一类混杂系统—切换系统的定性，稳定性分析
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摘要
切换系统是一类离散和连续的动态系统相组合的混杂系统。它可以看作是由几个微分方程及作用在其中的切换规则构成。近十年来,许多人对这类系统关注并深入研究,取得了很多结果。这些结果大都集中在稳定性方面。本文系统地研究了时滞切换系统的稳定性,主要探讨了离散时滞和分布时滞的连续和离散切换系统的稳定性问题。将已有的成果给予了完善。
本文主要分为以下四个方面的内容:
1.研究了一类带离散和分布时滞的线性切换系统的一致渐近稳定性,通过构造Lyapunov泛函,利用Riccati方程方法,给出在时滞充分小的条件下系统稳定性的充分条件。在结论之后,给出例子说明了系统的可行性。
2.进一步研究了带离散和分布时滞的线性切换系统的指数稳定性和L_2增益性能。应用线性矩阵方法给出了带离散和分布时滞的线性切换系统的指数稳定的与时滞有关的充分条件,并讨论了系统的L_2-增益性能。
3.研究了一类带时滞的离散切换控制系统平凡解的渐近稳定性的新标准,主要运用第二Laypunov方法,条件是以线性不等式的形式给出,结果是与时滞无关的。
4.进一步研究了一类带时滞的不确定离散切换控制系统平凡解的渐近稳定性的新标准,通过构造Laypunov泛函,运用线性不等式方法得到系统渐近稳定的结果,并通过几个例子验证结果的可行性。
Hybrid system is a combination of discrete and continuous dynamical systems. These systems arise as models for phenomena, which cannot be described by exclusively continuous, or exclusively discrete processes. There are many perfect results on the stability of switching systems.
This paper deals with the problem of stabilization of switched systems with time delays, and focuses on the stabilization of switched systems with discrete time delay and distributed time delay. It makes the existing results more perfect.
The paper is organized as follows:
1. This paper deals with the problem of a type of linear switching systems with discrete and continuous time delay. By constructing Lyapunov functional under a condition on the time delay, we show it stabilizes the system for sufficiently small delays.
2. In this paper, we study stability and L_2 -gain for a class of switched systems with discrete time delays and distributed time delays. Sufficient conditions for exponential stability and weighted L_2 -gain are developed. These conditions are delay-dependent and are shown in the form of linear matrix inequalities. The state decay estimate is given. Several examples illustrate the effectiveness of the proposed method.
3. In this paper, we obtain some criteria for determining the asymptotic stability of the zero solution for some classes of delay-difference system-discrete time switched systems, and delay-independent robust stability and stabilization of it. We mainly used a discrete version of the second Lyapunov method.
4. This article gives a novel criterion for the asymptotic stabilization of the zero solutions of a class of switched systems with delays in control input. By constructing Lyapunov functional, we obtained the criterion which is expressed in terms of matrix inequalities. The results are more advanced than the references. Numerical examples are presented to illustrate the effectiveness of the theoretical results.
本文主要分为以下四个方面的内容:
1.研究了一类带离散和分布时滞的线性切换系统的一致渐近稳定性,通过构造Lyapunov泛函,利用Riccati方程方法,给出在时滞充分小的条件下系统稳定性的充分条件。在结论之后,给出例子说明了系统的可行性。
2.进一步研究了带离散和分布时滞的线性切换系统的指数稳定性和L_2增益性能。应用线性矩阵方法给出了带离散和分布时滞的线性切换系统的指数稳定的与时滞有关的充分条件,并讨论了系统的L_2-增益性能。
3.研究了一类带时滞的离散切换控制系统平凡解的渐近稳定性的新标准,主要运用第二Laypunov方法,条件是以线性不等式的形式给出,结果是与时滞无关的。
4.进一步研究了一类带时滞的不确定离散切换控制系统平凡解的渐近稳定性的新标准,通过构造Laypunov泛函,运用线性不等式方法得到系统渐近稳定的结果,并通过几个例子验证结果的可行性。
Hybrid system is a combination of discrete and continuous dynamical systems. These systems arise as models for phenomena, which cannot be described by exclusively continuous, or exclusively discrete processes. There are many perfect results on the stability of switching systems.
This paper deals with the problem of stabilization of switched systems with time delays, and focuses on the stabilization of switched systems with discrete time delay and distributed time delay. It makes the existing results more perfect.
The paper is organized as follows:
1. This paper deals with the problem of a type of linear switching systems with discrete and continuous time delay. By constructing Lyapunov functional under a condition on the time delay, we show it stabilizes the system for sufficiently small delays.
2. In this paper, we study stability and L_2 -gain for a class of switched systems with discrete time delays and distributed time delays. Sufficient conditions for exponential stability and weighted L_2 -gain are developed. These conditions are delay-dependent and are shown in the form of linear matrix inequalities. The state decay estimate is given. Several examples illustrate the effectiveness of the proposed method.
3. In this paper, we obtain some criteria for determining the asymptotic stability of the zero solution for some classes of delay-difference system-discrete time switched systems, and delay-independent robust stability and stabilization of it. We mainly used a discrete version of the second Lyapunov method.
4. This article gives a novel criterion for the asymptotic stabilization of the zero solutions of a class of switched systems with delays in control input. By constructing Lyapunov functional, we obtained the criterion which is expressed in terms of matrix inequalities. The results are more advanced than the references. Numerical examples are presented to illustrate the effectiveness of the theoretical results.
引文
[1]张霄力,刘玉忠,赵军.一类离散切换系统的渐近稳定性.控制理论与应用,2002,19(5):774-776
[2]孙文安,董世山,邹开其,陈刚.线性周期切换系统的渐近稳定性.辽宁工程技术大学学报,2006,25(1):158-160
[3]V.F.Montagner,V.J.S.Leite,R.C.L.F.Oliveira,P.L.D.Peres.State feedback control of switched linear systems:An LMI approach.Computational and applied mathematics,2006,194:192-206
[4]Sehjeong Kim,Sue Ann Campbell,and Xinzhi Liu.Stability of a class of linear switching systems with time delay.IEEE Transactions on circuits and systems-regular papers,2006,53(1):384-393
[5]孙洪飞,赵军,高晓东.带有时滞摄动的线性切换系统的稳定性.控制与决策,2002,17(4):431-434
[6]Zhai G.,Sun Y.,Chen X.,et al.Stability and L_2 gain analysis for switched symmetric systems with time delay.Proceedinga of the American Control Conference,Denver.Colorado,2003:2682-2687
[7]陈松林,姚郁.一类时滞切换系统的稳定性和镇定.黑龙江大学自然科学学报,2006,23(2):206-210
[8]Chang-hong Wang,Li-xian Zhang,Hui-jun Gao,Li-guang Wu.Delay-dependent stability and stabilization of a class of linear switched time-varying delay systems.Proceedings of the fourth international conference on machine learning and cybernetics,Guang zhou,2005:18-21
[9]王仁明,关治洪,刘新芝.具有输入时滞切换系统的稳定性分析与控制.三峡大学学报(自然科学版),2004,26(2):170-172
[10]卢建宁,赵光宙.离散时滞切换系统稳定性分析.信息与控制,2005,34(3):381-384
[11]Gollu A,Varaiya PD..Hybrid Dynastic Systems.Proc.Of the 28~(th) CDC.,1989:2708-2713
[12]Back A.,Guckenheimer J.,Myers M.A..Dynastic sirradaticm facility for Hybrid Systems.Hybrid Systems,1993:39-53
[13]Varaiya P.,P.Smaa,Cars OH Smaa Road:problem of control.IEEE Trans.On Automatic Control,1993,38:195-207
[14]Peleties P.,DeCaclo R.,Asymptotic stability of M·switched Systems using Lyapunov-like function.Proe.American Control Conf.,Boston,1991:1679-1684
[15]Pettersson S,Lennartson B..Stability and robustness for hybrid systems.Proc.of the 35th IEEE Conference on Control and Deeisiort,1996:1202-1207
[16]Bo Hu,Xuping Xu,Michel,A.N.,Antsaklis P.J..Stability analysis for a class of nonlinear switched systems.Proc.of 38th Conference on Decision & Control,1999:4374-4379
[17]王仁明,关治洪,刘新芝.一类非线性切换系统的稳定性分析.系统工程与电子技术,2004,26(1):68-71
[18]董学平,王执铨.分布参数切换系统的稳定性分析.信息与控制,2006,35(4):503-507
[19]Vladimir B.,Kolmanovskii,Jean-Pierre Richard.Stability of some linear systems with delays.IEEE Transactions on automatic control,1999,44(5):984-989
[20]Pettersson S..Synthesis of switched linear systems.Proceedings of the 42~(nd) IEEE Conference on Decision and Control.Maui,America,IEEE,2003:5283-5288
[21]Daafouz J.,Riedinger P.,Iung C..Stability analysis and control synthesis for switched systems:a switched Lyapunov Function approach.IEEE Transactions on Automatic Control,2002,47(11):1883-1887
[22]Mignone D.,Fermri-Trecate G,Morari M.,Stability and stabilization of piecewise aft'me and hybrid systems:an LMI approach.Proceedings of the 39~(th) IEEE Conference on Decision and Control,Sydney,Australia:IEEE,2000:504-509
[23]张霓.参数不确定线性混杂系统的鲁棒控制及应用.杭州:浙江大学,2003
[24]G.Zhai,H.Lin,A.N.Michel,K.Yasuda.Stability analysis for switched systems with continuous-time and discrete-time subsystems.Proc.American Control Conf.,Boston,MA,Jun.2004:4555-4560
[25]G.Zhai,H.Lin,X.Xu,A.N.Michel.Stability analysis and design of switched normal systems,Proc.43rd IEEE Conf.Decision and Control,Atlantis,Bahamas,2004:3253-3258
[26]Guisheng Zhai,Derong Liu,Joe lmae.Lie Algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems.Trensactions on circuits and systems—Ⅱ:express briefs,2006,53(2)
[27]A.K.Gelig,A.N.Churilov.Stability and oscillations of nonlinear pulse-modulated systems.Birkhauser,Boston,1998
[28]JingYao,Zhi-Hong Guan,Guanrong Chen,Daniel W.C.Ho..Stability robust stabiliza-tion and H_∞ control of singular-impulsive systems via switching control.Systems & ControlLetters,2006,55:879-886
[29]Amol Sasane.Stability of switching infinite-dimensional systems.Automatica,2005,41:75-78
[30]廖晓昕.稳定性的理论、方法和应用.武汉:华中科技大学出版社,2002:42-65
[31]D.Q.Cao,Ping He.Stability criteria of linear neutral systems with a single delay.Applied Mathematics and Computation,2004,148:135-143
[32]俞立.鲁棒控制-线性矩阵不等式处理方法.北京:清华大学出版社,2002
[33]Boyd S.,El Ghaoui,L.Feron,et at.Linear matrix inequalities in systems and control theory Philadelphia:SIAM,1994
[34]R.A.DeCarlo,M.S.Branicky,S.Pettersson,B.Lennartson.Perspectives and results on the stability and stability of hybrid systems.Proe.IEEE 2000,88(7):1069-1082
[35]Zhai G.S.,Sun Y.,Chen X.K.,Anthony N.M..Stability and L_2-gain analysis for switched symmetric systems with time delay.American control conference,2003,Denver,CO,USA.:2682-2687
[36]Xi-Ming Sun,Jun Zhao,David J.Hill.Stability and L_2-gain analysis for switched delay systems:A delay-dependent method.Automatica,2006,42:1769-1774
[37]Yuan Gong Sun,Long Wang,Guangming Xie.Delay-dependent robust stability and H_∞ control for uncertain discrete-time switched systems with mode-dependent time delays.Applied Mathematics and Computation,2007,187(2):1228-1237
[38]王翼.离散控制系统.北京:科学出版社,1987
[39]Zhang X.,Liu Y.,Zhao J..Asymptotic stability of a class of discrete switched systems.Control theory and applications,2002,19(5):774-776
[40]Dongsheng Du,Shaosheng Zhou,Baoyong Zhang.Generalized H_2 output feedback controller design for uncertain discrete-time switched systems via switched Lyapunov functions.Nonlinear Analysis,2006,65:2135-2146
[41]Yuangong Sun,Long Wang,Guangming Xie.Delay-dependent robust stability and stabilization for discrete-time switched systems with mode-dependent time-varying delays.Applied Mathema-tics and Computation,2006,180:428-435
[42]Guisheng Zhai,Derong Liu,Joe Imae,Tomoaki Kobayashi.Lie Algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems.IEEE Transactions on circuits and systems-Ⅱ:express briefs,2006,53(2):152-156
[43]Guisheng Zhai,Xuping Xu,Hai Lin,Anthony N.Michel.Analysis and design of switched normal systems.Nonlinear Analysis,2006,65:2248-2259
[44]Zhang Xiao-li,Liu Yu-zhong,Zhao Jun.Asymptotic stability of a class of discrete switched systems, Control theory and applications. 2002,19(5):774-776
[45] Amol Sasane. Stability of switching infinite-dimensional systems. Automatica, 2005,41:75-78
[2]孙文安,董世山,邹开其,陈刚.线性周期切换系统的渐近稳定性.辽宁工程技术大学学报,2006,25(1):158-160
[3]V.F.Montagner,V.J.S.Leite,R.C.L.F.Oliveira,P.L.D.Peres.State feedback control of switched linear systems:An LMI approach.Computational and applied mathematics,2006,194:192-206
[4]Sehjeong Kim,Sue Ann Campbell,and Xinzhi Liu.Stability of a class of linear switching systems with time delay.IEEE Transactions on circuits and systems-regular papers,2006,53(1):384-393
[5]孙洪飞,赵军,高晓东.带有时滞摄动的线性切换系统的稳定性.控制与决策,2002,17(4):431-434
[6]Zhai G.,Sun Y.,Chen X.,et al.Stability and L_2 gain analysis for switched symmetric systems with time delay.Proceedinga of the American Control Conference,Denver.Colorado,2003:2682-2687
[7]陈松林,姚郁.一类时滞切换系统的稳定性和镇定.黑龙江大学自然科学学报,2006,23(2):206-210
[8]Chang-hong Wang,Li-xian Zhang,Hui-jun Gao,Li-guang Wu.Delay-dependent stability and stabilization of a class of linear switched time-varying delay systems.Proceedings of the fourth international conference on machine learning and cybernetics,Guang zhou,2005:18-21
[9]王仁明,关治洪,刘新芝.具有输入时滞切换系统的稳定性分析与控制.三峡大学学报(自然科学版),2004,26(2):170-172
[10]卢建宁,赵光宙.离散时滞切换系统稳定性分析.信息与控制,2005,34(3):381-384
[11]Gollu A,Varaiya PD..Hybrid Dynastic Systems.Proc.Of the 28~(th) CDC.,1989:2708-2713
[12]Back A.,Guckenheimer J.,Myers M.A..Dynastic sirradaticm facility for Hybrid Systems.Hybrid Systems,1993:39-53
[13]Varaiya P.,P.Smaa,Cars OH Smaa Road:problem of control.IEEE Trans.On Automatic Control,1993,38:195-207
[14]Peleties P.,DeCaclo R.,Asymptotic stability of M·switched Systems using Lyapunov-like function.Proe.American Control Conf.,Boston,1991:1679-1684
[15]Pettersson S,Lennartson B..Stability and robustness for hybrid systems.Proc.of the 35th IEEE Conference on Control and Deeisiort,1996:1202-1207
[16]Bo Hu,Xuping Xu,Michel,A.N.,Antsaklis P.J..Stability analysis for a class of nonlinear switched systems.Proc.of 38th Conference on Decision & Control,1999:4374-4379
[17]王仁明,关治洪,刘新芝.一类非线性切换系统的稳定性分析.系统工程与电子技术,2004,26(1):68-71
[18]董学平,王执铨.分布参数切换系统的稳定性分析.信息与控制,2006,35(4):503-507
[19]Vladimir B.,Kolmanovskii,Jean-Pierre Richard.Stability of some linear systems with delays.IEEE Transactions on automatic control,1999,44(5):984-989
[20]Pettersson S..Synthesis of switched linear systems.Proceedings of the 42~(nd) IEEE Conference on Decision and Control.Maui,America,IEEE,2003:5283-5288
[21]Daafouz J.,Riedinger P.,Iung C..Stability analysis and control synthesis for switched systems:a switched Lyapunov Function approach.IEEE Transactions on Automatic Control,2002,47(11):1883-1887
[22]Mignone D.,Fermri-Trecate G,Morari M.,Stability and stabilization of piecewise aft'me and hybrid systems:an LMI approach.Proceedings of the 39~(th) IEEE Conference on Decision and Control,Sydney,Australia:IEEE,2000:504-509
[23]张霓.参数不确定线性混杂系统的鲁棒控制及应用.杭州:浙江大学,2003
[24]G.Zhai,H.Lin,A.N.Michel,K.Yasuda.Stability analysis for switched systems with continuous-time and discrete-time subsystems.Proc.American Control Conf.,Boston,MA,Jun.2004:4555-4560
[25]G.Zhai,H.Lin,X.Xu,A.N.Michel.Stability analysis and design of switched normal systems,Proc.43rd IEEE Conf.Decision and Control,Atlantis,Bahamas,2004:3253-3258
[26]Guisheng Zhai,Derong Liu,Joe lmae.Lie Algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems.Trensactions on circuits and systems—Ⅱ:express briefs,2006,53(2)
[27]A.K.Gelig,A.N.Churilov.Stability and oscillations of nonlinear pulse-modulated systems.Birkhauser,Boston,1998
[28]JingYao,Zhi-Hong Guan,Guanrong Chen,Daniel W.C.Ho..Stability robust stabiliza-tion and H_∞ control of singular-impulsive systems via switching control.Systems & ControlLetters,2006,55:879-886
[29]Amol Sasane.Stability of switching infinite-dimensional systems.Automatica,2005,41:75-78
[30]廖晓昕.稳定性的理论、方法和应用.武汉:华中科技大学出版社,2002:42-65
[31]D.Q.Cao,Ping He.Stability criteria of linear neutral systems with a single delay.Applied Mathematics and Computation,2004,148:135-143
[32]俞立.鲁棒控制-线性矩阵不等式处理方法.北京:清华大学出版社,2002
[33]Boyd S.,El Ghaoui,L.Feron,et at.Linear matrix inequalities in systems and control theory Philadelphia:SIAM,1994
[34]R.A.DeCarlo,M.S.Branicky,S.Pettersson,B.Lennartson.Perspectives and results on the stability and stability of hybrid systems.Proe.IEEE 2000,88(7):1069-1082
[35]Zhai G.S.,Sun Y.,Chen X.K.,Anthony N.M..Stability and L_2-gain analysis for switched symmetric systems with time delay.American control conference,2003,Denver,CO,USA.:2682-2687
[36]Xi-Ming Sun,Jun Zhao,David J.Hill.Stability and L_2-gain analysis for switched delay systems:A delay-dependent method.Automatica,2006,42:1769-1774
[37]Yuan Gong Sun,Long Wang,Guangming Xie.Delay-dependent robust stability and H_∞ control for uncertain discrete-time switched systems with mode-dependent time delays.Applied Mathematics and Computation,2007,187(2):1228-1237
[38]王翼.离散控制系统.北京:科学出版社,1987
[39]Zhang X.,Liu Y.,Zhao J..Asymptotic stability of a class of discrete switched systems.Control theory and applications,2002,19(5):774-776
[40]Dongsheng Du,Shaosheng Zhou,Baoyong Zhang.Generalized H_2 output feedback controller design for uncertain discrete-time switched systems via switched Lyapunov functions.Nonlinear Analysis,2006,65:2135-2146
[41]Yuangong Sun,Long Wang,Guangming Xie.Delay-dependent robust stability and stabilization for discrete-time switched systems with mode-dependent time-varying delays.Applied Mathema-tics and Computation,2006,180:428-435
[42]Guisheng Zhai,Derong Liu,Joe Imae,Tomoaki Kobayashi.Lie Algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems.IEEE Transactions on circuits and systems-Ⅱ:express briefs,2006,53(2):152-156
[43]Guisheng Zhai,Xuping Xu,Hai Lin,Anthony N.Michel.Analysis and design of switched normal systems.Nonlinear Analysis,2006,65:2248-2259
[44]Zhang Xiao-li,Liu Yu-zhong,Zhao Jun.Asymptotic stability of a class of discrete switched systems, Control theory and applications. 2002,19(5):774-776
[45] Amol Sasane. Stability of switching infinite-dimensional systems. Automatica, 2005,41:75-78