Stability Analysis of Non-autonomous Switched Systems Based on Time-varying Scalar Functions
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- 英文论文题名：Stability Analysis of Non-autonomous Switched Systems Based on Time-varying Scalar Functions
- 论文作者：Mengmeng Qiu ; Junjie Lu ; Weijie Feng ; Zhikun She
- 英文论文作者：Mengmeng Qiu ; Junjie Lu ; Weijie Feng ; Zhikun She ; School of Mathematics and Systems Science ; Beihang University
- 年：2017
- 作者机构：School of Mathematics and Systems Science,Beihang University;
- 英文论文关键词：Switched systems ; Time-varying scalar functions ; Asymptotic stability ; Exponential stability ; Uniformly exponential stability
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（B）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（B）
- 语种：英文
- 分类号：O231
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：262k
- 原文格式：D
- 会议级别：全国
摘要
In this paper, we investigate asymptotic stability, exponential stability and uniformly exponential stability of non-autonomous switched systems, respectively. Starting with non-autonomous switched linear systems,we firstly provide necessary and sufficient conditions for its asymptotic stability and exponential stability based on the existence of an asymptotically stable function and an exponentially stable function. Successively, a sufficient condition for uniformly exponential stability of a non-autonomous switched linear system is proposed by the existence of an uniformly exponentially stable function. Based on this sufficient condition, we further derive a sufficient condition for local uniformly exponential stability of non-autonomous switched nonlinear systems. In the end, an illustrative example is given to show the applicability of our theoretical results.
In this paper, we investigate asymptotic stability, exponential stability and uniformly exponential stability of non-autonomous switched systems, respectively. Starting with non-autonomous switched linear systems,we firstly provide necessary and sufficient conditions for its asymptotic stability and exponential stability based on the existence of an asymptotically stable function and an exponentially stable function. Successively, a sufficient condition for uniformly exponential stability of a non-autonomous switched linear system is proposed by the existence of an uniformly exponentially stable function. Based on this sufficient condition, we further derive a sufficient condition for local uniformly exponential stability of non-autonomous switched nonlinear systems. In the end, an illustrative example is given to show the applicability of our theoretical results.
In this paper, we investigate asymptotic stability, exponential stability and uniformly exponential stability of non-autonomous switched systems, respectively. Starting with non-autonomous switched linear systems,we firstly provide necessary and sufficient conditions for its asymptotic stability and exponential stability based on the existence of an asymptotically stable function and an exponentially stable function. Successively, a sufficient condition for uniformly exponential stability of a non-autonomous switched linear system is proposed by the existence of an uniformly exponentially stable function. Based on this sufficient condition, we further derive a sufficient condition for local uniformly exponential stability of non-autonomous switched nonlinear systems. In the end, an illustrative example is given to show the applicability of our theoretical results.
引文
[1]B.Zhou,On asymptotic stability of linear time-varying systems,Automatica 68(2016)266-276.
[2]D.Liberzon,A.S.Morse,Basic problems in stability and design of switched systems,IEEE Control Systems19(5)(1999)59-70.
[3]D.Liberzon,Switching in systems and control,Springer Science&Business Media,2003.
[4]H.Ye,A.N.Michel,L.Hou,Stability theory for hybrid dynamical systems,IEEE Transactions on Automatic Control 43(4)(1998)461-474.
[5]H.Khalil,Nonlinear systems(3rd Ed.),Prentice Hall,2001.
[6]H.Lin,P.J.Antsaklis,Stability and stabilizability of switched linear systems:a survey of recent results,IEEE Transactions on Automatic Control 54(2)(2009)308-322.
[7]J.Liu,X.Liu,W.C.Xie,Uniform stability of switched nonlinear systems,Nonlinear Analysis:Hybrid Systems 3(4)(2009)441-454.
[8]J.Suo,J.Sun,Convergence domain for time-varying switched systems,Systems&Control Letters 78(2015)19-24.
[9]J.Lu,Z.She,Sufficient and necessary conditions for discrete-time nonlinear switched systems with uniform local exponential stability,International Journal of Systems Science 47(15)(2016)3561-3572.
[10]J.L.Mancilla-Aguilar,H.Haimovich,R.A.Garcia,Global stability results for switched systems based on weak Lyapunov functions,IEEE Transactions on Automatic Control(2017).
[11]M.S.Branicky,Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,IEEETransactions on Automatic Control 43(4)(1998)475-482.
[12]M.Hajiahmadi,B.De Schutter,H.Hellendoorn,Design of stabilizing switching laws for mixed switched affine systems,IEEE Transactions on Automatic Control 61(6)(2016)1676-1681.
[13]P.Niamsup,Controllability approach to H∞control problem of linear time-varying switched systems,Nonlinear Analysis:Hybrid Systems 2(3)(2008)875-886.
[14]Q.Liang,Z.She,L.Wang,H.Su,General Lyapunov functions for consensus of nonlinear multi-agent systems,IEEE Transactions on Circuits and Systems II:Express Briefs(2017).(Doi:10.1109/TCSII.2017.2647744)
[15]R.Shorten,F.Wirth,O.Mason,K.Wulff,C.King,Stability criteria for switched and hybrid systems,SIAMreview 49(4)(2007)545-592.
[16]S.P.Boyd,L.El Ghaoui,E.Feron,V.Balakrishnan,Linear matrix inequalities in system and control theory,Vol.15,SIAM,1994.
[17]S.Ratschan,Z.She,Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions,SIAM Journal on Control and Optimization 48(7)(2010)4377-4394.
[18]T.C.Lee,Z.P.Jiang,On uniform asymptotic stability of switched nonlinear time-varying systems,in:Proceedings of the 2007 American Control Conference,2007,pp.669-674.
[19]W.J.Rugh,Linear system theory(2nd ed.),Prentice Hall,1996.
[20]X.Xu,P.J.Antsaklis,Stabilization of second-order LTI switched systems,International Journal of Control,73(14)(2000)1261-1279.
[21]X.Zhao,L.Zhang,P.Shi,M.Liu,Stability and stabilization of switched linear systems with modedependent average dwell time,IEEE Transactions on Automatic Control 57(7)(2012)1809-1815.
[22]Y.Orlov,Extended invariance principle for nonautonomous switched systems,IEEE Transactions on Automatic Control 48(8)(2003)1448-1452.
[23]Y.E.Wang,X.M.Sun,F.Mazenc,Stability of switched nonlinear systems with delay and disturbance,Automatica 69(2016)78-86.
[24]Z.Sun,Stabilizing switching design for switched linear systems:A state-feedback path-wise switching approach,Automatica 45(7)(2009)1708-1714.
[25]Z.She and B.Xue.Computing an invariance kernel with target by computing Lyapunov-like functions.IETControl Theory and Applications,7(15):1932-1940,2013.
[26]Z.She,B.Xue.Discovering multiple Lyapunov functions for switched hybrid systems,SIAM Journal on Control and Optimization,52(5):3312-3340,2014.
1 tr(A)denotes the trace of matrix A.
[2]D.Liberzon,A.S.Morse,Basic problems in stability and design of switched systems,IEEE Control Systems19(5)(1999)59-70.
[3]D.Liberzon,Switching in systems and control,Springer Science&Business Media,2003.
[4]H.Ye,A.N.Michel,L.Hou,Stability theory for hybrid dynamical systems,IEEE Transactions on Automatic Control 43(4)(1998)461-474.
[5]H.Khalil,Nonlinear systems(3rd Ed.),Prentice Hall,2001.
[6]H.Lin,P.J.Antsaklis,Stability and stabilizability of switched linear systems:a survey of recent results,IEEE Transactions on Automatic Control 54(2)(2009)308-322.
[7]J.Liu,X.Liu,W.C.Xie,Uniform stability of switched nonlinear systems,Nonlinear Analysis:Hybrid Systems 3(4)(2009)441-454.
[8]J.Suo,J.Sun,Convergence domain for time-varying switched systems,Systems&Control Letters 78(2015)19-24.
[9]J.Lu,Z.She,Sufficient and necessary conditions for discrete-time nonlinear switched systems with uniform local exponential stability,International Journal of Systems Science 47(15)(2016)3561-3572.
[10]J.L.Mancilla-Aguilar,H.Haimovich,R.A.Garcia,Global stability results for switched systems based on weak Lyapunov functions,IEEE Transactions on Automatic Control(2017).
[11]M.S.Branicky,Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,IEEETransactions on Automatic Control 43(4)(1998)475-482.
[12]M.Hajiahmadi,B.De Schutter,H.Hellendoorn,Design of stabilizing switching laws for mixed switched affine systems,IEEE Transactions on Automatic Control 61(6)(2016)1676-1681.
[13]P.Niamsup,Controllability approach to H∞control problem of linear time-varying switched systems,Nonlinear Analysis:Hybrid Systems 2(3)(2008)875-886.
[14]Q.Liang,Z.She,L.Wang,H.Su,General Lyapunov functions for consensus of nonlinear multi-agent systems,IEEE Transactions on Circuits and Systems II:Express Briefs(2017).(Doi:10.1109/TCSII.2017.2647744)
[15]R.Shorten,F.Wirth,O.Mason,K.Wulff,C.King,Stability criteria for switched and hybrid systems,SIAMreview 49(4)(2007)545-592.
[16]S.P.Boyd,L.El Ghaoui,E.Feron,V.Balakrishnan,Linear matrix inequalities in system and control theory,Vol.15,SIAM,1994.
[17]S.Ratschan,Z.She,Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions,SIAM Journal on Control and Optimization 48(7)(2010)4377-4394.
[18]T.C.Lee,Z.P.Jiang,On uniform asymptotic stability of switched nonlinear time-varying systems,in:Proceedings of the 2007 American Control Conference,2007,pp.669-674.
[19]W.J.Rugh,Linear system theory(2nd ed.),Prentice Hall,1996.
[20]X.Xu,P.J.Antsaklis,Stabilization of second-order LTI switched systems,International Journal of Control,73(14)(2000)1261-1279.
[21]X.Zhao,L.Zhang,P.Shi,M.Liu,Stability and stabilization of switched linear systems with modedependent average dwell time,IEEE Transactions on Automatic Control 57(7)(2012)1809-1815.
[22]Y.Orlov,Extended invariance principle for nonautonomous switched systems,IEEE Transactions on Automatic Control 48(8)(2003)1448-1452.
[23]Y.E.Wang,X.M.Sun,F.Mazenc,Stability of switched nonlinear systems with delay and disturbance,Automatica 69(2016)78-86.
[24]Z.Sun,Stabilizing switching design for switched linear systems:A state-feedback path-wise switching approach,Automatica 45(7)(2009)1708-1714.
[25]Z.She and B.Xue.Computing an invariance kernel with target by computing Lyapunov-like functions.IETControl Theory and Applications,7(15):1932-1940,2013.
[26]Z.She,B.Xue.Discovering multiple Lyapunov functions for switched hybrid systems,SIAM Journal on Control and Optimization,52(5):3312-3340,2014.
1 tr(A)denotes the trace of matrix A.