一类切换系统的稳定性分析
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摘要
混杂系统是由离散事件动态系统与连续时间动态系统相互混和,相互作用而形成的统一动态系统。切换系统是混杂系统中重要的分支之一,本质上是一类非线性系统。切换系统可以看成是将非线性系统分成若干个线性子系统,通过切换控制在各个子系统之间切换,将逻辑动态和连续动态结合起来。但由于切换的引入,使整个系统的动态特性和各个子系统的动态特性有着很大的差异。因此,对于切换系统的研究是一个非常具有理论意义与应用价值的研究方向,而且近年来随着计算机技术的迅速发展得到了广泛的应用,切换系统的研究也成为控制领域研究的热点问题。
本文主要以理论分析研究为主,以李亚普诺夫稳定性理论为基础,首先从两个方面对一类特殊的线性切换系统一周期切换系统的稳定性进行分析。(1)对线性时变周期切换系统通过分析切换系统状态方程的解来研究系统的稳定性;(2)对离散线性周期切换系统,在schur稳定意义下,设计最小切换周期T,以保证整个切换系统是渐近稳定的。然后,对离散线性切换系统基于最小驻留时间讨论系统的稳定性;最后,针对离散线性切换系统中状态变量不可直接测量的情况,通过构造降维观测器,用观测器的估计值代替原系统中不可测量的状态变量的值,进一步给出包含观测器的状态反馈控制器和相应策略的设计方法。
The hybrid system is composed by discrete dynamic systems and continu-ous dynamic systems. It is formed a uniform dynamic systems by effected each other.Switched systems is an important class of hybrid systems.In fact,it is a class of noline system.It consists of some linear subsystems that integrate the logical and continuous dynamics by switching.Because of switching signal,there is a large different between the characteristic of Switched systems and every subsystems meanwhile,in recent years,the Switched systems gains the more extensive application with the rapid development of computer technique,therefore,the study of Switched systems will become a research direction with theory signification and application value.so it becomes the one hot spot issues of control field.
This paper deals mainly with theoretical analysis.Based on the theory of Lyapunov Stability.First,for a especial class linear switched systems—periodic switched systems,we analysis their stability from two parts.(1)For continuous linear time-varying periodic switched systems ,we study the stability by using the solve of state equation of switched systems and computer simulation. (2)For discrete linear periodic switched systems,Based on Schur stability ,the minimum switching periodic T are designed to guarantee the switched systems is asymptotically sta-ble.Then,for discrete linear switched systems ,Designed the minimum dwell time ,it could assure that the switched systems is asymptotically stable.Finally,for discrete linear switched systems, if the part of state value can not be measured directly, designing a Reduced-order State Observer,the switching strategy and sub-controller which use the observer value,are designed to guarantee the close-loop system is asymptotically stable.
本文主要以理论分析研究为主,以李亚普诺夫稳定性理论为基础,首先从两个方面对一类特殊的线性切换系统一周期切换系统的稳定性进行分析。(1)对线性时变周期切换系统通过分析切换系统状态方程的解来研究系统的稳定性;(2)对离散线性周期切换系统,在schur稳定意义下,设计最小切换周期T,以保证整个切换系统是渐近稳定的。然后,对离散线性切换系统基于最小驻留时间讨论系统的稳定性;最后,针对离散线性切换系统中状态变量不可直接测量的情况,通过构造降维观测器,用观测器的估计值代替原系统中不可测量的状态变量的值,进一步给出包含观测器的状态反馈控制器和相应策略的设计方法。
The hybrid system is composed by discrete dynamic systems and continu-ous dynamic systems. It is formed a uniform dynamic systems by effected each other.Switched systems is an important class of hybrid systems.In fact,it is a class of noline system.It consists of some linear subsystems that integrate the logical and continuous dynamics by switching.Because of switching signal,there is a large different between the characteristic of Switched systems and every subsystems meanwhile,in recent years,the Switched systems gains the more extensive application with the rapid development of computer technique,therefore,the study of Switched systems will become a research direction with theory signification and application value.so it becomes the one hot spot issues of control field.
This paper deals mainly with theoretical analysis.Based on the theory of Lyapunov Stability.First,for a especial class linear switched systems—periodic switched systems,we analysis their stability from two parts.(1)For continuous linear time-varying periodic switched systems ,we study the stability by using the solve of state equation of switched systems and computer simulation. (2)For discrete linear periodic switched systems,Based on Schur stability ,the minimum switching periodic T are designed to guarantee the switched systems is asymptotically sta-ble.Then,for discrete linear switched systems ,Designed the minimum dwell time ,it could assure that the switched systems is asymptotically stable.Finally,for discrete linear switched systems, if the part of state value can not be measured directly, designing a Reduced-order State Observer,the switching strategy and sub-controller which use the observer value,are designed to guarantee the close-loop system is asymptotically stable.
引文
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[29]Z.Sun,D.Zheng.On reachability and stabilization of switched linear systems,IEEE Transactions on Automatic,38,1221-1228,2002.
[30]G.Xie,D.Zheng,L.Wang.Controbility of linear swiched linear systems,IEEE Transactions on Automatic Control,46(2),291-295,2001.
[31]J.Ezzine,A.H.Haddad.Controllability and observability of hybrid systems.International Journal of Control,49(6),2045-2055,1989.
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[34]张霄力,赵军.任意切换下不确定线性切换系统的鲁棒稳定性[J].控制理论与应用.2002,28(5):859-861.
[35]张霓,吴铁军.参数摄动混杂离散系统的鲁棒稳定[J].控制理论与应用.2002,19(5):731-736
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[38]Bo Hu,Xuping Xu,Mickel Anetal.Stability analysis for aclass of nonlinear switched systems.Proceeding of 38h conferenceon Decision and Control,phoenix,Arizona USA,1999.4374-4379.
[39]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.
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[41]尤昌德.线性系统理论基础.电子工业出版社,1984.
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[43]景丽,高立群.线性切换系统稳定性及相关问题研究.东北大学博士学位论文(第四章),2004
[44]张瑞丰.精通Matlab6.5.中国水利水电出版社.2004
[45]王仁明,关治洪,刘新芝.线性时变切换系统及非线性扰动系统的稳定性分析.华中师范大学学报,vol.36,2002,NO.3:265-268,
[46]P.Peleties,R.Decarlo.Asymptotic stability of m-switched systems using Lyapunovlike Function[A].Proceedings of the American Control Conference[C],1991:1679-1684.
[47]张霄力,刘玉忠,赵军.一类离散切换系统的渐近称定性[J].控制理论与应用,2002,19(5):774-776.
[48]S.Pettersson,B.Lennartson.Stability and Robustness for Hybrid system Proceeding of the 35th CDC[C],1996:1202-1207.
[49]Zhai G,Hu.B,Yasuda.K,etal.Stability analysis of switched systems with stable and unstable subsystems:anaverage dwell time approch[J].Int J of Syatem Science,2002,32(8):1055-1061.
[50]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.辽宁工程大学学报,2006,vol 25,NO.1,pp.158-160.
[51]Wei Ni,Daizhan Cheng,Xiaoming Hu.Minimum dwell time for Stability and Stabilization of Switched linear systems.Proc.7th world congress on Intelligent Control and Automation June 25-27,2008,Chongqing,china.pp:4109-4115.
[52]D Cheng,L.Guo,L.Lin,Y.Wang.Stability of switched linear systems.IEEE,Trans on Automatic Control.,vol.50,No.5,2005.
[53]H.K.Khalil.Nonlinear systems.2nd,Prentice Hall,1996.
[54]J.C.Gromel,P.Colaneri,P.Bolzern.Dynamic output feedback control of Switched linear systems.IEEE,Trans on Automatic Control.vol.53,No.3,2008.PP:720-733.
[55]Meiyan Cong,Xiaowu Mu,Jianyin Fang.Stability of discrete-time Switched systems.Proc.7th world confess on Intelligent Control and Automation June 25-27,2008,Chongqing,china.pp:4100-4102
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[2]高春华,王慧,李平.混合系统建模,分析与综合:研究进展与展望[J].系统工程理论也实践,2002,11:15-20.
[3]徐心和.混杂系统的存在与研究[J].基础自动化,1996,1:1-5.
[4]A.Balluchi,L.Benvenuti.Automotive engine control and hybrid systems:challenges and opportunities,Proceedings of the IEEE,88(7),888-912,2000.
[5]M.Song,T.J.Tarn,N.Xi.Integration of task scheduling,action planning,and control in robotic manufacturing systems,Proceedings of the IEEE,88(7),1097-1107,2000.
[6]E.R.Westervelt,J.W.Grizzle,C.Candas.Switching and PI control of Walking motions of planar biped walkers,IEEE Transactions on Automatic Control,48(2),308-312,2003.
[7]J.Tan,N.Xi,Y.Wang.A Singularity-free motion control algorithm for robot manipulators-a hybrid systems approach,Automatic,40,1239-1245,2004.
[8]P.V.Zhivoglydov,R.H.Middleton.Networked Control Design for Linear Systems,Automatic,39,743-750,2003.
[9]P.Peletis,R.DeCarlo.Asymptotic Stability of m-Switched Systems Using Lyapunovlike Function[A].Proceedings of the American Control Confrence[C],1991:1679-1684.
[10]Liberzon D,Morse A S.Basic Problem in Stability and Design of Switched Systems [J],Control Systems Magazine,1999,19(5):59-70.
[11]C.A.Yfoulis,R.Shorten.A numerical tchnique for the stability of linear switched systems,International Journal of Control,77(11),1019-1039,2004.
[12]Braniky M S.Multiple Lyapunov Functions and other analysis tools for Switched and Hybrid Systems J.IEEE Trsns.on Automatic Control,1998,43(4):475-482.
[13]M.Johansson,A.Rantzer.Computation of piecewise quadratic Lyapunov function for hybrid systems,IEEE Transactions on Automatic Control,43(4),555-559,1998.
[14]A.S.Morse.Supervisory control of familis of linear set-point controllers,partl:exact matching,IEEE Transactions on Automatic Control,41,1413-1431,1996.
[15]J.P.Hespanha,A.S.Morse.Stability of switched systems by average dwell time,Proceedings of the 38th IEEE Conference on Decision and Control,2655-2600,1999.
[16]S.H.Lee,.H.Kim,J.T.Lim.A new stability of switched systems,Automatica,36,917-922,2000.
[17]G.Michaletzky,L.Gerencser.BIBO stability of switching sysems,IEEE Transactions on Automatic Control,47(11),1895-1898,2002.
[18]D.Liberzon.ISS and integral ISS disturbance attenuation with bounded control,Proceedings of the 38th IEEE Confrence on Decision and Control,2501-2506,1999.
[19]A.Hassibi,S.P.Boy,J.P.How.A class of Lyapunov function for analyzing hybrid systems,Proceeding of the American Control Conference,2455-2460,1999.
[20]D.Cheng.Stabilization of planar switched systems,System Control Letters,51,79-88,2004.
[21]董亚丽,王威,王颜刚.三阶切换系统的镇定.信息与控制,34(4),79-88,2005.
[22]Z.Sun.Stabilizability and insensitivity of switched systems,IEEE,Transactions on Automatic Control,49(7),1133-1137,2004.
[23]G.Zhai,B.Hu,K.Yasuda,A.N.Michel.Disturbance attenuaion properties of time-controlled switched systems,Journal of Franklin Institute,338,765-779,2001.
[24]Hedlund S,Rantzer A.Convex dynamic programming for hybrid systems.IEEE Transactions on Automatic Control,2002,47(9):1536-1540.
[25]翟海峰,苏宏业等.一类分段线性混杂系统的最优控制策略研究[J].控制理论与应用,2002,17(6):863-867.
[26]Shaikh M S.Optimal Control of Hybrid Systems:Theory and Algorithms.PhD thesis,McGill University.Montreal,Canada,2004.
[27]Shaikh M S.Caines P E.On the Optimal Control of Hybrid Systems:Optimization of Trajectories,Switching Times,and Location Schedules.Hybrid Systems:Computation and Control,Proceedings of the 6th International HSCC Workshop,2003:466-481.
[28]Rantzer A,Johansson M.Piece linear quadraic optimal control.IEEE Transactions on Automatic Control,2000,45(4),629-637.
[29]Z.Sun,D.Zheng.On reachability and stabilization of switched linear systems,IEEE Transactions on Automatic,38,1221-1228,2002.
[30]G.Xie,D.Zheng,L.Wang.Controbility of linear swiched linear systems,IEEE Transactions on Automatic Control,46(2),291-295,2001.
[31]J.Ezzine,A.H.Haddad.Controllability and observability of hybrid systems.International Journal of Control,49(6),2045-2055,1989.
[32]谢广明,郑大钟.一类线性切换系统能控性和能达性的充要条件.控制与决策,16(2),248-253,2001.
[33]汤伟,施颂椒,王孟效等.鲁棒控制理论中3种主要方法综述(一)[J].西北轻工业学院学报,2000,vol.18(4):54-59.
[34]张霄力,赵军.任意切换下不确定线性切换系统的鲁棒稳定性[J].控制理论与应用.2002,28(5):859-861.
[35]张霓,吴铁军.参数摄动混杂离散系统的鲁棒稳定[J].控制理论与应用.2002,19(5):731-736
[36]Peleties P,Decarlo R.Asymptotic.stability of m-switched systems using Lyapunovlike function.C.Proceeding American Control Conf,Boston.MA,1991.pp.1679-1648
[37]M.S.Branicky.Stability of switched and hybrid systems.in proc.IEEE Conf.Decision and control,Lake Buena Vista,FL,December 1994,pp.3498-3503.
[38]Bo Hu,Xuping Xu,Mickel Anetal.Stability analysis for aclass of nonlinear switched systems.Proceeding of 38h conferenceon Decision and Control,phoenix,Arizona USA,1999.4374-4379.
[39]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.
[40]段广仁.线性系统理论.哈尔滨工业大学出版社,2004.
[41]尤昌德.线性系统理论基础.电子工业出版社,1984.
[42]须田信英[日].自动控制中的矩阵理论.科学出版社,1979.
[43]景丽,高立群.线性切换系统稳定性及相关问题研究.东北大学博士学位论文(第四章),2004
[44]张瑞丰.精通Matlab6.5.中国水利水电出版社.2004
[45]王仁明,关治洪,刘新芝.线性时变切换系统及非线性扰动系统的稳定性分析.华中师范大学学报,vol.36,2002,NO.3:265-268,
[46]P.Peleties,R.Decarlo.Asymptotic stability of m-switched systems using Lyapunovlike Function[A].Proceedings of the American Control Conference[C],1991:1679-1684.
[47]张霄力,刘玉忠,赵军.一类离散切换系统的渐近称定性[J].控制理论与应用,2002,19(5):774-776.
[48]S.Pettersson,B.Lennartson.Stability and Robustness for Hybrid system Proceeding of the 35th CDC[C],1996:1202-1207.
[49]Zhai G,Hu.B,Yasuda.K,etal.Stability analysis of switched systems with stable and unstable subsystems:anaverage dwell time approch[J].Int J of Syatem Science,2002,32(8):1055-1061.
[50]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.辽宁工程大学学报,2006,vol 25,NO.1,pp.158-160.
[51]Wei Ni,Daizhan Cheng,Xiaoming Hu.Minimum dwell time for Stability and Stabilization of Switched linear systems.Proc.7th world congress on Intelligent Control and Automation June 25-27,2008,Chongqing,china.pp:4109-4115.
[52]D Cheng,L.Guo,L.Lin,Y.Wang.Stability of switched linear systems.IEEE,Trans on Automatic Control.,vol.50,No.5,2005.
[53]H.K.Khalil.Nonlinear systems.2nd,Prentice Hall,1996.
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