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摘要： A class of switched distributed parameter delay systems are considered in this paper. By means of multiple Lyapunov functions method， wirtinger's inequality， sufficient conditions of asymptotic stability and L2 gain are derived. These conditions are given in the form of linear matrix inequalities （LMIs） and arbitrary signal. The theoretical result is illustrated through matlab simulation.
關键词： distributed parameter system; switched system; time￣delay; arbitrary switching;L2 gain
中图分类号：TP13
文献标识码： A
The past decades has witnessed an enormous interest in switched systems due to theoretical research and practical application. A switched system can be described by a family of subsystems and a rule that orchestrates the switching between them [1]. For a discussion of various issues related to switched systems， the reader is referred to see the survey article[2]. Among them[3]，introduce multiple Lyapunov functions method as a tool for analyzing stability of switched systems. On the other hand， time￣delay phenomenon often appears in engineering control systems and it is frequently a source of instability. Switched systems with time￣delay have been extensively investigated by many authors[2].
Up to now， the overwhelming majority of switched systems results are available for systems governed by ordinary differential equations. Motivated by the fact that switched systems described by partial differential equations （i.e. distributed parameter switched systems） are more general， there is a real need to discuss such systems [4]. There are many related works in this fields [4-8].
The issue of distributed parameter switched delay systems is essentially more complicated. The LMIs technique has been shown an effective tool for the study of distributed parameter systems [9-13]. In this work， we extend the multiple Lyapunov method to distributed parameter switched delay systems. Our main contribution is to develop sufficient conditions of asymptotic stability and L2 gain analysis for a class of switched distributed paramete delay systems. These conditions are given in the form of LMIs and arbitrary switching.
5 Conclusion
Motivated by the fact that switched systems described by partial differential equations （i.e. distributed parameter switched systems） are more general， in this work， a class of distributed parameter switched delay systems have been studied. We have obtained asymptotic stable and L2 gain sufficient conditions which in the form of LMIs and arbitrary switching. The research considered L2 gain problem that is different from the previous works. Numerical example illustrated the effectiveness of the method. A potential extension by using average dwell time switching signal deserves further study. References
[1]LIBERZON D. Switching in systems and control， volume in series systems and control： foundations and applications[M]. Birkhauser， 2003.
[2]LIN H， ANTSAKLIS P J， Stability and stabilizability of switched linear systems： a survey of recent results[J]. IEEE Transactions on Automatic Control 2009， （54） ：308-322.
[3]BRANICKY M S， Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[J]. IEEE Transactions on Automatic Control 1998， （43）： 475-482.
[4]SASANE A， Stability of switching infinite￣dimensional systems[J]. Automatica， 2005， （41） ：75-78.
[5]AMIN S， HANTE F M， BAYEN A， Exponential stability of switched linear hyperbolic initial￣boundary value problems[J]. IEEE Transactions on Automatic Control， 2012， （57） ： 291-301.
[6]BAO L P， FEI S M， YU L， Exponential stability of linear distributed parameter switched systems with time￣delay[J]. Journal of Systems Science and Complexity， 2014， （27）： 263-275.
[7]YANG H， JIANG B. On stability of nonlinear and switched parabolic systems[J]. IET Control Theory and Applications， 2013， 7（5）：749-756.
[8]BAO L P， FEI S M， CHAI L. H￣infinity control of switched linear parabolic systems[J]. Electronic Journal of Differential Equations， 2014（131）： 1-10
[9]SELIVANOV A， FRIDMAN E. Distributed event￣triggered control of diffusion semilinear PDEs[J]. Automatica， 2016 ，（68）： 344-351.
[10]SU X， LIU X， SHI P et al. Sliding mode control of hybrid switched systems via an event￣triggered mechanism[J]. Automatica， 2018 （90）： 294-303.
[11]LUO Y P， DENG F Q， LMI￣based approach of robust control for uncertain distributed parameter control systems with time￣delay[J]. Control Theory Applications， 2006， （23）： 217-220.
[12]FRIDMAN E， ORLOV Y， An LMI approach to H￣infinity boundary control of semilinear parabolic and hyperbolic systems[J]. Automatica， 2009 （45） ：2060-2066.
[13]TAI Z X， LUN S X， Absolutely exponential stability of Lur’e distributed parameter control systems[J]. Applied Mathematics Letters， 2012， （25）： 232-236.
[14]WANG T， Stability in abstract functional differential equations[J]. Journal Mathematics Annual Application， 1994 （186） ： 835-861.
[15]BOYD S，GHAOUI L，FERON E，et al. Linear matrix inequalities in system and control theory[M]. SIAM， Philadelphia， PA， 1994.
關键词： distributed parameter system; switched system; time￣delay; arbitrary switching;L2 gain
中图分类号：TP13
文献标识码： A
The past decades has witnessed an enormous interest in switched systems due to theoretical research and practical application. A switched system can be described by a family of subsystems and a rule that orchestrates the switching between them [1]. For a discussion of various issues related to switched systems， the reader is referred to see the survey article[2]. Among them[3]，introduce multiple Lyapunov functions method as a tool for analyzing stability of switched systems. On the other hand， time￣delay phenomenon often appears in engineering control systems and it is frequently a source of instability. Switched systems with time￣delay have been extensively investigated by many authors[2].
Up to now， the overwhelming majority of switched systems results are available for systems governed by ordinary differential equations. Motivated by the fact that switched systems described by partial differential equations （i.e. distributed parameter switched systems） are more general， there is a real need to discuss such systems [4]. There are many related works in this fields [4-8].
The issue of distributed parameter switched delay systems is essentially more complicated. The LMIs technique has been shown an effective tool for the study of distributed parameter systems [9-13]. In this work， we extend the multiple Lyapunov method to distributed parameter switched delay systems. Our main contribution is to develop sufficient conditions of asymptotic stability and L2 gain analysis for a class of switched distributed paramete delay systems. These conditions are given in the form of LMIs and arbitrary switching.
5 Conclusion
Motivated by the fact that switched systems described by partial differential equations （i.e. distributed parameter switched systems） are more general， in this work， a class of distributed parameter switched delay systems have been studied. We have obtained asymptotic stable and L2 gain sufficient conditions which in the form of LMIs and arbitrary switching. The research considered L2 gain problem that is different from the previous works. Numerical example illustrated the effectiveness of the method. A potential extension by using average dwell time switching signal deserves further study. References
[1]LIBERZON D. Switching in systems and control， volume in series systems and control： foundations and applications[M]. Birkhauser， 2003.
[2]LIN H， ANTSAKLIS P J， Stability and stabilizability of switched linear systems： a survey of recent results[J]. IEEE Transactions on Automatic Control 2009， （54） ：308-322.
[3]BRANICKY M S， Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[J]. IEEE Transactions on Automatic Control 1998， （43）： 475-482.
[4]SASANE A， Stability of switching infinite￣dimensional systems[J]. Automatica， 2005， （41） ：75-78.
[5]AMIN S， HANTE F M， BAYEN A， Exponential stability of switched linear hyperbolic initial￣boundary value problems[J]. IEEE Transactions on Automatic Control， 2012， （57） ： 291-301.
[6]BAO L P， FEI S M， YU L， Exponential stability of linear distributed parameter switched systems with time￣delay[J]. Journal of Systems Science and Complexity， 2014， （27）： 263-275.
[7]YANG H， JIANG B. On stability of nonlinear and switched parabolic systems[J]. IET Control Theory and Applications， 2013， 7（5）：749-756.
[8]BAO L P， FEI S M， CHAI L. H￣infinity control of switched linear parabolic systems[J]. Electronic Journal of Differential Equations， 2014（131）： 1-10
[9]SELIVANOV A， FRIDMAN E. Distributed event￣triggered control of diffusion semilinear PDEs[J]. Automatica， 2016 ，（68）： 344-351.
[10]SU X， LIU X， SHI P et al. Sliding mode control of hybrid switched systems via an event￣triggered mechanism[J]. Automatica， 2018 （90）： 294-303.
[11]LUO Y P， DENG F Q， LMI￣based approach of robust control for uncertain distributed parameter control systems with time￣delay[J]. Control Theory Applications， 2006， （23）： 217-220.
[12]FRIDMAN E， ORLOV Y， An LMI approach to H￣infinity boundary control of semilinear parabolic and hyperbolic systems[J]. Automatica， 2009 （45） ：2060-2066.
[13]TAI Z X， LUN S X， Absolutely exponential stability of Lur’e distributed parameter control systems[J]. Applied Mathematics Letters， 2012， （25）： 232-236.
[14]WANG T， Stability in abstract functional differential equations[J]. Journal Mathematics Annual Application， 1994 （186） ： 835-861.
[15]BOYD S，GHAOUI L，FERON E，et al. Linear matrix inequalities in system and control theory[M]. SIAM， Philadelphia， PA， 1994.