下三角结构非线性切换系统的鲁棒镇定与干扰抑制
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摘要
切换系统是一类特殊而又重要的混杂系统,它有着重要的理论研究价值和广泛的工程应用背景,因此对切换系统的研究备受重视.由于切换系统中非线性的连续动态和离散的切换信号相互作用,使得切换系统的动态行为十分复杂,目前仍存在许多基本问题亟待解决.此外,切换系统的研究成果可为混杂系统的研究提供理论和方法上的借鉴和启示.对于控制系统的基本问题,切换系统的镇定问题、干扰抑制控制问题都是混杂系统控制领域中具有挑战性的研究课题.
本文主要利用Lyapunov理论、backstepping方法和adding one power integrator方法,在切换系统和非线性系统的分析和综合的基础上,系统地研究具有下三角结构的非线性切换系统的鲁棒镇定和干扰抑制问题.针对具有嵌入式下三角结构(nested lower triangular form)的非线性切换系统、具有下三角结构的高阶非线性切换系统中带有干扰、具有不确定性等情形,分别利用多Lyapunov函数方法、平均驻留时间方法,动态驻留时间方法,分别讨论鲁棒镇定和干扰抑制问题.本文主要工作包括如下几个方面：
(一)研究具有嵌入式下三角结构的非线性切换系统的镇定、鲁棒镇定问题.首先,利用backstepping方法同时设计各子系统的Lyapunov函数和相应的控制器.基于多Lyapunov函数方法,设计动态驻留时间的切换信号,使得闭环切换系统在此切换信号下是全局渐近稳定的.在驻留时间方法中,每个子系统被激活的最短时间是不变的.然而,在动态驻留时间方法中,被激活子系统的最小时间不仅依赖于切换时刻,还依赖于切换时刻前子系统的Lyapunov函数值和当前激活子系统的Lyapunov函数.激活的最小驻留时间是不断变化的,突破驻留时间方法的限制,这为设计问题提供了更大的空间和自由度.其次,在上述研究方法基础上,研究带有不确定项的具有嵌入式下三角结构的非线性切换系统的鲁棒镇定问题,其中不确定项应满足范数有界条件.
(二)研究一类具有嵌入式下三角结构的非线性切换系统的带有稳定性的干扰抑制问题.基于backstepping方法,递推设计各子系统的反馈控制器,同时得到各子系统的Lyapunov函数.基于多Lyapunov函数方法和平均驻留时间方法设计切换信号,使得闭环切换系统具有含权重的干扰抑制水平.
(三)研究一类具有下三角结构的高阶非线性切换系统的鲁棒镇定问题.其中各子系统不满足可反馈线性化的条件,且关于控制输入也不具有仿射形.应用adding one power integrator方法为每个子系统设计控制器,同时得到各子系统的Lyapunov函数,并利用多Lyapunov函数方法设计镇定系统的切换信号,使得整个闭环切换系统全局鲁棒镇定.
(四)对全文工作进行总结,同时展望下一步的工作.
As a special type and important class of hybrid systems, a switched system is of great significance both in theory research and engineer applications. Therefore, the study of switched systems has attracted increasing attention. The behavior of switched systems is very complicated because of the interaction between the continuous dynamics and discrete switching signals. So there are still many open problems which need to be studied. In addition, the research results of switched systems are also significant reference to hybrid systems in terms of theories and methods. As the basic problems of nonlinear control systems, both the stabilization and the disturbance attenuation properties of switched systems are challenging problems.
The Lyapunov theory, backstepping technique and the adding one power integrator technique are the major methods adopted in the study, and some other theories and methods of both switched systems and nonlinear control systems are also applied and extended to study the global robust stabilization and the disturbance attenuation properties for a class of switched nonlinear systems with lower triangular form. Based on multiple Lyapunov functions and dynamical dwell time, the stabilization and the robust stabilization are addressed for switched nonlinear systems with nested lower triangular form and uncertain switched systems with nested lower triangular form respectively. The disturbance attenuation properties are based on multiple Lyapunov functions and average dwell time. Meanwhile, the global robust stabilization is studied for high-order lower-triangular switch systems.
The main contributions of this thesis can be summarized as follows.
Firstly, we study the global stabilization and the global robust stabilization for a class of switched nonlinear systems with nested lower triangular form without uncertainty and with uncertainties respectively. The Lyapunov function and the state feedback controller for each subsystem will be simultaneously constructed by backstepping. Based on the multiple Lyapunov functions, a switching signal about dynamical dwell time is designed, which guarantees the closed-loop switched nonlinear system is global stable. According to the method above, the global robust stabilization will be considered for a class of uncertain switched nonlinear systems with nested lower triangular form. The uncertainties need to satisfy the norm-bounded conditions.
Secondly, we consider the problem of disturbance attenuation with stability for a class of switched nonlinear systems with nested lower triangular form. The state feedback controller for each subsystem and the Lyapunov function will be derived explicitly by backstepping. Under the switching signal of average dwell time, the closed-loop switched system has the weight disturbance attenuation from the disturbance input to the controlled output.
Thirdly, we address the robust stabilization for a class of switched nonlinear systems with high-order lower triangular form. All subsystems considered are feedback non-linearizable and non-affine in the control variables. Based on the method of adding one power integrator, the feedback control law and the Lyapunov function will be designed for each subsystem. The closed-loop switched nonlinear systems will be global robust approximation stabilize under the designed switching signal.
Finally, we summarize the whole dissertation and discuss the further research work.
本文主要利用Lyapunov理论、backstepping方法和adding one power integrator方法,在切换系统和非线性系统的分析和综合的基础上,系统地研究具有下三角结构的非线性切换系统的鲁棒镇定和干扰抑制问题.针对具有嵌入式下三角结构(nested lower triangular form)的非线性切换系统、具有下三角结构的高阶非线性切换系统中带有干扰、具有不确定性等情形,分别利用多Lyapunov函数方法、平均驻留时间方法,动态驻留时间方法,分别讨论鲁棒镇定和干扰抑制问题.本文主要工作包括如下几个方面：
(一)研究具有嵌入式下三角结构的非线性切换系统的镇定、鲁棒镇定问题.首先,利用backstepping方法同时设计各子系统的Lyapunov函数和相应的控制器.基于多Lyapunov函数方法,设计动态驻留时间的切换信号,使得闭环切换系统在此切换信号下是全局渐近稳定的.在驻留时间方法中,每个子系统被激活的最短时间是不变的.然而,在动态驻留时间方法中,被激活子系统的最小时间不仅依赖于切换时刻,还依赖于切换时刻前子系统的Lyapunov函数值和当前激活子系统的Lyapunov函数.激活的最小驻留时间是不断变化的,突破驻留时间方法的限制,这为设计问题提供了更大的空间和自由度.其次,在上述研究方法基础上,研究带有不确定项的具有嵌入式下三角结构的非线性切换系统的鲁棒镇定问题,其中不确定项应满足范数有界条件.
(二)研究一类具有嵌入式下三角结构的非线性切换系统的带有稳定性的干扰抑制问题.基于backstepping方法,递推设计各子系统的反馈控制器,同时得到各子系统的Lyapunov函数.基于多Lyapunov函数方法和平均驻留时间方法设计切换信号,使得闭环切换系统具有含权重的干扰抑制水平.
(三)研究一类具有下三角结构的高阶非线性切换系统的鲁棒镇定问题.其中各子系统不满足可反馈线性化的条件,且关于控制输入也不具有仿射形.应用adding one power integrator方法为每个子系统设计控制器,同时得到各子系统的Lyapunov函数,并利用多Lyapunov函数方法设计镇定系统的切换信号,使得整个闭环切换系统全局鲁棒镇定.
(四)对全文工作进行总结,同时展望下一步的工作.
As a special type and important class of hybrid systems, a switched system is of great significance both in theory research and engineer applications. Therefore, the study of switched systems has attracted increasing attention. The behavior of switched systems is very complicated because of the interaction between the continuous dynamics and discrete switching signals. So there are still many open problems which need to be studied. In addition, the research results of switched systems are also significant reference to hybrid systems in terms of theories and methods. As the basic problems of nonlinear control systems, both the stabilization and the disturbance attenuation properties of switched systems are challenging problems.
The Lyapunov theory, backstepping technique and the adding one power integrator technique are the major methods adopted in the study, and some other theories and methods of both switched systems and nonlinear control systems are also applied and extended to study the global robust stabilization and the disturbance attenuation properties for a class of switched nonlinear systems with lower triangular form. Based on multiple Lyapunov functions and dynamical dwell time, the stabilization and the robust stabilization are addressed for switched nonlinear systems with nested lower triangular form and uncertain switched systems with nested lower triangular form respectively. The disturbance attenuation properties are based on multiple Lyapunov functions and average dwell time. Meanwhile, the global robust stabilization is studied for high-order lower-triangular switch systems.
The main contributions of this thesis can be summarized as follows.
Firstly, we study the global stabilization and the global robust stabilization for a class of switched nonlinear systems with nested lower triangular form without uncertainty and with uncertainties respectively. The Lyapunov function and the state feedback controller for each subsystem will be simultaneously constructed by backstepping. Based on the multiple Lyapunov functions, a switching signal about dynamical dwell time is designed, which guarantees the closed-loop switched nonlinear system is global stable. According to the method above, the global robust stabilization will be considered for a class of uncertain switched nonlinear systems with nested lower triangular form. The uncertainties need to satisfy the norm-bounded conditions.
Secondly, we consider the problem of disturbance attenuation with stability for a class of switched nonlinear systems with nested lower triangular form. The state feedback controller for each subsystem and the Lyapunov function will be derived explicitly by backstepping. Under the switching signal of average dwell time, the closed-loop switched system has the weight disturbance attenuation from the disturbance input to the controlled output.
Thirdly, we address the robust stabilization for a class of switched nonlinear systems with high-order lower triangular form. All subsystems considered are feedback non-linearizable and non-affine in the control variables. Based on the method of adding one power integrator, the feedback control law and the Lyapunov function will be designed for each subsystem. The closed-loop switched nonlinear systems will be global robust approximation stabilize under the designed switching signal.
Finally, we summarize the whole dissertation and discuss the further research work.
引文
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[2]Decarlo RA, Branicky MS, Pettersson S, Lennartson B. Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE,2000,88(7):1069-1082
[3]Ooba T, Funahashi Y. On a common quadratic Lyapunov function for widely distant systems. IEEE Transactions on Automatic Control,1997,42(12):1697-1699
[4]Vu L, Liberzon D. Common Lyapunov functions for families of commuting nonlinear systems. Systems and Control Letters,2005,54(5):405-416
[5]Barkhordari Yazdi M, Jahed-Motlagh MR. Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization. Chemical Engineering Journal, 2009,155(3):838-843
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[11]Davoren JM, Nerode A. Logics for hybrid systems. Proceedings of the IEEE,2000,88 (7):985-1010
[12]Ji Z, Wang L, Guo X. Design of switching sequences for controllability realization of switched linear systems. Automatica,2007,43(4):662-668
[13]Laffey TJ, Smigoc H. Tensor conditions for the existence of a common solution to the Lyapunov equation. Linear Algebra and its Applications,2007,420 (2-3):672-685
[14]Gurvits L, Shorten R, Mason O. On the stability of switched positive linear systems. IEEE Transactions on Automatic Control,2007,52 (6):1099-1103
[15]Sun Z, Ge SS. Analysis and synthesis of switched linear control systems. Automatica,2005, 41(2):181-195
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[18]Bengea SC,DeCarlo RA.Optimal control of switching systems. Automatica,2005, 41(1):11-27
[19]Sun Y, Wang L, Xie G. 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
[20]Xie D, Wang L, Hao F, Xie G. LMI approach to L2-gain analysis and control synthesis of uncertain switched systems. IEEE Proceedings of Control Theory and Applications, 2004,151 (1):21-28
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[23]Zhao J, Hill DJ. Dissipativity theory for switched systems. IEEE Transactions on Automatic Control,2008,53(4):941-953
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[28]Zhao J, Hill DJ. On stability, L2-gain and Hm control for switched systems. Automatica, 2008,44 (5):1220-1232
[29]Morse AS. Supervisory control of families of linear set-point controllers Part 2. Robustness. IEEE Transactions on Automatic Control,1997,42 (11):1500-1515
[30]Zhai G, Hu B, Yasuda K, Michel AN. Stability analysis of switched systems with stable and unstable subsystems:An average dwell time approach. International Journal of Systems Science,2001,32 (8):1055-1061
[31]Vu L, Chatterjee D, Liberzon D. Input-to-state stability of switched systems and switching adaptive control. Automatica,2007,43 (4):639-646
[32]Xiang Z, Xiang W. Stability analysis of switched nonlinear systems based on dynamical dwell time approach. IEEE International Conference on Networking, Sensing and Control, 2008
[33]Cheng D. Stabilization of planar switched systems. Systems and Control Letters,2004,51 (2):79-88
[34]Moulay E, Bourdais R, Perruquetti W. Stabilization of nonlinear switched systems using control Lyapunov functions. Nonlinear Analysis:Hybrid Systems 2007,1 (4):482-490
[35]Wu J-L. Stabilizing controllers design for switched nonlinear systems in strict-feedback form. Automatica,2009,45 (4):1092-1096
[36]Lin H, Antsaklis PJ. Stability and stabilizability of switched linear systems:A survey of recent results. IEEE Transactions on Automatic Control,2009,54 (2):308-322
[37]付主木,费树岷,高爱云.切换系统的H∞控制.北京：科学出版社,2009
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[40]Wang M, Zhao J, Dimirovski MG.H∞ control for a class of cascade switched nonlinear systems. Asian Journal of Control,2008,10 (6):724-729
[41]Sun X-M, Zhao J, Hill DJ. Stability and L2-gain analysis for switched delay systems:A delay-dependent methord. Automatica,2006,42 (10):1769-1774
[42]Xie L. Output feedback H∞ control of systems with parameter uncertainty. International Journal of Control,1996,63 (4):741-750
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[2]Decarlo RA, Branicky MS, Pettersson S, Lennartson B. Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE,2000,88(7):1069-1082
[3]Ooba T, Funahashi Y. On a common quadratic Lyapunov function for widely distant systems. IEEE Transactions on Automatic Control,1997,42(12):1697-1699
[4]Vu L, Liberzon D. Common Lyapunov functions for families of commuting nonlinear systems. Systems and Control Letters,2005,54(5):405-416
[5]Barkhordari Yazdi M, Jahed-Motlagh MR. Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization. Chemical Engineering Journal, 2009,155(3):838-843
[6]Morse AS. Supervisory control of families of linear set-point controllers Part I. Exact matching. IEEE Transactions on Automatic Control,1996,41(10):1413-1431
[7]Balluchi A,Di Benedetto MD,Pinello C,Rossi C,Sangiovanni-Vincentelli A.Hybrid control in automotive applications:the cut-off control. Automatica.1999,35(3):519-535
[8]Narendra KS, Balakrishnan J. Adaptive control using multiple models. IEEE Transactions on Automatic Control,1997,42(2):171-187
[9]Branicky MS. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control,1998,43 (4):475-482
[10]Liberzon D. Switching in Systems and Control. Boston:Brikhauser,2003
[11]Davoren JM, Nerode A. Logics for hybrid systems. Proceedings of the IEEE,2000,88 (7):985-1010
[12]Ji Z, Wang L, Guo X. Design of switching sequences for controllability realization of switched linear systems. Automatica,2007,43(4):662-668
[13]Laffey TJ, Smigoc H. Tensor conditions for the existence of a common solution to the Lyapunov equation. Linear Algebra and its Applications,2007,420 (2-3):672-685
[14]Gurvits L, Shorten R, Mason O. On the stability of switched positive linear systems. IEEE Transactions on Automatic Control,2007,52 (6):1099-1103
[15]Sun Z, Ge SS. Analysis and synthesis of switched linear control systems. Automatica,2005, 41(2):181-195
[16]Zhao S, Zhao J, Dimirovski GM. Global robust H∞ control for non-minimum-phase uncertain nonlinear systems without strict triangular structure. July 4-8,2005, Prague
[17]Sun Z. Stabilizability and insensitivity of switched linear systems. IEEE Transactions on Automatic Control,2004,49 (7):1133-1137
[18]Bengea SC,DeCarlo RA.Optimal control of switching systems. Automatica,2005, 41(1):11-27
[19]Sun Y, Wang L, Xie G. 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
[20]Xie D, Wang L, Hao F, Xie G. LMI approach to L2-gain analysis and control synthesis of uncertain switched systems. IEEE Proceedings of Control Theory and Applications, 2004,151 (1):21-28
[21]Zhai G, Hu B, Yasuda K, Michel AN. Disturbance attenuation properties of time-controlled switched systems. Journal of the Franklin Institute,2001,338 (7):765-779
[22]Zhao J, Hill DJ. Passivity and stability of switched systems:A multiple storage function method. Systems and Control Letters,2008,57 (2):158-164
[23]Zhao J, Hill DJ. Dissipativity theory for switched systems. IEEE Transactions on Automatic Control,2008,53(4):941-953
[24]Narendra KS, Balakrishnan J. A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Transactions on Automatic Control,1994,39 (12):2469-2471
[25]Ye H, Michel AN, Hou L. Stability analysis of systems with impulse effects. IEEE Transactions on Automatic Control,1998,43(12):1719-1723
[26]Ooba T, Funahashi Y. Two conditions concerning common quadratic Lyapunov functions for linear systems. IEEE Transactions on Automatic Control,1997,42(5):719-722
[27]Ye H, Michel AN, Hou L. Stability theory for hybrid dynamical systems. IEEE Transactions on Automatic Control,1998,43 (4):461-474
[28]Zhao J, Hill DJ. On stability, L2-gain and Hm control for switched systems. Automatica, 2008,44 (5):1220-1232
[29]Morse AS. Supervisory control of families of linear set-point controllers Part 2. Robustness. IEEE Transactions on Automatic Control,1997,42 (11):1500-1515
[30]Zhai G, Hu B, Yasuda K, Michel AN. Stability analysis of switched systems with stable and unstable subsystems:An average dwell time approach. International Journal of Systems Science,2001,32 (8):1055-1061
[31]Vu L, Chatterjee D, Liberzon D. Input-to-state stability of switched systems and switching adaptive control. Automatica,2007,43 (4):639-646
[32]Xiang Z, Xiang W. Stability analysis of switched nonlinear systems based on dynamical dwell time approach. IEEE International Conference on Networking, Sensing and Control, 2008
[33]Cheng D. Stabilization of planar switched systems. Systems and Control Letters,2004,51 (2):79-88
[34]Moulay E, Bourdais R, Perruquetti W. Stabilization of nonlinear switched systems using control Lyapunov functions. Nonlinear Analysis:Hybrid Systems 2007,1 (4):482-490
[35]Wu J-L. Stabilizing controllers design for switched nonlinear systems in strict-feedback form. Automatica,2009,45 (4):1092-1096
[36]Lin H, Antsaklis PJ. Stability and stabilizability of switched linear systems:A survey of recent results. IEEE Transactions on Automatic Control,2009,54 (2):308-322
[37]付主木,费树岷,高爱云.切换系统的H∞控制.北京：科学出版社,2009
[38]Hespanha JP. Logic-based Switching Alogrithms in Control. New Haven:Yale University, 1998
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