含有离散子系统的切换系统的稳定性分析
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摘要
切换系统是一类由若干个连续或离散子系统或连续和离散子系统和一个作用在其上的切换法则组成的混合系统。它有着广泛的实际背景和复杂的动态行为,从而引起了国内外学者的普遍关注。
BZ化学反应为目前非线性化学动力学中最重要的试验研究和理论分析对象之一,该反应以结构简单,非线性动力学行为丰富而受到人们的极大关注。
在本文中,我们主要讨论含有离散子系统的切换系统的稳定性。主要研究内容概括如下:
1.对一个一般的非自治非线性离散切换系统,我们利用多重Lyapunov函数和离散系统的比较原理给出了该离散切换系统双测度稳定的一个结果,当离散子系统的个数有限时,我们在此基础上又给出了一个更细致的结果。接着我们对带有外部输入的非线性离散切换系统的输入状态稳定、具有权重的输入状态稳定、具有权重的可和输入状态稳定进行了研究,利用平均驻留时间方法分别给出使这个离散切换系统输入状态稳定、具有权重的输入状态稳定、具有权重的可和输入状态稳定的充分条件。作为上述输入状态稳定结果的一个应用,离散串联系统全局渐近稳定的充分条件很容易被给出。
2.对一类离散切换系统-串联离散切换系统,我们先利用Brower不动点定理给出了其存在周期轨道的充分条件,接着我们讨论周期轨道的稳定性,得到了周期轨道稳定的一个结果,此外,我们对这类离散切换系统的一个特殊情形-只有一个子系统的情形给出周期轨道最小正周期和个数的估计,并举例说明这些结果是无切换时独有的。
3.对一类新切换系统-由连续和离散子系统组成的切换系统,我们首先利用多重Lyapunov函数、连续和离散系统的比较原理给出了这类切换系统双测度稳定的一个结果,当连续和离散子系统的个数都有限时,我们在这个结果的基础上给出了一个更细致的结果,其次利用平均驻留时间方法我们得到了这类切换系统全局渐近稳定的充分条件,最后对由线性时不变的连续和离散子系统组成的切换系统我们给出其在周期切换信号下全局渐近稳定性的一个结果。
4.同样对这类新切换系统-由连续和离散子系统组成的切换系统,我们讨论了它的鲁棒稳定性。首先对这类切换系统在多胞型摄动下的稳定性进行了研究,获得其二次稳定的充要条件,最后对这类由连续和离散子系统组成的切换系统的一个非线性情形即离散子系统有非线性扰动的情形我们给出了其状态在任意切换律下的一个估计,并从这个状态估计中我们获得了其在任意切换律下全局渐近稳定的一个充分条件。
此外,我们还研究了一个典型的BZ化学反应系统的稳定性和最终有界集估计。首先利用小增益定理给出该化学反应系统全局渐近稳定的充分条件,最后通过构造一个恰当的Lyapunov函数,我们获得了该化学反应系统最终有界集的一个估计。
Switched systems are an important class of hybrid systems which consists of several continuous-time subsystems or discrete-time subsystems or continuous-time and discrete-time subsystems and a rule that orchestrates among them. Switched systems have broad applications and show complicated behaviours because of the interaction between the continuous dynamics and the switching signals. Hence switched systems attract a lot of authors' much attention.
The BZ reaction is currently one of the most arresting chemical experiments and objects of theoretic research in nonlinear chemistry. It has received much attention because the BZ reaction is the simplest reaction, and can show rich dynamical behaviours.
In this dissertation we mainly study stability of switched systems with discrete-time subsytems. The main results are summarized as follows:
1. For a general nonlinear nonautonomous discrete-time switched system, we first use multiple Lyapunov functions and the comparison principle of discrete systems to give a result on stability in terms of two measures. When the number of subsystems is finite, based on the result we present a detailed result on stability in terms of two measures. Subsequently we study several input-to-state stable(ISS)-type properties (input-to-state stable,1/λ-weighted input-to-state stable,1/λ-weighted sum input-tostate stable) of discrete-time switched systems with inputs, and we give some new results on several input-to-state stable(ISS)-type properties under average dwell-time switching. As an application of the result on input-to-state stability, sufficient conditions for the global asymptotic stability of cascade discrete-time switched systems are easily obtained.
2. For a class of discrete-time switched systems-cascade switched systems, by means of the Brower fixed point theorem we first obtain a result on the existence of periodic orbits of this class of switched systems, and then we present a result on stability of periodic orbits. Furthermore we consider a particular case of this class of switched systems that this class of switched systems has one subsystem, and give some results on the number and the minimum period of peridic orbits. Some examples show that the results are unique for the particular case .
3. For a new type of switched systems-switched systems composed of continuous-time subsystems and discrete-time subsystems proposed in the literature, by combining the method of myultiple Lyapunov functions with the comparison principle of continuous-time systems and discrete-time systems we first present a result on stability in terms of two measures for such type of switched systems. When the numbers of continuous-time and discrete-time sunsystems are finite, on the basis of the above result we obtain a detailed result on stability in terms of two measures. Then we give a result on global asymptotic stability of such type of switched systems under average dwell-time switching. Moreover, for a special case of such type of switched systems-switched sytems composed of continuous-time LTI(linear time invariant)and discrete-time LTI subsystems, a result on global asymptotic stability of the switched systems under periodic switching signals is obtained.
4. For switched systems composed of continuous-time subsystems and discrete-time subsystems, we discuss their robust stability. First we give a sufficient and necessary condition for quadratic stability of the linear case of the switched systems with polytopic pertubations, and then for a nonlinear case of the switched systems in which there are nonlinear pertubations in discrete-time subsystems we present an estimate of state under arbitrary switching laws, and from the estimate a sufficient condition for global asymptotic stability under arbitrary switching laws is easily obtained.
Moreover, we investigate stability of the typical BZ reaction system and estimates ofthe ultimate bounded set, by means of the small gain theorem we first present sufficient conditions for global asymptotic stability of an equilibrium point of the chemical system . Then we give an estimate of the ultimate bounded set of the chemical system.
BZ化学反应为目前非线性化学动力学中最重要的试验研究和理论分析对象之一,该反应以结构简单,非线性动力学行为丰富而受到人们的极大关注。
在本文中,我们主要讨论含有离散子系统的切换系统的稳定性。主要研究内容概括如下:
1.对一个一般的非自治非线性离散切换系统,我们利用多重Lyapunov函数和离散系统的比较原理给出了该离散切换系统双测度稳定的一个结果,当离散子系统的个数有限时,我们在此基础上又给出了一个更细致的结果。接着我们对带有外部输入的非线性离散切换系统的输入状态稳定、具有权重的输入状态稳定、具有权重的可和输入状态稳定进行了研究,利用平均驻留时间方法分别给出使这个离散切换系统输入状态稳定、具有权重的输入状态稳定、具有权重的可和输入状态稳定的充分条件。作为上述输入状态稳定结果的一个应用,离散串联系统全局渐近稳定的充分条件很容易被给出。
2.对一类离散切换系统-串联离散切换系统,我们先利用Brower不动点定理给出了其存在周期轨道的充分条件,接着我们讨论周期轨道的稳定性,得到了周期轨道稳定的一个结果,此外,我们对这类离散切换系统的一个特殊情形-只有一个子系统的情形给出周期轨道最小正周期和个数的估计,并举例说明这些结果是无切换时独有的。
3.对一类新切换系统-由连续和离散子系统组成的切换系统,我们首先利用多重Lyapunov函数、连续和离散系统的比较原理给出了这类切换系统双测度稳定的一个结果,当连续和离散子系统的个数都有限时,我们在这个结果的基础上给出了一个更细致的结果,其次利用平均驻留时间方法我们得到了这类切换系统全局渐近稳定的充分条件,最后对由线性时不变的连续和离散子系统组成的切换系统我们给出其在周期切换信号下全局渐近稳定性的一个结果。
4.同样对这类新切换系统-由连续和离散子系统组成的切换系统,我们讨论了它的鲁棒稳定性。首先对这类切换系统在多胞型摄动下的稳定性进行了研究,获得其二次稳定的充要条件,最后对这类由连续和离散子系统组成的切换系统的一个非线性情形即离散子系统有非线性扰动的情形我们给出了其状态在任意切换律下的一个估计,并从这个状态估计中我们获得了其在任意切换律下全局渐近稳定的一个充分条件。
此外,我们还研究了一个典型的BZ化学反应系统的稳定性和最终有界集估计。首先利用小增益定理给出该化学反应系统全局渐近稳定的充分条件,最后通过构造一个恰当的Lyapunov函数,我们获得了该化学反应系统最终有界集的一个估计。
Switched systems are an important class of hybrid systems which consists of several continuous-time subsystems or discrete-time subsystems or continuous-time and discrete-time subsystems and a rule that orchestrates among them. Switched systems have broad applications and show complicated behaviours because of the interaction between the continuous dynamics and the switching signals. Hence switched systems attract a lot of authors' much attention.
The BZ reaction is currently one of the most arresting chemical experiments and objects of theoretic research in nonlinear chemistry. It has received much attention because the BZ reaction is the simplest reaction, and can show rich dynamical behaviours.
In this dissertation we mainly study stability of switched systems with discrete-time subsytems. The main results are summarized as follows:
1. For a general nonlinear nonautonomous discrete-time switched system, we first use multiple Lyapunov functions and the comparison principle of discrete systems to give a result on stability in terms of two measures. When the number of subsystems is finite, based on the result we present a detailed result on stability in terms of two measures. Subsequently we study several input-to-state stable(ISS)-type properties (input-to-state stable,1/λ-weighted input-to-state stable,1/λ-weighted sum input-tostate stable) of discrete-time switched systems with inputs, and we give some new results on several input-to-state stable(ISS)-type properties under average dwell-time switching. As an application of the result on input-to-state stability, sufficient conditions for the global asymptotic stability of cascade discrete-time switched systems are easily obtained.
2. For a class of discrete-time switched systems-cascade switched systems, by means of the Brower fixed point theorem we first obtain a result on the existence of periodic orbits of this class of switched systems, and then we present a result on stability of periodic orbits. Furthermore we consider a particular case of this class of switched systems that this class of switched systems has one subsystem, and give some results on the number and the minimum period of peridic orbits. Some examples show that the results are unique for the particular case .
3. For a new type of switched systems-switched systems composed of continuous-time subsystems and discrete-time subsystems proposed in the literature, by combining the method of myultiple Lyapunov functions with the comparison principle of continuous-time systems and discrete-time systems we first present a result on stability in terms of two measures for such type of switched systems. When the numbers of continuous-time and discrete-time sunsystems are finite, on the basis of the above result we obtain a detailed result on stability in terms of two measures. Then we give a result on global asymptotic stability of such type of switched systems under average dwell-time switching. Moreover, for a special case of such type of switched systems-switched sytems composed of continuous-time LTI(linear time invariant)and discrete-time LTI subsystems, a result on global asymptotic stability of the switched systems under periodic switching signals is obtained.
4. For switched systems composed of continuous-time subsystems and discrete-time subsystems, we discuss their robust stability. First we give a sufficient and necessary condition for quadratic stability of the linear case of the switched systems with polytopic pertubations, and then for a nonlinear case of the switched systems in which there are nonlinear pertubations in discrete-time subsystems we present an estimate of state under arbitrary switching laws, and from the estimate a sufficient condition for global asymptotic stability under arbitrary switching laws is easily obtained.
Moreover, we investigate stability of the typical BZ reaction system and estimates ofthe ultimate bounded set, by means of the small gain theorem we first present sufficient conditions for global asymptotic stability of an equilibrium point of the chemical system . Then we give an estimate of the ultimate bounded set of the chemical system.
引文
[1]王联,望慕秋.非线性常微分方程定性分析.哈尔滨:哈尔滨工业大学出版社,1987
[2]Shtesse,Pzanopolov,Ozerov.Control of multiple modular dc-to-dc power converters in conventional and dynamic sliding surfaces.IEEE Trans.Circuits Systems,1998,45(10):1091-1100
[3]蔡自兴.机器人学.北京:清华大学出版社,2000
[4]Pravin V.Smart cars on smart roads:problems of controls.IEEE Trans.on Automatic Control,1993,38(2):195-207
[5]Brockett R.W.Hybird modes for motion control systems,In Trentelman H L,Willems J C(Eds.),Essays in control.Boston:Birkhauser,1993
[6]Back A.,Guckenheimer J.,Myers M.A.Dynamical simulation facility for hybrid system.In R.L.Grossman,Nerode A,Ravn A P,H.Rishel(Eds.),Hybird systems,New York:Springer,1993
[7]Zhang L.J.,Cheng D.Z.,Li C.W.Disturbance decoupling of switched nonlinear systems.Proceedings of the 23rd Chinese Control Conference,2004:1591-1595
[8]Niconllin X.,Olivero A.,Sifakis J.,et al.An approach to description and analysis of hybrid systems,LNCS 736,Springer-Verlag,1993
[9]Chase C.,Serrano J.,Ramadge P.J.Periodicity chaos from switched flow systems:contrasting examples of discretely controlled continuous systems.IEEE Trans.on Automatic Control,1993,38(1):70-83
[10]Brockett R.W.Asymptotic stability and feekback stabilization,Differential Geometric Control Theory,Boston:Birkhauser,1983
[11]Yang X.S.,Li Q.D.Chaotic attractors in a simple hybrid system.International Journal of Bifurcation and Chaos,2002,12(10):2255-2256
[12]Yang X.S.,Li Q.D.Horseshoes in a new switching circuit via Wien-Bridge oscillator.International Journal of Bifurcation and Chaos,2005,15(7):2271-2275
[13]Yang X.S.,Li Q.D.Chaos generator via Wien-Bridge oscillator.Electronics letters,2002,38(13):623-625
[14]Alur R.,Courcoubetis C.,Halbwachs N.,et al.The algorithmic analysis of hybrid systems.Theoretical Computer Science,1995,138(1):3-34
[15]Shfindas S.,Evans R.J.,Savkin A.V.,Petersen I.R.Stability results for switched controller systems.Automatica,1999,35(4):553-564
[16]Morse A.S.Supervisory control of families of linear set-point controllers-part 1:Exact Matching.IEEE Transactions on Automatic Control,1996,41(10):1413-1431
[17]Morse A.S.Supervisory control of families of linear set-point controllers-part 2:Robustness.IEEE Transactions on Automatic Control,1997,42(11):1500-1515
[18]Billings K.Switchmode Power Supply Handbook,New York:McGraw-Hill,1989
[19]Whitaker J.C.The Electronics Handbook,West Plato Beach,FL:CRC Press,1966
[20]Allison A.,Abbott D.Some benefits of random variables in switched control systems,Microeletromics Journal,2000,31(7):515-522
[21]Kolmanovsky I.V.,McClamroch.Developments in nonholonomic control problems,Contr.Syst.Mag.,1995,15(6):20-36
[22]Hespanha M.S.Logic-Based switching algorithms in contor,PhD thesis,Yale University,New Haven,CT.1998
[23]Xu X.P.Analysis and design of switched systems.PhD thesis,Unversity of Notre Dame.2002
[24]孙文安.几类线性切换系统的鲁棒控制.博士论文,东北大学,2005
[25]Xie G.M.,Wang L.Periodical stabilization of switched linear systems.Journal of Computational and Applied Mathematics.2005,181(1):176-187
[26]Zhai G.S.,Lin H.,Michel A.N.,Yasuda K.Stability Analysis for Switched systems with Continuous-Time and Discrete-Time Subsystems.Proceedings of the 2004 American Control Conference,Boston,Massachusetts,USA,2004,5:4555-4560
[27]Zhai G.S.,Liu D.R.,Imae J.,Kobayashi T.Lie Algebraic Stability Analysis for Switched systems with Continuous-Time and Discrete-Time subsystems.IEEE Transactions on Automatic Control,2006,53(2):152-156
[28]Zhai G.S.,Xu X.P.,Lin H.,Liu D.R.An Extension of Lie Algebraic Stability Analysis for switched systems with Continuous-Time and Discrete-Time subsystems.Proceedings of the 2006 IEEE International Conference on Networking,Sensing and Control,2006:362-367
[29]Decarlo R.A.,Branicky M.S.,Pettersson S.,Lennartson B.Perspectives and Results on the Stability and Stabilizability of Hybrid Systems.Proceedings of the IEEE,2000,88(7):1069-1082
[30]Liberzon D.,Morse A.S.Basic problems in stability and design of switched systems.IEEE Control Systems Magazine,1999,19(5):59-70
[31]Bdondel V.,Megretski A.Unsolved problems in mathematical system and control theory.New Jersey:Princeton Universitiy Press,2004
[32]Narendra K.S.,Balakrishnan J.Common Lyapunov functions for stable LTI systems with commuting A-matrices.IEEE Transactions on Automatic Control,1994,39(12):2469-2471
[33]Liberzon D.Control using logic and switching,part Ⅰ:switching in systems and control.Tutorial workship,the 40th con%fence on Decision and Control,Orlando,FL,Dec 2001
[34]Liberzon D.,Hespanha J.P.,Morse A.S.Stability of switched systems:A Liealgebra condition.Systems and Control Letters,1999,37(3):117-122
[35]Agrachev A.A.,Liberzon D.Lie-algebraic stability criteria for switched systems.SIAM Journal on Control and Optimization,2002,40(1):253-269
[36]Mancilla-Aguilar J.L.Condition for the stability of switched nonlinear systems.IEEE Transactions on Automatic Control,2000,45(11):2077-2079
[37]Margaliot M.,Liberzon D.Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions.Systems and Control Letters,2006,55(1):8-16
[38]Zhai G.S.,Chen X.K.,Ikeda M.,Yasuda K.Stability and L_2 gain analysis of switched symmetric systems.Proceedings of the IEEE Conference on Decision and Control,2002,4:4395-4400
[39]Zhai G.S.,Lin H.,Xu X.P.et al.Analysis of switched normal discrete-time systems.Nonlinear Analysis,2007,66(8):1788-1799
[40]Dayawansa W.P.,Martin C.F.A converse Lyapunov theorem for a class of dynautical systems which undergo switching.IEEE Trans.on Automatic Control,1999,44(4):751-760
[41]Molchanov A.P.,Pyatnitskiy Y.S.Criteria of absolute stability of differential and difference inclusions encountered in control theory,Systems and control letters,1989,13(1):59-61
[42]Mancilla-Aguilar J.L.,Carcia R.A.On converse Lyapunov theorem for ISS and iISS switched nonlinear systems.Systems and Control Letters,2001,42(1):47-53
[43]Meilakhs A.M.Design of stable control systems subject to parametric perturbation.Automat.Remote Control,1978,39(10):1409-1418.
[44]Vu L.,Liberzon D.Common Lyapunov functions for families of commuting nonlinear systems.Systems and Control Letters,2005,54(5):405-416
[45]Mancilla-Aguilar J.L.,Carcia R.A.A converse Lyapunov theorem for nonlinear switched systems.Systems and Control Letters,2000,41(1):67-71
[46]Shim H.,Nob D.J.,Seo J.H.Common Lyapunov functions for exponentially stable nonlinear systems.Presented at the 4th SIAM Conference on Control and its Applications,Florida,1998
[47]Hespanha J.P.,Morse A.S.Stability of switched systems with average dwell-time.Proceedings of the 38th IEEE Conference on Decision and Control,Phoenix,AZ,USA,1999,3:2655-2660
[48]Hu B.,Xu X.P.,Michael A.N.,Antsaklis P.J.Stability analysis for a class of nonlinear switched systems.Proceedings of the 38th IEEE Conference on Decision and Control,Phoenix,AZ,USA,1999,5:4374-4379
[49]Zhai G.S.,Hu B.,Yasuda K.,Michal A.N.Stability analysis of switched systems with stable and unstable subsystems:An average dwell time approach.International Journal of Systems Science,2002,32(8):1055-1061
[50]Zhai G.S.,Hu B.,Yasuda K.,Michael A.N.Qualitative analysis of discretetime switched systems.Proceedings of the American Control Conferrence,2002,3:1880-1885
[51]Vu L.,Chatterjee D.,Liberzon D.Input-to-state stability of switched systems and switching adaptive control.Automatica,2007,43(4):639-646
[52]Persis C.D.,Santis R.D.,Morse A.S.Switched systems with state-dependent dwell-time.System and Control Letters,2003,50(4):291-302
[53]Peleties P.A.,Decarlo R.A.Asymptotic stability of m-switched systems using Lyapunov-like functions.Proceedings of the American Control Conference,Boston,MA,1991,2:1679-1684
[54]Branicky M.S.Stability of switched and hybrid systems,Proceedings of the 33rd IEEE Conference Decision and Control,LakeBuena Vista,FL,Dec,1994,4:3498-3503
[55]Branicky M.S.Multiple lyapunov function and other analysis tool for switched and hybrid systems,IEEE,Transactions on Automatic Control,1998,43(4):475-482
[56]Ye H,Michel A.N.,Lou L.Stability theory for hybrid dynamic1 systems,Proceedings of the 34th IEEE Conference on Decision and Control,1995,3:2679-2684
[57]Ye H.,Michel A.N.,Lou L.Stability theory for hybrid dynamicl systems,IEEE,Transactions on Automatic Control,1998,43(4):461-474
[58]Pettersson S.Analysis and design of hybrid systems.PhD thesis,Chalmers University to Technology,Sweden,1999
[59]Dogruel M.,Ozguner U.Stability of hybrid systems,IEEE International Symposium on Intelligent Control,1994,129-134
[60]Chatterjee D.,Liberzon D.Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions.SIAM Journal on Control and Optimization,2006,45(1):174-206
[61]Wicks M.A.,Peleties P.A.,Decarlo R.A.Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems.European Journal of Control,1998,4:140-147
[62]Wicks M.A.,Peleties P.A.,Decarlo R.A.Construction of piecewise Lyapunov functions for stabilizing switched systems.Proceedings of the 33th IEEE Conference on Decision and Control,Lake Buena Vista,FL,USA,1994,4:3492-3497
[63]Ferron E.Quadratic stabilizability of switched systems via state and output feedback.SIAM Journal on Control and Optimization,submitted(Technical Report CICS-P-468,MIT,1996)
[64]Zhai G.S.Quadratic stabilizability of discrete-time switched systems via state and output feedback.Proceedings of the 40th IEEE Conference on Decision and Control,Orlando,FL,2001,3:2165-2166
[65]Zhao J.,Dimirovski G.M.Quadratic stability of a class of switched nonlinear systems.IEEE Transactions on Automatic Control,2004,49(4):574-578
[66]孙文安,赵军.凸锥型不确定线性切换系统的二次镇定.控制理论与应用,2005,22(5):790-793
[67]Cheng D.Z.Stabilization of planar switching systems.Systems and Control Letters,2004,51(2):79-ss
[68]Zhao J.,Spong M.W.Hybrid Control for global stabilization of the cart-pendulum systems.Automatica,2001,37(12):1941-1951
[69]Branicky M.S.Multiple Lyapunov functions and other analysis tools for switched and hybrid systems.IEEE Transactions on Automatic Control,1998,43(4):475-482
[70]Hespanha J.P.Uniform stability of switched linear systems:Extensions of Lasell's invariance principle.IEEE Transactions on Automatic Control,2004,49(4):470-482
[71]Caines P.E.,Zhang J.On the adaptive control of jump parameter systems via nonlinear filtering.SIAM Journal on Control and Optimization,1995,33(6):1758-1777
[72]Cheng D.,Guo L.,Lin Y.,et al.Stabilization of switched linear systems.IEEE Transactions on Automatic Control,2005,50(8):1224-1228
[73]Cohen N.,Lewkowicz I.Rodman L.Exponential stability of triangular differential inclusion systems.Systems and Control Letters,1997,30(4):159-164
[74]Li Z.G.,Wen Z.Y.,Soh Y.C.Observer-based stabilization of switched linear systems.Automatica,2003,39(3):517-524
[75]王仁明,关治洪,刘新芝.一类非线性切换系统的稳定性分析.系统工程和电子技术,2004,26(1):68-71
[76]张霄力,刘玉忠,赵军.一类离散切换系统的渐近稳定.控制理论与应用,2002,19(5):774-776
[77]刘玉忠,赵军.一类非线性开关系统二次稳定性的充分必要条件.自动化学报,2002,28(4):596-600
[78]瞿长连,何苇,吴智铭.切换系统的稳定性及镇定控制器设计.信息与控制,2000,29(1):21-26
[79]谢广明、郑大钟.线性切换系统基于范数的系统镇定条件及算法.自动化学报,2001, 27(1):115-119
[80]Sun Y.G.,Wang L.,Xie G.M.Necessary and sufficient conditions for stabilization of discrete-time planar switched systems.Nonlinear Analysis,2006,65(5):1039-1049
[81]王泽宁,费树岷,冯纯伯.一类开关切换系统的输出反馈镇定.控制与决策,2003,18(2):169-172
[82]林相泽,田玉平.切换系统的不变性原理与不变集的状态反馈镇定.控制与决策,2005,20(2):127-131
[83]宗广灯,武玉强,杨洪勇.一类离散时间切换混杂系统鲁棒控制.控制理论与应用,2005,22(5):728-732
[84]卢建宁,张彦虎,赵光宙.离散时滞切换系统的无记忆状态反馈镇定.控制与决策,2006,21(8):941-944
[85]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
[86]陈松林,姚郁.一类离散时间线性切换系统的二次鲁棒镇定.信息与控制,2005,34(4):393-397
[87]高军伟,叶阳东,蒋秋华,史天运,贾利民.焦炉加热智能控制系统的研究与应用.信息与控制,2003,31(1):86-91
[88]程代展,郭宇骞.切换系统的进展.控制理论与应用,2005,22(6):954-960
[89]Ferino I.,Rombi E.Oscillating reaction,Catalysis Today,1999,52(2-3):291-305
[90]Belousov B.P.,Ref.Radiat.Med.1958.145
[91]Zhabotinskii A.M.Biofizika,1964,9:306
[92]冯长根,曾庆轩.化学震荡、混沌和化学波,北京理工大学出版社,2004
[93]Strzhak P.E.,Kawczynski A.L.Regularities in Complex Transient Oscillations in the Belousov-Zhabotinsky Reaction in a Batch Reactor.J.Phys.Chem.1995,99:10830-10833.
[94]Khavrus V.A.,Farkas H.,Strzhak P.E.Conditions for Mixed Mode Oscillations and Deterministic chaos in nonlinear chemical systems.Theoretical and Experimental Chemistry.2002,38(5):301-307.
[95]Huang Y.,Yang X.S.Numerical analysis on the complex dynamicas in a chemical system.Journal of Mathematical Chemistry.2006,39(2):377-387.
[96]Xie W.X.,Wen C.Y.,Li Z.G.Input-to-state stabilization of switched nonlinear systems.IEEE Transactions on Automatic Control,2001,46(7):1111-1116
[97]Lakshmikantham V.,liu X.Stability Analysis in Terms of Two Measures,World Scientific,Singapore,1995
[98]Lakshmikantham V.,Trigiante D.Theory of Difference equations:numerical methods and applications,Academic Press,New York:Marcel Dekker,2002
[99]Jiang Z.P.,Wang Y.Input-to-state stability for discrete-time nonlinear systems,Automatica,2001,37(6):857-869
[100]Savkin A.V.,Matveev A.S.A switched server systems of order with all its trajectories converging to limit cycles,Autornatica,2001,37(2):303-306
[101]Savkin A.V.,Matveev A.S.Existence and stability of periodic trajectories in switched server systems,Autornatica,2000,36(5):775-779
[102]Matveev A.S.,Savkin A.V.Limit cycles in switched single-server flow networks,Applied mathmatics letters,2002,15(2):251-256
[103]Agarwal R.,Meehan M.,O'Regan Donal,Fixed point theory and applications.London,Cambridge University Press,2001.
[104]Lakshmikantham V.,Leela S.Differential and integral inequalities:Theory and Application,Vol.1,Academic Press,New York,1969
[105]Zhai G.,Lin H.,et al.Quadratic stabilizability of switched systems with polytopic uncertainties,Int.J.Control,2003,76(7):747-753
[106]Hu B.,Xu X.P.,Michel A.N.,Antsaklis P.J.Stability Analysis for a class of nonlinear switched systems.Proceedings of the 38th Conference on Decision and Control,Phoenix,Arizon,USA,1999,5:4374-4379
[107]Isidori A.Nonlinear control theory.London,Springer-verlag,1999.
[2]Shtesse,Pzanopolov,Ozerov.Control of multiple modular dc-to-dc power converters in conventional and dynamic sliding surfaces.IEEE Trans.Circuits Systems,1998,45(10):1091-1100
[3]蔡自兴.机器人学.北京:清华大学出版社,2000
[4]Pravin V.Smart cars on smart roads:problems of controls.IEEE Trans.on Automatic Control,1993,38(2):195-207
[5]Brockett R.W.Hybird modes for motion control systems,In Trentelman H L,Willems J C(Eds.),Essays in control.Boston:Birkhauser,1993
[6]Back A.,Guckenheimer J.,Myers M.A.Dynamical simulation facility for hybrid system.In R.L.Grossman,Nerode A,Ravn A P,H.Rishel(Eds.),Hybird systems,New York:Springer,1993
[7]Zhang L.J.,Cheng D.Z.,Li C.W.Disturbance decoupling of switched nonlinear systems.Proceedings of the 23rd Chinese Control Conference,2004:1591-1595
[8]Niconllin X.,Olivero A.,Sifakis J.,et al.An approach to description and analysis of hybrid systems,LNCS 736,Springer-Verlag,1993
[9]Chase C.,Serrano J.,Ramadge P.J.Periodicity chaos from switched flow systems:contrasting examples of discretely controlled continuous systems.IEEE Trans.on Automatic Control,1993,38(1):70-83
[10]Brockett R.W.Asymptotic stability and feekback stabilization,Differential Geometric Control Theory,Boston:Birkhauser,1983
[11]Yang X.S.,Li Q.D.Chaotic attractors in a simple hybrid system.International Journal of Bifurcation and Chaos,2002,12(10):2255-2256
[12]Yang X.S.,Li Q.D.Horseshoes in a new switching circuit via Wien-Bridge oscillator.International Journal of Bifurcation and Chaos,2005,15(7):2271-2275
[13]Yang X.S.,Li Q.D.Chaos generator via Wien-Bridge oscillator.Electronics letters,2002,38(13):623-625
[14]Alur R.,Courcoubetis C.,Halbwachs N.,et al.The algorithmic analysis of hybrid systems.Theoretical Computer Science,1995,138(1):3-34
[15]Shfindas S.,Evans R.J.,Savkin A.V.,Petersen I.R.Stability results for switched controller systems.Automatica,1999,35(4):553-564
[16]Morse A.S.Supervisory control of families of linear set-point controllers-part 1:Exact Matching.IEEE Transactions on Automatic Control,1996,41(10):1413-1431
[17]Morse A.S.Supervisory control of families of linear set-point controllers-part 2:Robustness.IEEE Transactions on Automatic Control,1997,42(11):1500-1515
[18]Billings K.Switchmode Power Supply Handbook,New York:McGraw-Hill,1989
[19]Whitaker J.C.The Electronics Handbook,West Plato Beach,FL:CRC Press,1966
[20]Allison A.,Abbott D.Some benefits of random variables in switched control systems,Microeletromics Journal,2000,31(7):515-522
[21]Kolmanovsky I.V.,McClamroch.Developments in nonholonomic control problems,Contr.Syst.Mag.,1995,15(6):20-36
[22]Hespanha M.S.Logic-Based switching algorithms in contor,PhD thesis,Yale University,New Haven,CT.1998
[23]Xu X.P.Analysis and design of switched systems.PhD thesis,Unversity of Notre Dame.2002
[24]孙文安.几类线性切换系统的鲁棒控制.博士论文,东北大学,2005
[25]Xie G.M.,Wang L.Periodical stabilization of switched linear systems.Journal of Computational and Applied Mathematics.2005,181(1):176-187
[26]Zhai G.S.,Lin H.,Michel A.N.,Yasuda K.Stability Analysis for Switched systems with Continuous-Time and Discrete-Time Subsystems.Proceedings of the 2004 American Control Conference,Boston,Massachusetts,USA,2004,5:4555-4560
[27]Zhai G.S.,Liu D.R.,Imae J.,Kobayashi T.Lie Algebraic Stability Analysis for Switched systems with Continuous-Time and Discrete-Time subsystems.IEEE Transactions on Automatic Control,2006,53(2):152-156
[28]Zhai G.S.,Xu X.P.,Lin H.,Liu D.R.An Extension of Lie Algebraic Stability Analysis for switched systems with Continuous-Time and Discrete-Time subsystems.Proceedings of the 2006 IEEE International Conference on Networking,Sensing and Control,2006:362-367
[29]Decarlo R.A.,Branicky M.S.,Pettersson S.,Lennartson B.Perspectives and Results on the Stability and Stabilizability of Hybrid Systems.Proceedings of the IEEE,2000,88(7):1069-1082
[30]Liberzon D.,Morse A.S.Basic problems in stability and design of switched systems.IEEE Control Systems Magazine,1999,19(5):59-70
[31]Bdondel V.,Megretski A.Unsolved problems in mathematical system and control theory.New Jersey:Princeton Universitiy Press,2004
[32]Narendra K.S.,Balakrishnan J.Common Lyapunov functions for stable LTI systems with commuting A-matrices.IEEE Transactions on Automatic Control,1994,39(12):2469-2471
[33]Liberzon D.Control using logic and switching,part Ⅰ:switching in systems and control.Tutorial workship,the 40th con%fence on Decision and Control,Orlando,FL,Dec 2001
[34]Liberzon D.,Hespanha J.P.,Morse A.S.Stability of switched systems:A Liealgebra condition.Systems and Control Letters,1999,37(3):117-122
[35]Agrachev A.A.,Liberzon D.Lie-algebraic stability criteria for switched systems.SIAM Journal on Control and Optimization,2002,40(1):253-269
[36]Mancilla-Aguilar J.L.Condition for the stability of switched nonlinear systems.IEEE Transactions on Automatic Control,2000,45(11):2077-2079
[37]Margaliot M.,Liberzon D.Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions.Systems and Control Letters,2006,55(1):8-16
[38]Zhai G.S.,Chen X.K.,Ikeda M.,Yasuda K.Stability and L_2 gain analysis of switched symmetric systems.Proceedings of the IEEE Conference on Decision and Control,2002,4:4395-4400
[39]Zhai G.S.,Lin H.,Xu X.P.et al.Analysis of switched normal discrete-time systems.Nonlinear Analysis,2007,66(8):1788-1799
[40]Dayawansa W.P.,Martin C.F.A converse Lyapunov theorem for a class of dynautical systems which undergo switching.IEEE Trans.on Automatic Control,1999,44(4):751-760
[41]Molchanov A.P.,Pyatnitskiy Y.S.Criteria of absolute stability of differential and difference inclusions encountered in control theory,Systems and control letters,1989,13(1):59-61
[42]Mancilla-Aguilar J.L.,Carcia R.A.On converse Lyapunov theorem for ISS and iISS switched nonlinear systems.Systems and Control Letters,2001,42(1):47-53
[43]Meilakhs A.M.Design of stable control systems subject to parametric perturbation.Automat.Remote Control,1978,39(10):1409-1418.
[44]Vu L.,Liberzon D.Common Lyapunov functions for families of commuting nonlinear systems.Systems and Control Letters,2005,54(5):405-416
[45]Mancilla-Aguilar J.L.,Carcia R.A.A converse Lyapunov theorem for nonlinear switched systems.Systems and Control Letters,2000,41(1):67-71
[46]Shim H.,Nob D.J.,Seo J.H.Common Lyapunov functions for exponentially stable nonlinear systems.Presented at the 4th SIAM Conference on Control and its Applications,Florida,1998
[47]Hespanha J.P.,Morse A.S.Stability of switched systems with average dwell-time.Proceedings of the 38th IEEE Conference on Decision and Control,Phoenix,AZ,USA,1999,3:2655-2660
[48]Hu B.,Xu X.P.,Michael A.N.,Antsaklis P.J.Stability analysis for a class of nonlinear switched systems.Proceedings of the 38th IEEE Conference on Decision and Control,Phoenix,AZ,USA,1999,5:4374-4379
[49]Zhai G.S.,Hu B.,Yasuda K.,Michal A.N.Stability analysis of switched systems with stable and unstable subsystems:An average dwell time approach.International Journal of Systems Science,2002,32(8):1055-1061
[50]Zhai G.S.,Hu B.,Yasuda K.,Michael A.N.Qualitative analysis of discretetime switched systems.Proceedings of the American Control Conferrence,2002,3:1880-1885
[51]Vu L.,Chatterjee D.,Liberzon D.Input-to-state stability of switched systems and switching adaptive control.Automatica,2007,43(4):639-646
[52]Persis C.D.,Santis R.D.,Morse A.S.Switched systems with state-dependent dwell-time.System and Control Letters,2003,50(4):291-302
[53]Peleties P.A.,Decarlo R.A.Asymptotic stability of m-switched systems using Lyapunov-like functions.Proceedings of the American Control Conference,Boston,MA,1991,2:1679-1684
[54]Branicky M.S.Stability of switched and hybrid systems,Proceedings of the 33rd IEEE Conference Decision and Control,LakeBuena Vista,FL,Dec,1994,4:3498-3503
[55]Branicky M.S.Multiple lyapunov function and other analysis tool for switched and hybrid systems,IEEE,Transactions on Automatic Control,1998,43(4):475-482
[56]Ye H,Michel A.N.,Lou L.Stability theory for hybrid dynamic1 systems,Proceedings of the 34th IEEE Conference on Decision and Control,1995,3:2679-2684
[57]Ye H.,Michel A.N.,Lou L.Stability theory for hybrid dynamicl systems,IEEE,Transactions on Automatic Control,1998,43(4):461-474
[58]Pettersson S.Analysis and design of hybrid systems.PhD thesis,Chalmers University to Technology,Sweden,1999
[59]Dogruel M.,Ozguner U.Stability of hybrid systems,IEEE International Symposium on Intelligent Control,1994,129-134
[60]Chatterjee D.,Liberzon D.Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions.SIAM Journal on Control and Optimization,2006,45(1):174-206
[61]Wicks M.A.,Peleties P.A.,Decarlo R.A.Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems.European Journal of Control,1998,4:140-147
[62]Wicks M.A.,Peleties P.A.,Decarlo R.A.Construction of piecewise Lyapunov functions for stabilizing switched systems.Proceedings of the 33th IEEE Conference on Decision and Control,Lake Buena Vista,FL,USA,1994,4:3492-3497
[63]Ferron E.Quadratic stabilizability of switched systems via state and output feedback.SIAM Journal on Control and Optimization,submitted(Technical Report CICS-P-468,MIT,1996)
[64]Zhai G.S.Quadratic stabilizability of discrete-time switched systems via state and output feedback.Proceedings of the 40th IEEE Conference on Decision and Control,Orlando,FL,2001,3:2165-2166
[65]Zhao J.,Dimirovski G.M.Quadratic stability of a class of switched nonlinear systems.IEEE Transactions on Automatic Control,2004,49(4):574-578
[66]孙文安,赵军.凸锥型不确定线性切换系统的二次镇定.控制理论与应用,2005,22(5):790-793
[67]Cheng D.Z.Stabilization of planar switching systems.Systems and Control Letters,2004,51(2):79-ss
[68]Zhao J.,Spong M.W.Hybrid Control for global stabilization of the cart-pendulum systems.Automatica,2001,37(12):1941-1951
[69]Branicky M.S.Multiple Lyapunov functions and other analysis tools for switched and hybrid systems.IEEE Transactions on Automatic Control,1998,43(4):475-482
[70]Hespanha J.P.Uniform stability of switched linear systems:Extensions of Lasell's invariance principle.IEEE Transactions on Automatic Control,2004,49(4):470-482
[71]Caines P.E.,Zhang J.On the adaptive control of jump parameter systems via nonlinear filtering.SIAM Journal on Control and Optimization,1995,33(6):1758-1777
[72]Cheng D.,Guo L.,Lin Y.,et al.Stabilization of switched linear systems.IEEE Transactions on Automatic Control,2005,50(8):1224-1228
[73]Cohen N.,Lewkowicz I.Rodman L.Exponential stability of triangular differential inclusion systems.Systems and Control Letters,1997,30(4):159-164
[74]Li Z.G.,Wen Z.Y.,Soh Y.C.Observer-based stabilization of switched linear systems.Automatica,2003,39(3):517-524
[75]王仁明,关治洪,刘新芝.一类非线性切换系统的稳定性分析.系统工程和电子技术,2004,26(1):68-71
[76]张霄力,刘玉忠,赵军.一类离散切换系统的渐近稳定.控制理论与应用,2002,19(5):774-776
[77]刘玉忠,赵军.一类非线性开关系统二次稳定性的充分必要条件.自动化学报,2002,28(4):596-600
[78]瞿长连,何苇,吴智铭.切换系统的稳定性及镇定控制器设计.信息与控制,2000,29(1):21-26
[79]谢广明、郑大钟.线性切换系统基于范数的系统镇定条件及算法.自动化学报,2001, 27(1):115-119
[80]Sun Y.G.,Wang L.,Xie G.M.Necessary and sufficient conditions for stabilization of discrete-time planar switched systems.Nonlinear Analysis,2006,65(5):1039-1049
[81]王泽宁,费树岷,冯纯伯.一类开关切换系统的输出反馈镇定.控制与决策,2003,18(2):169-172
[82]林相泽,田玉平.切换系统的不变性原理与不变集的状态反馈镇定.控制与决策,2005,20(2):127-131
[83]宗广灯,武玉强,杨洪勇.一类离散时间切换混杂系统鲁棒控制.控制理论与应用,2005,22(5):728-732
[84]卢建宁,张彦虎,赵光宙.离散时滞切换系统的无记忆状态反馈镇定.控制与决策,2006,21(8):941-944
[85]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
[86]陈松林,姚郁.一类离散时间线性切换系统的二次鲁棒镇定.信息与控制,2005,34(4):393-397
[87]高军伟,叶阳东,蒋秋华,史天运,贾利民.焦炉加热智能控制系统的研究与应用.信息与控制,2003,31(1):86-91
[88]程代展,郭宇骞.切换系统的进展.控制理论与应用,2005,22(6):954-960
[89]Ferino I.,Rombi E.Oscillating reaction,Catalysis Today,1999,52(2-3):291-305
[90]Belousov B.P.,Ref.Radiat.Med.1958.145
[91]Zhabotinskii A.M.Biofizika,1964,9:306
[92]冯长根,曾庆轩.化学震荡、混沌和化学波,北京理工大学出版社,2004
[93]Strzhak P.E.,Kawczynski A.L.Regularities in Complex Transient Oscillations in the Belousov-Zhabotinsky Reaction in a Batch Reactor.J.Phys.Chem.1995,99:10830-10833.
[94]Khavrus V.A.,Farkas H.,Strzhak P.E.Conditions for Mixed Mode Oscillations and Deterministic chaos in nonlinear chemical systems.Theoretical and Experimental Chemistry.2002,38(5):301-307.
[95]Huang Y.,Yang X.S.Numerical analysis on the complex dynamicas in a chemical system.Journal of Mathematical Chemistry.2006,39(2):377-387.
[96]Xie W.X.,Wen C.Y.,Li Z.G.Input-to-state stabilization of switched nonlinear systems.IEEE Transactions on Automatic Control,2001,46(7):1111-1116
[97]Lakshmikantham V.,liu X.Stability Analysis in Terms of Two Measures,World Scientific,Singapore,1995
[98]Lakshmikantham V.,Trigiante D.Theory of Difference equations:numerical methods and applications,Academic Press,New York:Marcel Dekker,2002
[99]Jiang Z.P.,Wang Y.Input-to-state stability for discrete-time nonlinear systems,Automatica,2001,37(6):857-869
[100]Savkin A.V.,Matveev A.S.A switched server systems of order with all its trajectories converging to limit cycles,Autornatica,2001,37(2):303-306
[101]Savkin A.V.,Matveev A.S.Existence and stability of periodic trajectories in switched server systems,Autornatica,2000,36(5):775-779
[102]Matveev A.S.,Savkin A.V.Limit cycles in switched single-server flow networks,Applied mathmatics letters,2002,15(2):251-256
[103]Agarwal R.,Meehan M.,O'Regan Donal,Fixed point theory and applications.London,Cambridge University Press,2001.
[104]Lakshmikantham V.,Leela S.Differential and integral inequalities:Theory and Application,Vol.1,Academic Press,New York,1969
[105]Zhai G.,Lin H.,et al.Quadratic stabilizability of switched systems with polytopic uncertainties,Int.J.Control,2003,76(7):747-753
[106]Hu B.,Xu X.P.,Michel A.N.,Antsaklis P.J.Stability Analysis for a class of nonlinear switched systems.Proceedings of the 38th Conference on Decision and Control,Phoenix,Arizon,USA,1999,5:4374-4379
[107]Isidori A.Nonlinear control theory.London,Springer-verlag,1999.