{align}
+&R_0 CZ + \sum_{i=1}^{n-1} (R_i - AR_{i-1}) CZB^i - AR_{n-1} CZB^n &&(7) \\
+&= CZ \sum_{i=0}^{n} \alpha_i B^i = CZ f_{(I,A)}(B) = CY. &&(8)
+\end{align}
+$$
+
+Thus, the matrices X and Y given in (4) satisfy the matrix equation (2).
+
+Secondly, we show the completeness of solution (4). It follows from Theorem 6 of [20] that there are *rp* degrees of freedom in the solution of matrix equation (2), while solution (4) has exactly *rp* parameters represented by the elements of the free matrix Z. Therefore, in the following we only need to show that all the parameters in the matrix Z contribute to the solution. To do this, it suffices to show that the mapping Z $\rightarrow$ (X, Y) defined by (5) is injective. This is true since $f_{(I,A)}(B)$ is nonsingular under the condition of {$s$ | det$(I - sA) = 0$} $\cap$ $\lambda(B) = \emptyset$. The proof is thus completed. $\square$

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+In [21], we can find the following well-known generalized Faddeev-Leverrier algorithm:
+
+$$
+\begin{align}
+R_k &= R_{k-1}A + \alpha_k I_n, & R_0 &= I_n, & k = 1, 2, \dots, n, \nonumber \\
+\alpha_k &= \frac{\operatorname{trace}(R_{k-1}A)}{k}, & \alpha_0 &= 1, & k = 1, 2, \dots, n, \tag{8}
+\end{align}
+$$
+
+where $\alpha_i$, $i = 0, 1, 2, \dots, n - 1$, are the coefficients of the characteristic polynomial of the matrix $A$, and $R_i$, $i = 0, 1, \dots, n-1$, are the coefficient matrices of the adjoint matrix $\text{adj}(sI_n - A)$.
+
+**Theorem 2.** Given matrices $A \in R^{n \times n}$, $B \in R^{p \times p}$, $C \in R^{r \times p}$, let
+
+$$
+f_{(I,A)}(s) = \det(I - sA) = \alpha_n s^n + \cdots + \alpha_1 s + \alpha_0, \quad \alpha_0 = 1. \tag{9}
+$$
+
+Then the matrices X and Y given by (4) have the following equivalent form:
+
+$$
+X = \sum_{j=0}^{n-1} \sum_{k=0}^{j} \alpha_k A^{j-k} CZB^j,
+$$
+
+$$
+Y = Z f_{(I,A)}(B).
+$$
+
+Proof. According to (8), the following is easily obtained:
+
+$$
+\begin{align*}
+& R_0 = I, \\
+& R_1 = \alpha_1 I + A, \\
+& R_2 = \alpha_2 I + \alpha_1 A + A^2, \\
+& \vdots \\
+& R_{n-1} = \alpha_{n-1} I + \alpha_{n-2} A + \dots + A^{n-1}.
+\end{align*}
+$$
+
+This relation can be compactly expressed as
+
+$$
+R_j = \sum_{k=0}^{j} \alpha_k A^{j-k}, \quad \alpha_0 = 1, \quad j = 1, 2, \dots, n-1. \tag{12}
+$$
+
+Substituting this into the expression of X in (10) and record-
+ing the sum, we have
+
+$$
+\begin{align}
+X &= \sum_{j=0}^{n-1} R_j CZB^j = \sum_{j=0}^{n-1} \left( \sum_{k=0}^{j} \alpha_k A^{j-k} \right) CZB^j \tag{13} \\
+&= \sum_{j=0}^{n-1} \sum_{k=0}^{j} \alpha_k A^{j-k} CZB^j. \notag
+\end{align}
+$$
+
+Combining this with Theorem 1 gives the conclusion. $\square$
+
+2.2. Real Representation of a Quaternion Matrix. For any quaternion matrix $A = A_1 + A_2 i + A_3 j + A_4 k \in Q^{m \times n}$,
+
+$A_l \in R^{m \times n} (l = 1, 2, 3, 4)$, the real representation matrix of quaternion matrix $A$ can be defined as
+
+$$
+A_{\sigma} = \begin{bmatrix}
+A_1 & A_2 & -A_3 & A_4 \\
+A_2 & -A_1 & -A_4 & -A_3 \\
+A_3 & -A_4 & A_1 & A_2 \\
+A_4 & A_3 & A_2 & -A_1
+\end{bmatrix} \in R^{4m \times 4n}. \quad (14)
+$$
+
+For a *m×n* quaternion matrix *A*, we define *A*<sub>*σ*</sub><sup>*t*</sup> = (*A*<sub>*σ*</sub>)<sup>*t*</sup>.
+
+In addition, if we let
+
+$$
+P_t = \begin{bmatrix} I_t & 0 & 0 & 0 \\ 0 & -I_t & 0 & 0 \\ 0 & 0 & I_t & 0 \\ 0 & 0 & 0 & -I_t \end{bmatrix}, \quad Q_t = \begin{bmatrix} 0 & -I_t & 0 & 0 \\ I_t & 0 & 0 & 0 \\ 0 & 0 & 0 & I_t \\ 0 & 0 & -I_t & 0 \end{bmatrix}, \\
+S_t = \begin{bmatrix} 0 & 0 & 0 & -I_t \\ 0 & 0 & I_t & 0 \\ 0 & -I_t & 0 & 0 \\ I_t & 0 & 0 & 0 \end{bmatrix}, \quad R_t = \begin{bmatrix} 0 & 0 & I_t & 0 \\ 0 & 0 & 0 & I_t \\ -I_t & 0 & 0 & 0 \\ 0 & -I_t & 0 & 0 \end{bmatrix},
+$$
+
+in which $I_t$ is a $t \times t$ identity matrix, then $P_t$, $Q_t$, $S_t$, $R_t$ are unitary matrices.
+
+The real representation has the following properties,
+which are given in [13].
+
+**Proposition 3.** Let $A, B \in Q^{m \times n}, C \in Q^{n \times s}, a \in R$. Then
+
+$$
+(1) (A+B)_\sigma = A_\sigma+B_\sigma, (aA)_\sigma = aA_\sigma, (AC)_\sigma = A_\sigma P_n C_\sigma =
+A_\sigma(\hat{C})_\sigma P_s;
+$$
+
+(2) $A = B \Leftrightarrow A_\sigma = B_\sigma$;
+
+(3) $Q_m^{-1} A_\sigma Q_n = -A_\sigma$, $R_m^{-1} A_\sigma R_n = A_\sigma$, $S_m^{-1} A_\sigma S_n = -A_\sigma$,
+$P_m^{-1} A_\sigma P_n = (\hat{A})_\sigma$;
+
+(4) the quaternion matrix $A$ is nonsingular if and only if $A_\sigma$ is nonsingular, and the quaternion matrix $A$ is an unitary matrix if and only if $A_\sigma$ is an orthogonal matrix;
+
+(5) if $A \in Q^{m \times m}$, then $A_{\sigma}^{2k} = ((\widehat{A})_{\sigma})^{k} P_{m}$;
+
+(6) $A \in Q^{m \times m}$, $B \in Q^{n \times n}$, $C \in Q^{m \times n}$, and $k+l$ is even, then
+
+$$
+A_{\sigma}^{k} C_{\sigma} B_{\sigma}^{l}
+$$
+
+$$
+= \begin{cases}
+((\widehat{A}\widehat{A})^s (\widehat{A}\widehat{C}\widehat{B}) (\widehat{B}\widehat{B})^t)_{\sigma}, & k = 2s + 1, l = 2t + 1, \\
+((\widehat{A}\widehat{A})^s C(\widehat{B}\widehat{B})^t)_{\sigma}, & k = 2s, l = 2t.
+\end{cases}
+\tag{16}
+$$
+
+**Proposition 4.** If $\lambda$ is a characteristic value of $A_\sigma$, then so are $\pm\lambda$, $\pm\bar{\lambda}$.
+
+For any $A \in Q^{m \times m}$, let the characteristic polynomial of the real representation matrix $A_\sigma$ be $f_{(I,A_\sigma)}(\lambda) = \det(I_{4m} - \lambda A_\sigma) = \sum_{k=0}^{2m} a_{2k} \lambda^{2k}$, and define $h_{A_\sigma}(\lambda) = \lambda^{4m} f_{(I,A_\sigma)}(\lambda^{-1}) = \sum_{k=0}^{2m} a_{2k} \lambda^{2(2m-k)}$. So by Propositions 3 and 4 we have the following proposition.

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+**Proposition 5.** Let $A \in Q^{m \times m}, B \in Q^{n \times n}$. Then
+
+(1) $f_{(I,A_\sigma)}(\lambda)$ is a real polynomial, and $f_{(I,A_\sigma)}(\lambda) = \sum_{k=0}^{2m} a_{2k} \lambda^{2k}$;
+
+(2) $h_{A_\sigma}(\lambda)$ is a real polynomial, and $h_{A_\sigma}(\lambda) = \sum_{k=0}^{2m} a_{2k} \lambda^{2(2m-k)}$;
+
+(3) $h_{A_\sigma}(B_\sigma) = (g_{A_\sigma}(\tilde{B}))_\sigma P_n$, $f_{(I,A_\sigma)}(B_\sigma) = (p_{A_\sigma}(\tilde{B}))_\sigma P_n$, in which $g_{A_\sigma}(\lambda) = \sum_{k=0}^{2m} a_{2k} \lambda^{m-k}$, $p_{A_\sigma}(\lambda) = \sum_{k=0}^{2m} a_{2k} \lambda^k$ are real polynomials.
+
+*Proof.* By Proposition 4, we easily know that $a_k$ is a real number, and $a_{2k+1} = 0$. For any $k$, by Proposition 3, we have $B_\sigma^{2k} = ((\tilde{B}\tilde{B})^k)_\sigma P_n$, so we can obtain the result (3). $\square$
+
+**2.3. On Solutions to the Quaternion j-Conjugate Matrix Equation $X-A\tilde{X}B=CY$.** In this subsection, we discuss the solution of the following quaternion matrix equation:
+
+$$X - A\tilde{X}B = CY, \quad (17)$$
+
+by means of real representation, where $A \in Q^{n \times n}, B \in Q^{p \times p}$, and $C \in Q^{n \times r}$ are known matrices, $X \in Q^{n \times p}$ and $Y \in Q^{r \times p}$ are unknown matrices.
+
+We first define the real representation of quaternion matrix equation (17) by
+
+$$V - A_{\sigma}VB_{\sigma} = C_{\sigma}P_{r}W. \quad (18)$$
+
+According to (1) in Proposition 3, the quaternion matrix equation (17) is equivalent to the following equation:
+
+$$ (X - A\tilde{X}B)_{\sigma} = X_{\sigma} - A_{\sigma}X_{\sigma}B_{\sigma}. \quad (19) $$
+
+Therefore, the matrix equation (17) can be converted into
+
+$$X_{\sigma} - A_{\sigma}X_{\sigma}B_{\sigma} = C_{\sigma}P_{r}Y_{\sigma}. \quad (20)$$
+
+Thus, we have the following conclusion.
+
+**Proposition 6.** Given the quaternion matrices $A \in Q^{n \times n}, B \in Q^{p \times p}$ and $C \in Q^{n \times r}$, then the quaternion matrix equation (17) has a solution $(X,Y)$ if and only if the real representation matrix equation (18) has a solution $(V,W) = (X_\sigma, Y_\sigma)$.
+
+**Theorem 7.** Let $A \in Q^{n \times n}, B \in Q^{p \times p}$, and $C \in Q^{n \times r}$. Then quaternion matrix equation (17) has a solution $(X,Y)$ if and only if real representation matrix equation (18) has a solution $(V,W)$. Furthermore, if $(V,W)$ is a solution to (18), then
+
+the following quaternion matrices are solutions to quaternion matrix equation (17):
+
+$$ 
+\begin{aligned}
+X &= \frac{1}{16} [I_n \ iI_n \ jI_n \ kI_n] \\
+& \quad \times (V - Q_n^{-1}VQ_p + R_n^{-1}VR_p - S_n^{-1}VS_p) 
+\begin{bmatrix}
+I_p \\
+-iI_p \\
+-jI_p \\
+-kI_p
+\end{bmatrix}, \\
+Y &= \frac{1}{16} [I_r \ iI_r \ jI_r \ kI_r] \\
+& \quad \times (W - Q_n^{-1}WQ_p + R_n^{-1}WR_p - S_n^{-1}WS_p) 
+\begin{bmatrix}
+I_p \\
+-iI_p \\
+-jI_p \\
+-kI_p
+\end{bmatrix}.
+\end{aligned}
+\quad (21)
+$$
+
+*Proof.* By (3) of Proposition 3, the quaternion matrix equation (18) is equivalent to
+
+$$V - R_n^{-1} A_\sigma R_n V R_p^{-1} B_\sigma R_p = R_n^{-1} C_\sigma R_r P_r W. \quad (22)$$
+
+After multiplying the two sides of quaternion matrix equation (22) by $R_p^{-1}$, we can obtain
+
+$$VR_p^{-1} - R_n^{-1}A_\sigma R_n VR_p^{-1}B_\sigma = R_n^{-1}C_\sigma R_r P_r WR_p^{-1}. \quad (23)$$
+
+Before multiplying the two sides of quaternion matrix equation (23) by $R_n$, we have
+
+$$R_n V R_p^{-1} - A_\sigma R_n V R_p^{-1} B_\sigma = C_\sigma R_r P_r W R_p^{-1}. \quad (24)$$
+
+Noting that $R_p^{-1} = -R_p$, $R_r P_r = P_r R_r$, we give
+
+$$R_n^{-1}V R_p - A_\sigma R_n^{-1}V R_p B_\sigma = C_\sigma P_r R_r^{-1}W R_p. \quad (25)$$
+
+This shows that if $(V,W)$ is a real solution of matrix equation (18), then $(R_n^{-1}V R_p, R_r^{-1}W R_p)$ is also a real solution of quaternion matrix equation (18). In addition, according to (3) of Proposition 3, the quaternion matrix equation (18) is also equivalent to
+
+$$V - Q_n A_\sigma Q_n V Q_p B_\sigma Q_p = Q_n C_\sigma Q_r P_r W. \quad (26)$$
+
+After multiplying the two sides of quaternion matrix equation (26) by $Q_p^{-1}$, we have
+
+$$VQ_p^{-1} - Q_n A_\sigma Q_n VQ_p B_\sigma = Q_n C_\sigma Q_r P_r WQ_p^{-1}. \quad (27)$$
+
+Noting that $Q_p^{-1} = -Q_p$, $Q_r P_r = -P_r Q_r$, before multiplying the two sides of the quaternion matrix equation (27) by $Q_n^{-1}$, gives
+
+$$(-Q_n^{-1}VQ_p) - A_\sigma (-Q_n^{-1}VQ_p) B_\sigma = C_\sigma P_p (-Q_r^{-1}WQ_p). \quad (28)$$

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+**Corollary 10.** Given quaternion matrices $A \in Q^{n \times n}$, $B \in Q^{p \times p}$, and $C \in Q^{n \times p}$, let
+
+$$ 
+\begin{align} 
+f_{(I,A_\sigma)} (s) &= \det (I_{4n} - sA_\sigma) = \sum_{k=0}^{2n} a_{2k} s^{2k}, \tag{37} \\ 
+p_{A_\sigma} (s) &= \sum_{k=0}^{2n} a_{2k} s^k. 
+\end{align} 
+$$
+
+If X is a solution of equation (36), then
+
+$$ 
+\begin{align} 
+Xp_{A_{\sigma}}(\hat{BB}) &= \sum_{k=0}^{2n-1} \sum_{j=k}^{2n-1} \alpha_{2k} (A\hat{A})^{j-k} C(\hat{BB})^j \tag{38} \\ 
+&\quad + \sum_{k=0}^{2n-1} \sum_{j=k}^{2n-1} (A\hat{A})^{j-k} A\hat{C}B(\hat{BB})^j. 
+\end{align} 
+$$
+
+*Proof.* If X is a solution of equation (36), then $Y = X_\sigma$ is a solution of the equation $X_\sigma - A_\sigma X_\sigma B_\sigma = C_\sigma$. By Theorem 3 in [22] and Proposition 3, we have
+
+$$ X_{\sigma} f_{(I, A_{\sigma})} (B_{\sigma}) = \sum_{k=0}^{2n-1} \sum_{j=2k}^{4n-1} \alpha_{2k} A_{\sigma}^{j-2k} C_{\sigma} B_{\sigma}^{j}. \quad (39) $$
+
+By Proposition 5, $f_{(I,A_σ)}(s)$ is a real polynomial and
+$f_{(I,A_σ)}(B_σ) = (p_{A_σ}(B̂))_σ P_p$. So from Proposition 3 and (39),
+we have
+
+$$ 
+\begin{align}
+[Xp_{A_\sigma}(B\hat{B})]_\sigma & \\
+&= X_\sigma [p_{A_\sigma}(B\hat{B})]_\sigma P_p \nonumber \\
+&= X_\sigma f_{(I,A_\sigma)}(B_\sigma) = \sum_{k=0}^{2n-1} \sum_{j=2k}^{4n-1} \alpha_{2k} A_\sigma^{j-2k} C_\sigma B_\sigma^j \nonumber \\
+&= \sum_{k=0}^{2n-1} \alpha_{2k} \left[ \sum_{j=k}^{2n-1} A_\sigma^{2j-2k} C_\sigma B_\sigma^{2j} \right. \nonumber \\
+&\qquad \left. + \sum_{j=k}^{2n-1} A_\sigma^{2j+1-2k} C_\sigma B_\sigma^{2j+1} \right] \nonumber \\
+&= \sum_{k=0}^{2n-1} \alpha_{2k} \left[ \sum_{j=k}^{2n-1} \left((A\hat{A})^{j-k}\right)_\sigma P_n C_\sigma \left((\hat{B}\bar{B})^j\right)_\sigma P_p \right. \nonumber \\
+&\qquad \left. + \sum_{j=k}^{2n-1} \left((A\hat{A})^{j-k}\right)_\sigma P_n A_\sigma C_\sigma B_\sigma \left((\hat{B}\bar{B})^j\right)_\sigma P_p \right] \nonumber
+\end{align}
+$$
+
+$$ 
+\begin{align}
+&= \sum_{k=0}^{2n-1} \alpha_{2k} \nonumber \\
+&\quad \times \left[ \sum_{j=k}^{2n-1} \left( (A\hat{A})^{j-k} C(\hat{B}\bar{B})^j \right)_\sigma \right. \nonumber \\
+&\qquad \left. + \sum_{j=k}^{2n-1} \left( (A\hat{A})^{j-k} A\hat{C}B(\hat{B}\bar{B})^j \right)_\sigma \right] \nonumber \\
+&= \sum_{k=0}^{2n-1} \sum_{j=k}^{2n-1} \alpha_{2k} \left( (A\hat{A})^{j-k} C(\hat{B}\bar{B})^j \right)_\sigma \nonumber \\
+&\quad + \sum_{k=0}^{2n-1} \sum_{j=k}^{2n-1} \left( (A\hat{A})^{j-k} A\hat{C}B(\hat{B}\bar{B})^j \right)_\sigma . \tag{40}
+\end{align}
+$$
+
+Thus, the first conclusion has been proved. With this the second conclusion is obviously true.
+□
+
+In the following, we provide an equivalent statement of Theorem 7.
+
+**Corollary 11.** Given quaternion matrices $A \in Q^{n \times n}$, $B \in Q^{p \times p}$, and $C \in Q^{n \times p}$, let
+
+$$ 
+\begin{align}
+f_{(I, A_{\sigma})}(s) &= \det(I_{4n} - sA_{\sigma}) = \sum_{k=0}^{2n} a_{2k}s^{2k}, && (41) \\
+p_{A_{\sigma}}(s) &= \sum_{k=0}^{2n} a_{2k}s^k. && 
+\end{align}
+$$
+
+If X is a solution of (36), then
+
+$$ 
+\begin{align}
+Xp_{A_\sigma}(\hat{BB}) &= Q_c(A\hat{A}, C, 2n)S_p(I, A_\sigma)Q_o(\hat{BB}, I_p, 2n) \nonumber \\
+&\quad + Q_c(A\hat{A}, A, 2n)S_n(I, A_\sigma)Q_o(\hat{BB}, C\hat{B}, 2n). 
+\end{align}
+\tag{42}
+$$
+
+**3. Complex Conjugate Matrix Equation**
+
+$$ X - A\bar{X}B = CY $$
+
+In this section, we study the solution to the complex matrix equation
+
+$$ X - A\bar{X}B = CY, \quad (43) $$
+
+where $A \in C^{n \times n}$, $B \in C^{p \times p}$, and $C \in C^{n \times r}$. Next, we define
+real representation of complex matrix as follows.
+
+For any complex matrix $A = A_1 + A_2 i \in C^{m \times n}$, $A_l \in R^{m \times n}$ ($l = 1, 2$), we define a real representation of a complex matrix as
+
+$$ A_{\sigma} = \begin{bmatrix} A_1 & A_2 \\ A_2 & -A_1 \end{bmatrix} \in R^{2m \times 2n}. \quad (44) $$

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+Then the real matrix $A_\sigma$ is called real representation of complex matrix A.
+
+Let
+
+$$P_t = \begin{bmatrix} I_t & 0 \\ 0 & -I_t \end{bmatrix}, \quad Q_t = \begin{bmatrix} 0 & I_t \\ -I_t & 0 \end{bmatrix}, \quad (45)$$
+
+in which $I_t$ is $t \times t$ identity matrix. Then $P_t$, $Q_t$ are unitary matrices. The real presentation has the following properties, which are given by Jiang and Wei [14].
+
+**Proposition 12.** Consider the following.
+
+(1) If $A, B \in C^{m \times n}$, $a \in R$, then $(A + B)_\sigma = A_\sigma + B_\sigma$, $(aA)_\sigma = aA_\sigma$, $P_m A_\sigma P_n = (\bar{A})_\sigma$;
+
+(2) let $A \in C^{m \times n}$, $C \in C^{n \times s}$, $a \in R$, then $(AC)_\sigma = A_\sigma P_n C_\sigma$;
+
+(3) if $A \in C^{m \times m}$, then complex matrix A is nonsingular if and only if $A_\sigma$ is nonsingular;
+
+(4) if $A \in C^{m \times m}$, then $A_\sigma^{2k} = ((\overline{A}\bar{A})^k)_\sigma P_m$;
+
+(5) if $A \in C^{m \times n}$, then $Q_m A_\sigma Q_n = A_\sigma$.
+
+Actually, since complex matrix is a special case of quater-
+nion matrix, in this case, we also have the following similar
+results. Because the proofs are similar to Section 2 and are
+omitted.
+
+**Theorem 13.** Given complex matrices $A \in C^{n \times n}$, $B \in C^{p \times p}$, and $C \in C^{n \times r}$. Let
+
+$$f_{(I,A_{\sigma})}(s) = \det(I_{2n} - sA_{\sigma}) = \sum_{k=0}^{n} a_{2k}s^{2k}, 
+\\
+p_{A_{\sigma}}(s) = \sum_{k=0}^{n} a_{2k}s^{k}. 
+\tag{46}$$
+
+Then the solution to the matrix equation (43) is given by
+
+$$X = \sum_{k=0}^{n-1} \sum_{s=k}^{n-1} \alpha_{2k} (\overline{A}\overline{A})^{s-k} CZ(\overline{B}\overline{B})^s \\
++ \sum_{k=0}^{n-1} \sum_{s=k}^{n-1} \alpha_{2k} (\overline{A}\overline{A})^{s-k} \overline{A}\overline{C}\overline{Z}\overline{B}(\overline{B}\overline{B})^s, 
+\tag{47}$$
+
+$$Y = Z p_{A_{\sigma}}(\overline{B}\overline{B}).$$
+
+In the following, we provide an equivalent statement of
+Theorem 13.
+
+**Theorem 14.** Given complex matrices $A \in C^{n \times n}$, $B \in C^{p \times p}$, and $C \in C^{n \times p}$, let
+
+$$f_{(I,A_{\sigma})}(s) = \det(sI_{2n} - A_{\sigma}) = \sum_{k=0}^{n} a_{2k}s^{2k}, 
+\\
+p_{A_{\sigma}}(s) = \sum_{k=0}^{n} a_{2k}s^{k}. 
+\tag{48}$$
+
+Then the matrices X and Y given by (47) have the following equivalent form:
+
+$$X = Q_c (A\bar{A}, C, n) S_r (I, A_\sigma) Q_o (\bar{B}B, Z, n) + Q_c (A\bar{A}, A\bar{C}, n) S_r (I, A_\sigma) Q_o (\bar{B}B, Z\bar{B}, n), 
+\quad (49)$$
+
+$$Y = Z p_{A_{\sigma}}(\bar{B}\bar{B}).$$
+
+Finally, we consider the solution to the so-called Kalman-Yakubovich-conjugate matrix
+
+$$X - A\bar{X}B = C. 
+\tag{50}$$
+
+Based on the main result proposed above, we have the
+following conclusions regarding matrix equation (50).
+
+**Theorem 15.** Given the complex matrices $A \in C^{n \times n}$, $B \in C^{p \times p}$, and $C \in C^{n \times p}$, let
+
+$$f_{(I,A_{\sigma})}(s) = \det(sI_{2n} - A_{\sigma}) = \sum_{k=0}^{n} a_{2k}s^{2k}, 
+\\
+p_{A_{\sigma}}(s) = \sum_{k=0}^{n} a_{2k}s^{k}. 
+\tag{51}$$
+
+(1) If X is a solution of (50), then
+
+$$X p_{A_{\sigma}}(\overline{B}\overline{B}) = \sum_{k=0}^{n-1} \sum_{j=k}^{n-1} \alpha_{2k} (\overline{A}\overline{A})^{j-k} C(\overline{B}\overline{B})^j \\
++ \sum_{k=0}^{n-1} \sum_{j=k}^{n-1} \alpha_{2k} (\overline{A}\overline{A})^{j-k} A\overline{C}\overline{B}(\overline{B}\overline{B})^j. 
+\tag{52}$$
+
+(2) If X is the unique solution of (50), then
+
+$$X = \left[ \begin{array}{c}
+\sum_{k=0}^{n-1} \sum_{j=k}^{n-1} \alpha_{2k} (\overline{A}\overline{A})^{j-k} C(\overline{B}\overline{B})^j \\
++ \sum_{k=0}^{n-1} \sum_{j=k}^{n-1} \alpha_{2k} (\overline{A}\overline{A})^{j-k} A\overline{C}\overline{B}(\overline{B}\overline{B})^j
+\end{array} \right] \\
+\times [p_{A_\sigma}(\overline{B}\overline{B})]^{-1}. 
+\tag{53}$$
+
+**Theorem 16.** Given the complex matrices $A \in C^{n \times n}$, $B \in C^{p \times p}$, and $C \in C^{n \times p}$, let
+
+$$f_{(I,A_{\sigma})}(s) = \det(sI_{2n} - A_{\sigma}) = \sum_{k=0}^{n} a_{2k}s^{2k}, 
+\\
+p_{A_{\sigma}}(s) = \sum_{k=0}^{n} a_{2k}s^{k}. 
+\tag{54}$$
+
+(1) If X is a solution of (50), then
+
+$$X p_{A_{\sigma}}(\overline{B}\overline{B}) = Q_c (A\bar{A}, C, n) S_p (I, A_{\sigma}) Q_o (\overline{B}\overline{B}, I_p, n) + Q_c (A\bar{A}, A, n) S_n (I, A_{\sigma}) Q_o (\overline{B}\overline{B}, C\overline{B}, n). 
+\tag{55}$$

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+(2) If X is the unique solution of (50), then
+
+$$
+\begin{equation}
+\begin{aligned}
+X = {}& [Q_c (\overline{A}\bar{A}, C, n) S_p (I, A_\sigma) Q_o (\overline{B}B, I_p, n) \\
+      & + Q_c (\overline{A}\bar{A}, A, n) S_n (I, A_\sigma) Q_o (\overline{B}B, \overline{C}B, n)] \quad (56) \\
+      & \times [p_{A_\sigma}(\overline{B}B)]^{-1}.
+\end{aligned}
+\end{equation}
+$$
+
+4. Illustrative Example
+
+In this section, we give an example to obtain the solution of
+complex conjugate matrix equation $X - A\overline{X}B = CY$.
+
+*Example 1.* Consider Yakubovich-conjugate matrix equation in the form of (43) with the following parameters:
+
+$$
+\begin{align}
+A &= \begin{bmatrix} 1+i & 2i \\ 4 & 0 \end{bmatrix}, &
+B &= \begin{bmatrix} 3 & 4+i \\ 1 & -2i \end{bmatrix}, \tag{57} \\
+C &= \begin{bmatrix} 3 & 2i \\ 2-i & 4 \end{bmatrix}. \notag
+\end{align}
+$$
+
+According to the definition of real representation of a
+complex matrix, we have
+
+$$
+A_{\sigma} = \begin{bmatrix} 1 & 0 & 1 & 2 \\ 4 & 0 & 0 & 0 \\ 1 & 2 & -1 & 0 \\ 0 & 0 & -4 & 0 \end{bmatrix}. \qquad (58)
+$$
+
+By some simple computations, we have
+
+$$
+f_{(I,A_{\sigma})}(\lambda) = 64\lambda^4 - 2\lambda^2 + 1,
+$$
+
+$p_{A_{\sigma}}(\lambda) = 64\lambda^2 - 2\lambda + 1,$
+
+$$
+S_2(A_{\sigma}) = \begin{bmatrix} I_2 & 2I_2 \\ 0 & I_2 \end{bmatrix}, \quad I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},
+$$
+
+$$
+Q_c(A\bar{A}, C, 2) = \begin{bmatrix} 3 & 2i & 8+18i & -8-4i \\ 2-i & 4 & 4-28i & 8-24i \end{bmatrix},
+$$
+
+$$
+Q_o(\overline{BB}, Z, 2) = \begin{bmatrix} 1 & i \\ -1 & 1 \\ 11 + 2i & 9 + 3i \\ -10 + 3i & -2 + 6i \end{bmatrix},
+$$
+
+$$
+Q_c(A\bar{A}, A\bar{C}, 2) = \begin{bmatrix} 1+7i & 2+6i & -30-2i & -60+12i \\ 12 & -8i & 32-72i & -32+16i \end{bmatrix},
+$$
+
+$$
+Q_o(\overline{BB}, \overline{Z}B, 2) = \begin{bmatrix} 3-i & 2+i \\ -2 & -4-3i \\ 42-9i & 40-15i \\ -32-15i & -49-18i \end{bmatrix}. \quad (59)
+$$
+
+Choose
+
+$$
+Z = \begin{bmatrix} 1 & i \\ -1 & 1 \end{bmatrix}, \tag{60}
+$$
+
+then it follows from Theorem 14 that the solution of (43) is
+
+$$
+X = \begin{bmatrix} 659 + 840i & 1649 + 1118i \\ 1350 - 3683i & 1611 - 4132i \end{bmatrix},
+$$
+
+$$
+Y = \begin{bmatrix} 10603 + 2684i & 12078 - 133i \\ -9261 + 4026i & -6843 + 8052i \end{bmatrix}.
+$$
+
+5. Conclusions
+
+In the present paper, by means of the real representation of a quaternion matrix, we study the quaternion matrix equation $X - A\overline{X}B = CY$. Compared to our previous results [10], there are no requirements on the coefficient matrix $A$. Explicit solutions to this quaternion matrix equation are established by application of the real representation of a quaternion matrix. As a special case of quaternion $j$-conjugate matrix equation, complex conjugate matrix equation $X - A\overline{X}B = CY$ is also considered and the explicit solutions to complex conjugate are proposed. In addition, the equivalent forms of the explicit solutions are given.
+
+**Conflict of Interests**
+
+The authors declare that there is no conflict of interests regarding the publication of this paper.
+
+Acknowledgments
+
+The authors are very grateful to the anonymous reviewers and the editor for their helpful comments and suggestions which have helped us in improving the quality of this paper. This project is granted financial support from the NNSF (nos. 61374025, 61174141, 11171226, 11301247) of China, the Post-doctoral Science Foundation of China (no. 2013M541900), the Research Awards Young and Middle-Aged Scientists of Shandong Province (BS2011SF009, BS2011DX019), and the Excellent Youth Foundation of Shandong's Natural Scientific Committee (JQ201219).
+
+References
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+
+[2] B. H. Kwon and M. J. Youn, “Eigenvalue-generalized eigen-vector assignment by output feedback,” *IEEE Transactions on Automatic Control*, vol. 32, no. 5, pp. 417–421, 1987.
+
+[3] D. G. Luenberger, “An introduction to observers,” *IEEE Transactions on Automatic Control*, vol. 16, pp. 596–602, 1971.
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+[4] C.-C. Tsui, “New approach to robust observer design,” *International Journal of Control*, vol. 47, no. 3, pp. 745–751, 1988.
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+[5] J. Chen, R. J. Patton, and H.-Y. Zhang, “Design of unknown input observers and robust fault detection filters,” *International Journal of Control*, vol. 63, no. 1, pp. 85–105, 1996.
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+[6] J. Park and G. Rizzoni, “An eigenstructure assignment algorithm for the design of fault detection filters,” *IEEE Transactions on Automatic Control*, vol. 39, no. 7, pp. 1521–1524, 1994.

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+
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+
+[9] C. Song and G. Chen, "On solutions of matrix equation $XF - AX = C$ and $XF - \overline{A}X = C$ over quaternion field," *Journal of Applied Mathematics and Computing*, vol. 37, no. 1-2, pp. 57-68, 2011.
+
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samples/texts/4083386/page_2.md

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+## 1.2 Case 6b.2 Varying the frequency at a given distance
+
+Figure 3: RCS of a wire on the ground / varying the frequency
+
+- The calculation will be performed at 10m
+
+- Frequency band [100 MHz – 1 GHz] - frequency step : 10 MHz
+
+a) The wire is laid on the ground surface. The distance between the wire center at any point along it, to the ground is consequently equal to its radius i.e. 0.75 mm
+
+b) The wire is slightly buried. The distance between the wire center at any point along it, to the ground surface is 1.5 cm.
+
+## 2. Results to be provided
+
+d (in meters), $E_{\theta\theta}$, $E_{\phi\phi}$ (in V/m) for the case 6b.1
+frequency (in GHz), $E_{\theta\theta}$, $E_{\phi\phi}$ (in V/m) for the case 6b.2
+
+Each result will be provided as an ASCII file.
+They will be named
+BuriedTargets_Case_6b.i_CompanyName.txt