Brief paperAsynchronous control for 2-D switched systems with mode-dependent average dwell time☆
Introduction
Over the past few decades, two-dimensional (2-D) systems have been attracting considerable attention due to their extensive practical applications such as multi-dimensional digital filtering, linear image processing, electricity transmission, gas absorption and water stream heating (Dymkov and Dymkou, 2012, Kaczorek, 1985). So far, quantities of results on the issues of analysis and synthesis for 2-D systems have been reported (Du et al., 2001, Hu and Liu, 2005, Li and Gao, 2013, Li et al., 2012, Shaker and Shaker, 2014, Shaker and Tahavori, 2010, Wu et al., 2012, Yang et al., 2006). To mention a few, the stability analysis problems for 2-D linear and nonlinear systems are studied, respectively, in Hu and Liu (2005) and Shaker and Tahavori (2010). In Duan et al., 2014a, Duan et al., 2014b, state feedback controllers are designed for a class of 2-D systems to guarantee exponential stability. In Du et al. (2001), the authors address the robust stabilization for 2-D systems. In addition, mixed control problem for 2-D systems is solved in Yang et al. (2006).
On the other hand, it is well known that many systems are subject to abrupt changes, such as flight control systems, power electronics and chaos generators (Mitsubori and Saito, 1997, Tse and Bernardo, 2002). A very common way to describe these characteristics is to assume the system is operating in different modes with a switching signal regulating the behavior. As an important class of hybrid systems, switched systems consist of a finite number of subsystems and a switching signal which governs the switching among them. Owing to the significance both in theoretical development and practical applications, switched systems have been widely investigated (Andersen et al., 2014, Guan et al., 2016, Hespanha, 2004, Jiang et al., 2015, Li et al., 2014, Xiang and Xiao, 2011, Zhang et al., 2015, Zhao et al., 2008, Zhou, 2013). In many physical processes, since the complexity of the system, it is necessary to establish model by a 2-D switching representation. For example, heat flux switching and modulating in 2-D thermal transistors can be modeled by 2-D switched systems (Lo, Wang, & Li, 2008). Recently, several results about 2-D switched systems have been obtained. In Kaczorek (2012), conditions for the asymptotic stability of the positive 2-D switched systems are established. Stability and stabilization for 2-D switched systems is investigated in Duan and Xiang (2013) and Xiang and Huang (2013). It is worth mentioning that all above results are based on the Roesser model. In general, 2-D systems can be represented by different models such as the Roesser model, Attasi model and Fornasini–Marchesini (FM) model. Since the Roesser model and Attasi model can be viewed as a particular case of the second FM model, it would be more general to consider a 2-D switched system in FM type.
Similarly to 1-D switched system, one popular assumption is that the controller’s switching is synchronized with the subsystem’s switching in 2-D switched systems. However, it rarely holds since there is always a delay in the controller when the switching happens. For instance, when the system and the controller connect via a communication channel, it requires time to identify the active system and apply the corresponding controller (Wang, Zhao, & Jiang, 2013). A number of reports about 1-D asynchronously switched systems are available (Wang et al., 2012, Wang et al., 2013). However, to the best of the authors’ knowledge, the asynchronism between controllers and subsystems was never considered before due to the complexity of 2-D Fornasini–Marchesini local state-space (FMLSS) systems. Since the disturbance is inevitable in actual control systems, it would be more practical to take performance into consideration (Wang & Liu, 2003). Motivated by discussions above, our objective in this paper is to address control for 2-D asynchronously switched systems described by FMLSS model. To be specific, we concentrate on stabilization problem under asynchronous switching. When the controller is mismatched with the corresponding subsystem, we allow the Lyapunov function to increase in some extent as asynchronous switching happens. But the whole state responses will still be bounded by the curve with certain exponential stability parameters. Then we devote to design mode-dependent controllers and admissible switching signals such that the corresponding system is stable and has weighted performance with asymptotic stability under asynchronous switching.
The rest of this paper is organized as follows. Section 2 reviews system description, preliminaries and necessary definitions. In Section 3, the stabilization problem for 2-D asynchronously switched systems is taken into consideration. In Section 4, sufficient conditions which guarantee stability of 2-D asynchronously switched systems with performance are presented. Two examples are provided to illustrate the effectiveness of the proposed methods in Section 5. Finally, we conclude the paper in Section 6.
Notations: The notation used in this paper is fairly standard. The superscript ‘’ stands for matrix transposition. denotes the -dimensional Euclidean space, and represents the set of nonnegative integers. In addition, in symmetric block matrices or long matrix expressions, we use a ‘’ as an ellipsis for the terms that are introduced by symmetry and stands for a block-diagonal matrix. refers to the Euclidean vector norm. and 0 represent identify matrix and zero matrix with appropriate dimensions. The notation means that is real symmetric and positive definite. and denote the minimum and maximum eigenvalues of matrix , respectively. The -norm of a 2-D signal belongs to if .
Section snippets
Problem formulation and preliminaries
Consider a class of 2-D switched discrete-time systems described by the FMLSS model: where is the state vector of the system, is the control input, is the controlled output, and is the exogenous disturbance input which belongs to . , is the switching
Stabilization
In this section, our objective is to design a mode-dependent stabilization controller for 2-D switched system (1) such that the corresponding closed-loop system is stable. Theorem 1 For any , , let , and be given constants, if there exist matrices , , , and , such that
control
Based on the results in Theorem 1, we can solve the control problem. Theorem 2 For any , , let , and be given constants, if there exist matrices , , , , , and a scalar such that (19), (20) hold with , ,
Illustrative examples
In this section, our objective is to design a mode-dependent stabilization controller for 2-D switched system (1) such that the corresponding closed-loop system is stable. Example 1 Firstly, we present a practical example borrowed from Cichy, Galkowski, and Rogers (2012) and Wu et al. (2015). It is known that some dynamical processes such as gas absorption, water stream heating and air drying can be modeled by the Darboux equation:
Conclusions
In this paper, we have investigated the control problem for a class of 2-D switched systems represented by FMLSS model under asynchronous switching. Based on the MDADT technique, improved results of stabilization index have been established at first, which are less strict than the ADT switching. This paper also takes asynchronous switching into consideration, which is widespread in practice. Then, a sufficient condition for the asymptotic stability has been presented with weighted
Zhongyang Fei received the B.E. degree and the M.S. degree in Control Science and Engineering from Harbin Institute of Technology, PR China in 2007 and 2009 respectively; and the Ph.D. degree in Mechanical Engineering and Material Science Department, Washington University in St. Louis, USA in 2013. His research interests involve 2-D systems, complex networks, robust control, time-delay systems and switched systems.
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Zhongyang Fei received the B.E. degree and the M.S. degree in Control Science and Engineering from Harbin Institute of Technology, PR China in 2007 and 2009 respectively; and the Ph.D. degree in Mechanical Engineering and Material Science Department, Washington University in St. Louis, USA in 2013. His research interests involve 2-D systems, complex networks, robust control, time-delay systems and switched systems.
Shuang Shi received the B.E. degree in Automation from Harbin Institute of Technology, Harbin, China, in 2015. He is now studying for the Ph.D. degree in Control Theory and Control Engineering from Harbin Institute of Technology. His current research interests include switched systems and robust control.
Chang Zhao received the bachelor degree in Control Science and Engineering from Harbin Institute of Technology, PR China in 2015. Currently, she is pursuing her master degree in HIT. Her research interests involve two dimensional systems, control.
Ligang Wu received the B.S. degree in Automation from Harbin University of Science and Technology, China in 2001; the M.E. degree in Navigation Guidance and Control from Harbin Institute of Technology, China in 2003; the Ph.D. degree in Control Theory and Control Engineering from Harbin Institute of Technology, China in 2006. From January 2006 to April 2007, he was a Research Associate in the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong. From September 2007 to June 2008, he was a Senior Research Associate in the Department of Mathematics, City University of Hong Kong, Hong Kong. From December 2012 to December 2013, he was a Research Associate in the Department of Electrical and Electronic Engineering, Imperial College London, London, UK. In 2008, he joined the Harbin Institute of Technology, China, as an Associate Professor, and was then promoted to a Professor in 2012. Dr. Wu was the winner of the National Science Fund for Distinguished Young Scholars in 2015. He received China Young Five Four Medal in 2016, and was named to the 2015 & 2016 Thomson Reuters Highly Cited Researchers lists.
Dr. Wu currently serves as an Associate Editor for a number of journals, including IEEE Transactions on Automatic Control, IEEE/ASME Transactions on Mechatronics, Information Sciences, Signal Processing, and IET Control Theory and Applications. He is also an Associate Editor for the Conference Editorial Board, IEEE Control Systems Society. He is also an Associate Editor for the Conference Editorial Board, IEEE Control Systems Society. Dr. Wu has published 5 research monographs and more than 120 research papers in international referred journals. His current research interests include switched systems, stochastic systems, computational and intelligent systems, multidimensional systems, sliding mode control, and flight control.
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This research is partially supported by the National Natural Science Foundation of China (61503094, 61525303), the Heilongjiang Outstanding Youth Science Fund (JC201406), the Fok Ying Tung Education Foundation (141059), the Fundamental Research Funds for the Central Universities (No. HIT.BRETIII.201515) and China Postdoctoral Science Foundation (2015M570292, 2016T90290 and LBH-Z14091). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Constantino M. Lagoa under the direction of Editor Richard Middleton.
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