一类切换系统的滤波器设计
|
摘要
在过去二十年中,混杂系统由于其广泛存在于工程和社会系统中而引起控制界极大的兴趣,切换系统是各种混杂系统中最为典型的一类混杂系统。切换系统由一组连续时间或者离散时间子系统和一个切换规则组成,切换规则决定在某一个时间段内激活哪一个子系统。本文依托国家自然科学基金项目,采用分段Lyapunov函数和平均驻留时间方法研究了切换系统的滤波器设计问题,主要内容如下：
研究了连续线性切换系统和满足Lipschitz条件的非线性时滞切换系统鲁棒H∞滤波器设计问题。基于分段Lyapunov函数和平均驻留时间方法研究了滤波误差系统满足H∞性能的指数稳定条件,并用线性矩阵不等式的形式给出了系统存在线性滤波器的充分条件。
考虑在实际应用中存在异步切换的情况,即滤波器实际的切换时刻超前或者滞后于系统的切换时刻,首次探讨了异步切换下的连续线性切换系统鲁棒L2-L∞滤波问题,基于分段Lyapunov函数和平均驻留时间方法,同时结合包含稳定与不稳定子系统的切换系统稳定分析方法,导出了滤波器的参数表达式。
研究了包含丢失测量离散时滞切换系统的鲁棒l2-l∞滤波器设计问题,丢失测量是指系统的测量输出并不是连续的而是包含丢失观测。目的是使设计的线性滤波器,在给定的l2-l∞噪声抑制性能和丢失概率下,保证滤波误差系统是指数均方稳定的。通过使用分段Lyapunov函数和平均驻留时间方法,给出了以线性矩阵不等式表示的充分条件。
探讨了异步切换下包含丢失测量的离散时滞切换系统的鲁棒H∞滤波问题,根据分段Lyapunov函数和平均驻留时间方法,提出了滤波误差系统存在指数均方稳定的充分条件,然后研究其H∞性能,以LMI的形式给出了存在期望滤波器参数解的表达式。
In the last two decades, hybrid systems have attracted a recurring interest in the control community for they exist widely in engineering and social systems. Among various kinds of hybrid systems, switched system is the most typical one. Switched systems consist of a family of continuous-time or discrete-time subsystems and a switching law, which defines a specific subsystem being activated during a certain interval of time. This work, which is supported by the National Natural Science Foundation of China, investigates the problem of filter design for switched systems by applying Lyapunov functions theory and average dwell time approach. The main contributions are as follows.
The robust H∞filtering problem for a class of continuous-time linear and nonlinear time-delays switched systems which satisfy Lipschitz conditions are investigated, based on piecewise Lyapunov functions theory and average dwell time approach, the condition that guarantee the filtering error system to be exponentially stable with a prescribed H∞performance is derived, and the existence conditions of linear filter are proposed in terms of LMI.
It is considered that in practical application, there exists asynchronous switching between the filter and the system, that is, the real switching instants of the filter will exceed or lag behind those of the system, we study the robust L2-L∞filtering problem for continuous-time switched linear systems under asynchronous switching. By using piecewise Lyapunov functions theory and average dwell time approach, and combining the stability analysis idea which is used to deal with the switched system with stable and unstable subsystem, then the filter parameter expression is derived.
We consider the problem of the robust l2-l∞filter design for discrete-time switched time-delay systems with missing measurements. The missing measurement means that the system's measurement output is not consecutive but contains missing observations. The objective is to design a linear filter to guarantee the filtering error system exponentially mean-square stable for a prescribed noise attenuation level in the l2-l∞sense and missing measurements probability. By using piecewise Lyapunov functions and average dwell time approach, a sufficient condition on the existence of filter is given in terms of LMIs.
The problem of robust H∞filtering for discrete-time switched time-delays systems with missing measurements and asynchronous switching is investigated. According to the piecewise Lyapunuov function and average dwell-time approach, we propose the sufficient condition for the existence of exponentially mean-square stable for the filtering error system also H∞performance is analyzed, and the solution of the desired filter parameters has been formulated in terms of a set of LMIs.
研究了连续线性切换系统和满足Lipschitz条件的非线性时滞切换系统鲁棒H∞滤波器设计问题。基于分段Lyapunov函数和平均驻留时间方法研究了滤波误差系统满足H∞性能的指数稳定条件,并用线性矩阵不等式的形式给出了系统存在线性滤波器的充分条件。
考虑在实际应用中存在异步切换的情况,即滤波器实际的切换时刻超前或者滞后于系统的切换时刻,首次探讨了异步切换下的连续线性切换系统鲁棒L2-L∞滤波问题,基于分段Lyapunov函数和平均驻留时间方法,同时结合包含稳定与不稳定子系统的切换系统稳定分析方法,导出了滤波器的参数表达式。
研究了包含丢失测量离散时滞切换系统的鲁棒l2-l∞滤波器设计问题,丢失测量是指系统的测量输出并不是连续的而是包含丢失观测。目的是使设计的线性滤波器,在给定的l2-l∞噪声抑制性能和丢失概率下,保证滤波误差系统是指数均方稳定的。通过使用分段Lyapunov函数和平均驻留时间方法,给出了以线性矩阵不等式表示的充分条件。
探讨了异步切换下包含丢失测量的离散时滞切换系统的鲁棒H∞滤波问题,根据分段Lyapunov函数和平均驻留时间方法,提出了滤波误差系统存在指数均方稳定的充分条件,然后研究其H∞性能,以LMI的形式给出了存在期望滤波器参数解的表达式。
In the last two decades, hybrid systems have attracted a recurring interest in the control community for they exist widely in engineering and social systems. Among various kinds of hybrid systems, switched system is the most typical one. Switched systems consist of a family of continuous-time or discrete-time subsystems and a switching law, which defines a specific subsystem being activated during a certain interval of time. This work, which is supported by the National Natural Science Foundation of China, investigates the problem of filter design for switched systems by applying Lyapunov functions theory and average dwell time approach. The main contributions are as follows.
The robust H∞filtering problem for a class of continuous-time linear and nonlinear time-delays switched systems which satisfy Lipschitz conditions are investigated, based on piecewise Lyapunov functions theory and average dwell time approach, the condition that guarantee the filtering error system to be exponentially stable with a prescribed H∞performance is derived, and the existence conditions of linear filter are proposed in terms of LMI.
It is considered that in practical application, there exists asynchronous switching between the filter and the system, that is, the real switching instants of the filter will exceed or lag behind those of the system, we study the robust L2-L∞filtering problem for continuous-time switched linear systems under asynchronous switching. By using piecewise Lyapunov functions theory and average dwell time approach, and combining the stability analysis idea which is used to deal with the switched system with stable and unstable subsystem, then the filter parameter expression is derived.
We consider the problem of the robust l2-l∞filter design for discrete-time switched time-delay systems with missing measurements. The missing measurement means that the system's measurement output is not consecutive but contains missing observations. The objective is to design a linear filter to guarantee the filtering error system exponentially mean-square stable for a prescribed noise attenuation level in the l2-l∞sense and missing measurements probability. By using piecewise Lyapunov functions and average dwell time approach, a sufficient condition on the existence of filter is given in terms of LMIs.
The problem of robust H∞filtering for discrete-time switched time-delays systems with missing measurements and asynchronous switching is investigated. According to the piecewise Lyapunuov function and average dwell-time approach, we propose the sufficient condition for the existence of exponentially mean-square stable for the filtering error system also H∞performance is analyzed, and the solution of the desired filter parameters has been formulated in terms of a set of LMIs.
引文
[1]Witsenhausen H. A class of hybrid-state continuous-time dynamic systems. IEEE Transactions on Automatic Control[J],1966,11(2):161～167.
[2]Stiver J. A, Antsaklis P. J. Modeling and analysis of hybrid control systems[C]//1992 Proceedings of the 31st IEEE Conference on Decision and Control,1992,4:3748～3751.
[3]Balluchi A, Benedetto M. D, Pinello C, Rossi C. Cut-off in engine control:a hybrid system approach[C]//Proceedings of the 36th IEEE Conference on Decision and Control,1997,5: 4720～4725.
[4]Hall J. G, Delemos R. Extended real time logic for hybrid systems controller design[C]// IEEE Colloquium on Hybrid Control for Real Time Systems, London,1996:4/1～4/2.
[5]Zhang W, Branicky M. S, Phillips S. M. Stability of networked control systems[J]. IEEE Control Systems Magazine,2001,21(1):84～99.
[6]Hiskens I. A. Stability of hybrid system limit cycles:application to the compass gait biped robot[C]//In Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL,2001,1:774～779.
[7]李秀改,岳红,高东杰.复杂工业过程新型控制方法—混杂系统控制理论的研究[J].化工自动化及仪表,2001,8(5)：1～8.
[8]吴锋,刘文煌,郑应平.混杂系统方法及其在过程控制中的应用[J].清华大学学报(自然科学版),1997,37(11)：77～81.
[9]Tomlin C, Pappas G. J, Sastry S. Conflict resolution for air traffic management:a study in multi-agent hybrid systems[J]. IEEE Transactions on Automatic Control,1998,43(4):509～521.
[10]郑刚,谭民,宋永华.混杂系统的研究进展[J].控制与决策,2004,19(1)：7～11,16.
[11]Antsaklis P. J, Nerode A. Hybrid control systems:an introductory discussion to the special issue[J]. IEEE Transactions on Automatic Control,1998,43(4):457～460.
[12]Persis C. D, Santis R. D, Morse A. S. Supervisory control with state-dependent dwell-time logic and constraints [J]. Automatica,2004,40(2):269～275.
[13]Lemmon M. D, He K. X, Markovsky I. Supervisory hybrid systems[J]. IEEE Control Systems Magazine,1999,19(4):42～55.
[14]王荣浩.一类时滞切换系统的分析与控制[D].南京理工大学硕士论文,2009.
[15]Liberzon D. Switching in systems and control[M]. Boston, Birkhauser,2003.
[16]邹洪波.切换线性系统稳定性若干问题研究[D].浙江大学博士学位论文,2007.
[17]Ooba T. Funahashi Y. On a common quadratic Lyapunov function for widely distant systems[J]. IEEE Transactions on Automatic Control,1997,42(12):1697～1699.
[18]赵佳,于志,申功璋.切换系统理论及其在飞行控制中的应用[c]／／中国航空学会控制与应用第十二届学术年会,2006,136～142.
[19]程代展,郭宇骞.切换系统进展[J].控制理论与应用,2005,22(6)：954～960.
[20]俞立.鲁棒控制一线性矩阵不等式处理方法,第1版[M].北京：清华大学出版社,2002.
[21]Liberzon D, Morse A. S. Basic problems in stability and design of switched systems[J]. IEEE Control Systems Magazine,1999,19(5):59～70.
[22]Liberzon D, Tempo R. Common Lyapunov functions and gradient algorithms[J]. IEEE Transactions on Automatic Control,2004,49(6):990～994.
[23]Narendra K. S, Balakrishnan J. A common Lyapunov function for stable LTI systems with commuting A-matrices[J]. IEEE Transactions on Automatic Control,1994,39(12): 2469～2471.
[24]Michel A. N, Hu B. Toward a stability theory of general hybrid dynamical systems[J]. Automatics,1999,35:371～384.
[25]Morse A. S. Supervisory control of families of linear set-point controllers. Part 1. Exact matching[J]. IEEE Transactions on Automatic Control,1996,41(10):1413～1431.
[26]Hespanha J. P, Morse A. S. Stability of switched systems with average dwell time[C]//In Proceedings of the 38th IEEE Conference on Decision and Control,1999,3:2655～2660.
[27]Liberzon D, Hespanha J. P, Morse A. S. Stability of switched systems:a Lie-algebraic condition[J]. Systems and Control Letters,1999,37:117～122.
[28]Peleties P, Decarlo R. A. Asymptotic stability of m-switched systems using Lypaunov-like functions[C]//Proceedings of American Control Conference, Boston, MA, USA,1991,1679～1684.
[29]Branicky M. S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[J]. IEEE Transactions on Automatic Control,1998,43(4):475～482.
[30]Daafouz J, Riedinger P, lung C. Stability analysis and control synthesis for switched systems:a switched Lyapunov function approach[J]. IEEE Transactions on Automatic Control,2002,47(11):1883-1887.
[31]Zhai G, Hu B, Yasuda K, Michel A. N. Disturbance attenuation properties of time-controlled switched systems[J]. Journal of the Franklin Institute,2001,338(7): 765～779.
[32]Zhai G, Hu B, Yasuda K, Michel A. N. Stability analysis of switched systems with stable and unstable subsystems:an average dwell time approach[C]//In Proceedings of the 2000 American Control Conference, Chicago, IL,2000,1(6):200～204.
[33]孙希明,齐丽,赵军.一类不确定线性系统的混杂状态反馈保成本控制[J].控制与决策,2005,20(4)：421～425.
[34]汪锐,赵军.一类不确定线性系统的混杂状态反馈可靠H∞控制[J].系统工程理论与实践,2006,26(12)：110～114.
[35]Bara G. I. Robust switched static output feedback control for discrete-time switched linear systems [C]//2007 46th IEEE Conference on Decision and Control, New Orleans, LA,2007,4986～4992.
[36]Ding D, Yang G. H∞ static output feedback control for discrete-time switched linear systems with average dwell time[C]//2009 American Control Conference, St. Louis, MO, USA,2009,2356～2361.
[37]Hong X, Xu H, Sun H. Fault-tolerant control of discrete-time switched systems[C]//2007 IEEE International Conference on Control and Automation, Guangzhou, China,2007, 3082～3086.
[38]Liu S, Zhou D. H. Reliable control for a class of switched systems with dwell time considerations against sensor faults[C]//2008 Chinese Control and Decision Conference, Yantai, Shandong,2008,490～495.
[39]Li L, Feng J, Dimirovski G. M, Zhao J. Reliable stabilization and H∞ control for switched systems with faulty actuators:an average dwell time approach[C]//2009 American Control Conference, St. Louis, MO, USA,2009,1072～1077.
[40]Li Q, Dimirovski G. M, Zhao J. Robust tracking control for switched linear systems with time-varying delays[J]. IET Control Theory & Applications,2008,2(6):449-457.
[41]Li Q, Dimirovski G. M, Zhao J. A. solution to the tracking control problem for switched linear systems with time-varying delays[C]//2008 47th IEEE Conference on Decision and Control, Cancun,2008,5348～5353.
[42]Chen W, Saif M. Observer design for linear switched control systems[C]//Proceeding of the 2004 American Control Conference, Boston,2004,6:5796～5801.
[43]Arash M, Ahmadreza M, Amir G, Aghdam, Peyman G. On observer design for a class of impulsive switched systems[C]//2008 American Control Conference, Seattle, WA, USA, 2008,4633～4639.
[44]Wu L, Zheng W. On H∞ model reduction of continuous time-delay switched systems[C]//Proceedings of the 2007 IEEE International Conference on Automation and Logistics, Jinan, China,2007,405-410.
[45]Zhang L, Shi P, Basin M. Model Reduction for switched linear parameter varying systems with average dwell time[C]//2008 American Control Conference, USA,2008, 3999-4004.
[46]Xiang Z. R, Wang R. H. Robust control for uncertain switched non-linear systems with time delay under asynchronous switching[J]. IET Control Theory Appl.,2009,3(8): 1041～1050.
[47]Xiang Z. R, Wang R. H. Robust stabilization of switched non-linear systems with time-varying delays under asynchronous switching[J]. J. Systems and Control Engineering. Proc. IMechE Vol.223 Part I:1111～1128.
[48]Wu L, Ho D. W. C. Reduced-order L2-L∞ filtering for a class of nonlinear switched stochastic systems[J]. IET Control Theory Appl.,2009,5(3):493～508.
[49]Shi P. Robust Kalman filtering for continuous-time systems with discrete-time measurements[J]. IMA J. Math. Control Inf.,1999,16(3):221～232.
[50]De Souza C. E, Trofino A. An LMI approach to the design of robust H2 filters. in recent advances on linear matrix inequality methods in Control[M], Philadelphia, PA,1999.
[51]Xu S. Robust H∞ filtering for a class of discrete-time uncertain nonlinear systems with state delay[J]. IEEE Trans. Circuits Syst. (I),2002,49:1853～1859.
[52]Zhang H, Chen Q, Yan H, Liu J. Robust H∞ filtering for switched stochastic system with missing measurements [J]. IEEE Transactions on Signal Processing,2009,57(9): 3466～3474.
[53]Zhao Y, Fu Y, Duan G. Robust passive filtering for switched systems with time-varying delays[C]//2007 Second IEEE Conference on Industrial Electronics and Applications, 2007:1350～1354.
[54]Liu X. Robust H∞ filtering for switched discrete-time systems with time-delays[C]// Proceedings of the 26th Chinese Control Conference,2007,660～664.
[55]Du D, Jiang B, Shi P, Zhou S. H∞ filtering of discrete-time switched systems with state delays via switched Lyapunov function approach[J]. IEEE Transactions on Automatic Control,2007,52(8):1520～1525.
[56]Zhang L, Shi P, Wang C, Gao H. Robust H∞ filtering for switched linear discrete-time Systems with polytopic uncertainties[J]. Int. J. Adapt. Control Signal Process,2006,20: 291～304.
[57]Hu K, Yuan J. Improved robust H∞ filtering for uncertain discrete-time switched systems[J]. IET Control Theory Appl.,2009,3(3):315～324.
[58]Zhang L, Shi P, Boukas E-K, Wang C. Robust l2-l∞ filtering for switched linear, discrete time-delay systems with polytopic uncertainties[J]. IET Control Theory Appl., 2007,1(3):722～730.
[59]Wu G, Wen Y. H∞ filtering for discrete-time systems with missing measurements [J]. Acta Automatica Sinca,2006,32(1):107-111.
[60]Wang Z, Yang F, Ho D. W. C, Liu X. Robust H∞ filtering for stochastic time-delay systems with missing measurements [J]. IEEE Transactions on Signal Processing,2006, 54(7):2579～2587.
[61]Xie D. M, Wang L, Hao F. Robust stability analysis and control synthesis for discrete-time uncertain switched systems[C]//Proceedings of the 42nd IEEE Conference on Decision and Control,2003,5:4812～4817.
[62]马克茂,王子才,史小平.一类非线性系统的观测器设计[J].控制理论与应用,1998,15(3)：443～446.
[63]Rajamani R. Observer for Lipschitz nonlinear systems[J]. IEEE Transactions on Automatic Control,1998,43(3):397～401.
[64]Zak S. H. On the stabilization and observation of nonlinear uncertain dynamic systems[J]. IEEE Transactions on Automatic Control,1990,35(5):604～607.
[65]Xiang Z. R, Wang R. H. Robust reliable control for uncertain switched nonlinear systems with time delay[C]//Proceedings of the 7th World Congress on Intelligent Control and Automation, China,2008,5487～5491.
[66]Wang C. H, Zhang L. X, Gao H. J, Wu L. G. Delay-dependent stability and stabilization of a class of linear switched time-varying delay systems[C]//Proceedings of 2005 International Conference on Machine Learning and Cybernetics, Guangzhou, China, 2005,2:917～922.
[67]Sun X., Zhao J, David J. H. Stability and L2-gain analysis for switched delay systems:a delay-dependent method[J]. Automatica,2006,42(10):1769～1774.
[68]Zhang Y, Liu X. Z, Shen X. M. Stability of switched systems with time delay[J]. Nonlinear Analysis:Hybrid Systems,2007,1(1):44～58.
[69]Halanay A. Differential Equations:Stability, Oscillations, Time Lags[M]. New York: Academic Press,1966.
[70]丛屾,费树岷,徐君祥.时滞切换系统稳定性分析与镇定—Lyapunov函数方法[J].控制理论与应用,2007,24(3)：465～470.
[71]Tseng C. S, Chen B. S. L∞-gain fuzzy control for nonlinear dynamic systems with persistent bounded disturbance[C]//Proceedings of 2004 IEEE International Conference on Fuzzy Systems,2004,2:783～788.
[72]Smith S. C, Peter S. Estimation with lossy measurements:jump estimators for jump systems[J]. IEEE Transactions on Automatic Control,2003,48(12):2163～2171.
[73]Tarn T, Yona R. Observers for nonlinear stochastic systems[J]. IEEE Transactions on Autobiatic Control,1976,21(4):441～448.
[74]Subramanian A, Sayed A. H. Multiobjective filter design for uncertain stochastic time-delay systems[J]. IEEE Transaction on Automatic Control,2004,49(1):149～154.
[75]Sun X, Wang D, Wang W, Yang G. Stability analysis and L2-gain of switched delay systems with stable and unstable subsystems[C]//22nd IEEE International Symposium on Intelligent Control Part of IEEE Multi-conference on Systems and Control, Singapore, 2007,208～213.
[2]Stiver J. A, Antsaklis P. J. Modeling and analysis of hybrid control systems[C]//1992 Proceedings of the 31st IEEE Conference on Decision and Control,1992,4:3748～3751.
[3]Balluchi A, Benedetto M. D, Pinello C, Rossi C. Cut-off in engine control:a hybrid system approach[C]//Proceedings of the 36th IEEE Conference on Decision and Control,1997,5: 4720～4725.
[4]Hall J. G, Delemos R. Extended real time logic for hybrid systems controller design[C]// IEEE Colloquium on Hybrid Control for Real Time Systems, London,1996:4/1～4/2.
[5]Zhang W, Branicky M. S, Phillips S. M. Stability of networked control systems[J]. IEEE Control Systems Magazine,2001,21(1):84～99.
[6]Hiskens I. A. Stability of hybrid system limit cycles:application to the compass gait biped robot[C]//In Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL,2001,1:774～779.
[7]李秀改,岳红,高东杰.复杂工业过程新型控制方法—混杂系统控制理论的研究[J].化工自动化及仪表,2001,8(5)：1～8.
[8]吴锋,刘文煌,郑应平.混杂系统方法及其在过程控制中的应用[J].清华大学学报(自然科学版),1997,37(11)：77～81.
[9]Tomlin C, Pappas G. J, Sastry S. Conflict resolution for air traffic management:a study in multi-agent hybrid systems[J]. IEEE Transactions on Automatic Control,1998,43(4):509～521.
[10]郑刚,谭民,宋永华.混杂系统的研究进展[J].控制与决策,2004,19(1)：7～11,16.
[11]Antsaklis P. J, Nerode A. Hybrid control systems:an introductory discussion to the special issue[J]. IEEE Transactions on Automatic Control,1998,43(4):457～460.
[12]Persis C. D, Santis R. D, Morse A. S. Supervisory control with state-dependent dwell-time logic and constraints [J]. Automatica,2004,40(2):269～275.
[13]Lemmon M. D, He K. X, Markovsky I. Supervisory hybrid systems[J]. IEEE Control Systems Magazine,1999,19(4):42～55.
[14]王荣浩.一类时滞切换系统的分析与控制[D].南京理工大学硕士论文,2009.
[15]Liberzon D. Switching in systems and control[M]. Boston, Birkhauser,2003.
[16]邹洪波.切换线性系统稳定性若干问题研究[D].浙江大学博士学位论文,2007.
[17]Ooba T. Funahashi Y. On a common quadratic Lyapunov function for widely distant systems[J]. IEEE Transactions on Automatic Control,1997,42(12):1697～1699.
[18]赵佳,于志,申功璋.切换系统理论及其在飞行控制中的应用[c]／／中国航空学会控制与应用第十二届学术年会,2006,136～142.
[19]程代展,郭宇骞.切换系统进展[J].控制理论与应用,2005,22(6)：954～960.
[20]俞立.鲁棒控制一线性矩阵不等式处理方法,第1版[M].北京：清华大学出版社,2002.
[21]Liberzon D, Morse A. S. Basic problems in stability and design of switched systems[J]. IEEE Control Systems Magazine,1999,19(5):59～70.
[22]Liberzon D, Tempo R. Common Lyapunov functions and gradient algorithms[J]. IEEE Transactions on Automatic Control,2004,49(6):990～994.
[23]Narendra K. S, Balakrishnan J. A common Lyapunov function for stable LTI systems with commuting A-matrices[J]. IEEE Transactions on Automatic Control,1994,39(12): 2469～2471.
[24]Michel A. N, Hu B. Toward a stability theory of general hybrid dynamical systems[J]. Automatics,1999,35:371～384.
[25]Morse A. S. Supervisory control of families of linear set-point controllers. Part 1. Exact matching[J]. IEEE Transactions on Automatic Control,1996,41(10):1413～1431.
[26]Hespanha J. P, Morse A. S. Stability of switched systems with average dwell time[C]//In Proceedings of the 38th IEEE Conference on Decision and Control,1999,3:2655～2660.
[27]Liberzon D, Hespanha J. P, Morse A. S. Stability of switched systems:a Lie-algebraic condition[J]. Systems and Control Letters,1999,37:117～122.
[28]Peleties P, Decarlo R. A. Asymptotic stability of m-switched systems using Lypaunov-like functions[C]//Proceedings of American Control Conference, Boston, MA, USA,1991,1679～1684.
[29]Branicky M. S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[J]. IEEE Transactions on Automatic Control,1998,43(4):475～482.
[30]Daafouz J, Riedinger P, lung C. Stability analysis and control synthesis for switched systems:a switched Lyapunov function approach[J]. IEEE Transactions on Automatic Control,2002,47(11):1883-1887.
[31]Zhai G, Hu B, Yasuda K, Michel A. N. Disturbance attenuation properties of time-controlled switched systems[J]. Journal of the Franklin Institute,2001,338(7): 765～779.
[32]Zhai G, Hu B, Yasuda K, Michel A. N. Stability analysis of switched systems with stable and unstable subsystems:an average dwell time approach[C]//In Proceedings of the 2000 American Control Conference, Chicago, IL,2000,1(6):200～204.
[33]孙希明,齐丽,赵军.一类不确定线性系统的混杂状态反馈保成本控制[J].控制与决策,2005,20(4)：421～425.
[34]汪锐,赵军.一类不确定线性系统的混杂状态反馈可靠H∞控制[J].系统工程理论与实践,2006,26(12)：110～114.
[35]Bara G. I. Robust switched static output feedback control for discrete-time switched linear systems [C]//2007 46th IEEE Conference on Decision and Control, New Orleans, LA,2007,4986～4992.
[36]Ding D, Yang G. H∞ static output feedback control for discrete-time switched linear systems with average dwell time[C]//2009 American Control Conference, St. Louis, MO, USA,2009,2356～2361.
[37]Hong X, Xu H, Sun H. Fault-tolerant control of discrete-time switched systems[C]//2007 IEEE International Conference on Control and Automation, Guangzhou, China,2007, 3082～3086.
[38]Liu S, Zhou D. H. Reliable control for a class of switched systems with dwell time considerations against sensor faults[C]//2008 Chinese Control and Decision Conference, Yantai, Shandong,2008,490～495.
[39]Li L, Feng J, Dimirovski G. M, Zhao J. Reliable stabilization and H∞ control for switched systems with faulty actuators:an average dwell time approach[C]//2009 American Control Conference, St. Louis, MO, USA,2009,1072～1077.
[40]Li Q, Dimirovski G. M, Zhao J. Robust tracking control for switched linear systems with time-varying delays[J]. IET Control Theory & Applications,2008,2(6):449-457.
[41]Li Q, Dimirovski G. M, Zhao J. A. solution to the tracking control problem for switched linear systems with time-varying delays[C]//2008 47th IEEE Conference on Decision and Control, Cancun,2008,5348～5353.
[42]Chen W, Saif M. Observer design for linear switched control systems[C]//Proceeding of the 2004 American Control Conference, Boston,2004,6:5796～5801.
[43]Arash M, Ahmadreza M, Amir G, Aghdam, Peyman G. On observer design for a class of impulsive switched systems[C]//2008 American Control Conference, Seattle, WA, USA, 2008,4633～4639.
[44]Wu L, Zheng W. On H∞ model reduction of continuous time-delay switched systems[C]//Proceedings of the 2007 IEEE International Conference on Automation and Logistics, Jinan, China,2007,405-410.
[45]Zhang L, Shi P, Basin M. Model Reduction for switched linear parameter varying systems with average dwell time[C]//2008 American Control Conference, USA,2008, 3999-4004.
[46]Xiang Z. R, Wang R. H. Robust control for uncertain switched non-linear systems with time delay under asynchronous switching[J]. IET Control Theory Appl.,2009,3(8): 1041～1050.
[47]Xiang Z. R, Wang R. H. Robust stabilization of switched non-linear systems with time-varying delays under asynchronous switching[J]. J. Systems and Control Engineering. Proc. IMechE Vol.223 Part I:1111～1128.
[48]Wu L, Ho D. W. C. Reduced-order L2-L∞ filtering for a class of nonlinear switched stochastic systems[J]. IET Control Theory Appl.,2009,5(3):493～508.
[49]Shi P. Robust Kalman filtering for continuous-time systems with discrete-time measurements[J]. IMA J. Math. Control Inf.,1999,16(3):221～232.
[50]De Souza C. E, Trofino A. An LMI approach to the design of robust H2 filters. in recent advances on linear matrix inequality methods in Control[M], Philadelphia, PA,1999.
[51]Xu S. Robust H∞ filtering for a class of discrete-time uncertain nonlinear systems with state delay[J]. IEEE Trans. Circuits Syst. (I),2002,49:1853～1859.
[52]Zhang H, Chen Q, Yan H, Liu J. Robust H∞ filtering for switched stochastic system with missing measurements [J]. IEEE Transactions on Signal Processing,2009,57(9): 3466～3474.
[53]Zhao Y, Fu Y, Duan G. Robust passive filtering for switched systems with time-varying delays[C]//2007 Second IEEE Conference on Industrial Electronics and Applications, 2007:1350～1354.
[54]Liu X. Robust H∞ filtering for switched discrete-time systems with time-delays[C]// Proceedings of the 26th Chinese Control Conference,2007,660～664.
[55]Du D, Jiang B, Shi P, Zhou S. H∞ filtering of discrete-time switched systems with state delays via switched Lyapunov function approach[J]. IEEE Transactions on Automatic Control,2007,52(8):1520～1525.
[56]Zhang L, Shi P, Wang C, Gao H. Robust H∞ filtering for switched linear discrete-time Systems with polytopic uncertainties[J]. Int. J. Adapt. Control Signal Process,2006,20: 291～304.
[57]Hu K, Yuan J. Improved robust H∞ filtering for uncertain discrete-time switched systems[J]. IET Control Theory Appl.,2009,3(3):315～324.
[58]Zhang L, Shi P, Boukas E-K, Wang C. Robust l2-l∞ filtering for switched linear, discrete time-delay systems with polytopic uncertainties[J]. IET Control Theory Appl., 2007,1(3):722～730.
[59]Wu G, Wen Y. H∞ filtering for discrete-time systems with missing measurements [J]. Acta Automatica Sinca,2006,32(1):107-111.
[60]Wang Z, Yang F, Ho D. W. C, Liu X. Robust H∞ filtering for stochastic time-delay systems with missing measurements [J]. IEEE Transactions on Signal Processing,2006, 54(7):2579～2587.
[61]Xie D. M, Wang L, Hao F. Robust stability analysis and control synthesis for discrete-time uncertain switched systems[C]//Proceedings of the 42nd IEEE Conference on Decision and Control,2003,5:4812～4817.
[62]马克茂,王子才,史小平.一类非线性系统的观测器设计[J].控制理论与应用,1998,15(3)：443～446.
[63]Rajamani R. Observer for Lipschitz nonlinear systems[J]. IEEE Transactions on Automatic Control,1998,43(3):397～401.
[64]Zak S. H. On the stabilization and observation of nonlinear uncertain dynamic systems[J]. IEEE Transactions on Automatic Control,1990,35(5):604～607.
[65]Xiang Z. R, Wang R. H. Robust reliable control for uncertain switched nonlinear systems with time delay[C]//Proceedings of the 7th World Congress on Intelligent Control and Automation, China,2008,5487～5491.
[66]Wang C. H, Zhang L. X, Gao H. J, Wu L. G. Delay-dependent stability and stabilization of a class of linear switched time-varying delay systems[C]//Proceedings of 2005 International Conference on Machine Learning and Cybernetics, Guangzhou, China, 2005,2:917～922.
[67]Sun X., Zhao J, David J. H. Stability and L2-gain analysis for switched delay systems:a delay-dependent method[J]. Automatica,2006,42(10):1769～1774.
[68]Zhang Y, Liu X. Z, Shen X. M. Stability of switched systems with time delay[J]. Nonlinear Analysis:Hybrid Systems,2007,1(1):44～58.
[69]Halanay A. Differential Equations:Stability, Oscillations, Time Lags[M]. New York: Academic Press,1966.
[70]丛屾,费树岷,徐君祥.时滞切换系统稳定性分析与镇定—Lyapunov函数方法[J].控制理论与应用,2007,24(3)：465～470.
[71]Tseng C. S, Chen B. S. L∞-gain fuzzy control for nonlinear dynamic systems with persistent bounded disturbance[C]//Proceedings of 2004 IEEE International Conference on Fuzzy Systems,2004,2:783～788.
[72]Smith S. C, Peter S. Estimation with lossy measurements:jump estimators for jump systems[J]. IEEE Transactions on Automatic Control,2003,48(12):2163～2171.
[73]Tarn T, Yona R. Observers for nonlinear stochastic systems[J]. IEEE Transactions on Autobiatic Control,1976,21(4):441～448.
[74]Subramanian A, Sayed A. H. Multiobjective filter design for uncertain stochastic time-delay systems[J]. IEEE Transaction on Automatic Control,2004,49(1):149～154.
[75]Sun X, Wang D, Wang W, Yang G. Stability analysis and L2-gain of switched delay systems with stable and unstable subsystems[C]//22nd IEEE International Symposium on Intelligent Control Part of IEEE Multi-conference on Systems and Control, Singapore, 2007,208～213.