切换广义系统的稳定性与鲁棒控制
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摘要
切换广义系统是最近几年才发展起来的一个控制领域的研究方向，由于系统自身结构的复杂性，切换广义系统的研究相较于正常切换系统的研究要困难的多。目前切换系统的研究成果大多是针对正常切换系统展开的，对子系统全是广义系统的切换系统，研究成果相对较少。切换广义系统存在正则性、脉冲模去除、切换时刻状态相容等问题，与正常切换系统相比，它的分析与设计要更加复杂和困难，许多理论问题和实际应用问题亟待解决。因此，对切换广义系统展开进一步的研究和探索具有重要的理论意义和实际意义。
本文以切换广义系统为研究对象，分别对系统的Lyapunov稳定性，有限时间稳定性，输入-状态稳定性，以及系统的保成本控制和H_∞控制展开研究，采用的主要方法有公共Lyapunov函数方法结合凸组合技术，多Lyapunov函数方法，切换Lyapunov函数方法，平均驻留时间方法等。研究成果具有一定的创新性，对切换广义系统理论的发展是一个补充。全文共分七章，展开的主要工作概括如下：
1）讨论了切换广义系统的Lyapunov渐近稳定性问题。首先，基于时滞区间分割的思想，提出一种新型的多Lyapunov泛函，结合自由权矩阵方法和积分方法，对结构更为一般的含有不确定性和时滞的切换广义系统（即各个子系统导数矩阵不尽相同的切换广义系统）的稳定性问题进行了研究，给出了系统正则、无脉冲并且渐近稳定的新的判定条件；然后，分别对导数矩阵不尽相同的连续切换广义系统和离散切换广义系统，引入比例-导数反馈控制的思想，设计比例-导数反馈控制器，分别采用多Lyapunov函数方法和切换Lyapunov函数方法，给出了切换广义系统可镇定的充分条件。
2）讨论了切换广义系统的有限时间控制问题。首先，把正常系统有限时间稳定和有限时间有界的概念推广到连续切换广义系统上；然后，基于状态转移矩阵方法，给出了系统有限时间稳定的充分必要条件，但是这种方法不便于验证和控制器设计；为了解决这个问题，基于平均驻留时间方法结合多Lyapunov方法，给出了连续时间切换广义系统有限时间稳定和有限时间有界的充分条件；最后，基于切换Lyapunov函数方法，得到了在任意切换下离散时间切换广义系统的一致有限时间稳定和一致有限时间有界的充分条件。
3）讨论了一类非线性切换广义系统的输入-状态稳定性问题。从输入-状态稳定性的定义出发，利用平均驻留时间方法，结合Gronwall-Bellman不等式，分两类情况讨论了非线性切换广义系统的输入-状态稳定性问题，第一类情况是子系统全是输入-状态稳定的情形，第二类是子系统不全是输入-状态稳定的情形。在定理的证明过程中，无需构造系统输入-状态稳定的Lyapunov函数，也不需要设计控制输入的具体结构，只需要控制输入有界即可，通过设计适当的切换规则，就可以保证系统整体是输入-状态稳定的，与以往的控制器设计方法相比，在一定程度上大大降低了控制器设计的难度。
4）讨论了切换广义系统的保成本控制和鲁棒H_∞控制问题。首先，针对一类切换广义时滞系统，分别采用公共Lyapunov函数方法结合凸组合技术，以及多Lyapunov函数方法，给出了系统最优保成本控制器的设计方法；然后，采用多Lyapunov函数方法，分别利用状态反馈和静态输出反馈设计控制器，讨论了切换广义系统的H_∞控制问题，给出了控制器和切换规则的具体设计方法，与已有研究成果相比，研究对象更一般化，采用的研究方法保守性更小。
Switched descriptor system is a new research field of control theory developed in recentyears. Due to the complexity of the structure of the systems, the study of them is moredifficult than the study of the normal switched systems. At present, the study results of theswitched systems are mostly focused on normal switched systems, and the results are relativefewer for the systems whose subsystems are all descriptor systems. Compared with thenormal switched systems, there exists regularity, pulse, compatibility of the state at switchingtime and other issues in switched descriptor systems. So its analysis and design is morecomplex and more difficult, and many theoretical and practical problems need to be solved.Therefore, it has important theoretical and practical significance for the further researches andexplorations of switched descriptor systems.
Switched descriptor systems are the study object in this dissertation, and main researchesare focused on Lyapunov stability, finity-time stability, input-to-state stability, guranteed costcontrol and H-infinity control. The main methods adopted in this dissertation are commonLyapunov function method combined with convex combination technology, multipleLyapunov function method, switched Lyapunov function approach, and the average dwelltime method. The research results have a certain innovation, and these results are thesupplements for the study of the switched descriptor systems. The study works of thisdissertation are divided into seven chapters, and the main contents are as follows:
1) The problem of Lyapunov asymptotic stability for switched descriptor systems isdiscussed. First of all, based on the idea of partitioning the time delay, a new Lyapunovfunctional is proposed, and combined with the free weighting matrix method and integralmethod，the problem of stability of uncertain switched descriptor systems with time-delay (thederivative matrices of the subsystems are not all same) is considered, some new sufficientconditions, which can guarantee the system regular, impulse-free and asymptotically stable,are given. Then, introducing the idea of proportional-derivative feedback, and designingproportional-derivative feedback controllers, sufficient conditions of stabilizability are givenfor the continuous switched descriptor systems and discrete switched descriptor systemswhose derivative matrices of the subsystems are not all same.
2) The problem of finite-time control for switched descriptor systems is considered. Firstof all, the definitions of finite-time stability and finite-time boundedness of the normalsystems are extended to continuous and discrete switched descriptor systems. Then, based onthe state transition matrix method, the necessary and sufficient conditions of finite-timestability of the system are given, but this method is not convenient to design the controllers.Based on the average dwell time method and multiple Lyapunov method, sufficient conditionsare given to guarantee continuous switched descriptor systems finite time stable and finitetime bounded. Based on the switched Lyapunov function approach, sufficient conditions arederived, under which, discrete switched descriptor systems are regular, causal, uniformfinite-time bounded and uniform finite-time stable under arbitrary switching law.
3) The input-to-state stability problem is discussed for a class of nonlinear switcheddescriptor systems. The problem is divided into two cases. For the first case, each subsystemis input-to-state stable, and for the second case, some subsystems are not input-to-state stable,even unstable. Based on the dwell time approach and the Gronwall-Bellman inequality, thesufficient conditions are derived to guarantee the systems to be input-to-state stable from thedefinition of input-to-state stability. Compared with existing methods, it is not necessary toconstruct an input-state stability control Lyapunov function, and not required to design thespecific structure of the control input. Through designing the appropriate switching rule, thewhole system can be input-to-state stable. It can be seen that the difficulty of controllers'design is reduced to some extent.
4) The problem of guaranteed cost control and robust H-infinity control for switcheddescriptor systems is considered. First of all, by using the common Lyapunov function methodcombined with a convex combination technology, and multiple Lyapunov function method,the design method of optimal guaranteed cost controllers is given for a class of switcheddescriptor systems with time-delay. Compared with existing results, the methods used in thisdissertation are less conservative. Then, by the multiple Lyapunov function method, the statefeedback controllers and static output feedback controllers are designed respectively todiscuss the H-infinity control of the switched descriptor systems. The specific design methodsof controllers and switching law are obtained. Compared to the existed results, the model ofthe study is more general, and the methods adopted in this dissertation are less conservative.
本文以切换广义系统为研究对象，分别对系统的Lyapunov稳定性，有限时间稳定性，输入-状态稳定性，以及系统的保成本控制和H_∞控制展开研究，采用的主要方法有公共Lyapunov函数方法结合凸组合技术，多Lyapunov函数方法，切换Lyapunov函数方法，平均驻留时间方法等。研究成果具有一定的创新性，对切换广义系统理论的发展是一个补充。全文共分七章，展开的主要工作概括如下：
1）讨论了切换广义系统的Lyapunov渐近稳定性问题。首先，基于时滞区间分割的思想，提出一种新型的多Lyapunov泛函，结合自由权矩阵方法和积分方法，对结构更为一般的含有不确定性和时滞的切换广义系统（即各个子系统导数矩阵不尽相同的切换广义系统）的稳定性问题进行了研究，给出了系统正则、无脉冲并且渐近稳定的新的判定条件；然后，分别对导数矩阵不尽相同的连续切换广义系统和离散切换广义系统，引入比例-导数反馈控制的思想，设计比例-导数反馈控制器，分别采用多Lyapunov函数方法和切换Lyapunov函数方法，给出了切换广义系统可镇定的充分条件。
2）讨论了切换广义系统的有限时间控制问题。首先，把正常系统有限时间稳定和有限时间有界的概念推广到连续切换广义系统上；然后，基于状态转移矩阵方法，给出了系统有限时间稳定的充分必要条件，但是这种方法不便于验证和控制器设计；为了解决这个问题，基于平均驻留时间方法结合多Lyapunov方法，给出了连续时间切换广义系统有限时间稳定和有限时间有界的充分条件；最后，基于切换Lyapunov函数方法，得到了在任意切换下离散时间切换广义系统的一致有限时间稳定和一致有限时间有界的充分条件。
3）讨论了一类非线性切换广义系统的输入-状态稳定性问题。从输入-状态稳定性的定义出发，利用平均驻留时间方法，结合Gronwall-Bellman不等式，分两类情况讨论了非线性切换广义系统的输入-状态稳定性问题，第一类情况是子系统全是输入-状态稳定的情形，第二类是子系统不全是输入-状态稳定的情形。在定理的证明过程中，无需构造系统输入-状态稳定的Lyapunov函数，也不需要设计控制输入的具体结构，只需要控制输入有界即可，通过设计适当的切换规则，就可以保证系统整体是输入-状态稳定的，与以往的控制器设计方法相比，在一定程度上大大降低了控制器设计的难度。
4）讨论了切换广义系统的保成本控制和鲁棒H_∞控制问题。首先，针对一类切换广义时滞系统，分别采用公共Lyapunov函数方法结合凸组合技术，以及多Lyapunov函数方法，给出了系统最优保成本控制器的设计方法；然后，采用多Lyapunov函数方法，分别利用状态反馈和静态输出反馈设计控制器，讨论了切换广义系统的H_∞控制问题，给出了控制器和切换规则的具体设计方法，与已有研究成果相比，研究对象更一般化，采用的研究方法保守性更小。
Switched descriptor system is a new research field of control theory developed in recentyears. Due to the complexity of the structure of the systems, the study of them is moredifficult than the study of the normal switched systems. At present, the study results of theswitched systems are mostly focused on normal switched systems, and the results are relativefewer for the systems whose subsystems are all descriptor systems. Compared with thenormal switched systems, there exists regularity, pulse, compatibility of the state at switchingtime and other issues in switched descriptor systems. So its analysis and design is morecomplex and more difficult, and many theoretical and practical problems need to be solved.Therefore, it has important theoretical and practical significance for the further researches andexplorations of switched descriptor systems.
Switched descriptor systems are the study object in this dissertation, and main researchesare focused on Lyapunov stability, finity-time stability, input-to-state stability, guranteed costcontrol and H-infinity control. The main methods adopted in this dissertation are commonLyapunov function method combined with convex combination technology, multipleLyapunov function method, switched Lyapunov function approach, and the average dwelltime method. The research results have a certain innovation, and these results are thesupplements for the study of the switched descriptor systems. The study works of thisdissertation are divided into seven chapters, and the main contents are as follows:
1) The problem of Lyapunov asymptotic stability for switched descriptor systems isdiscussed. First of all, based on the idea of partitioning the time delay, a new Lyapunovfunctional is proposed, and combined with the free weighting matrix method and integralmethod，the problem of stability of uncertain switched descriptor systems with time-delay (thederivative matrices of the subsystems are not all same) is considered, some new sufficientconditions, which can guarantee the system regular, impulse-free and asymptotically stable,are given. Then, introducing the idea of proportional-derivative feedback, and designingproportional-derivative feedback controllers, sufficient conditions of stabilizability are givenfor the continuous switched descriptor systems and discrete switched descriptor systemswhose derivative matrices of the subsystems are not all same.
2) The problem of finite-time control for switched descriptor systems is considered. Firstof all, the definitions of finite-time stability and finite-time boundedness of the normalsystems are extended to continuous and discrete switched descriptor systems. Then, based onthe state transition matrix method, the necessary and sufficient conditions of finite-timestability of the system are given, but this method is not convenient to design the controllers.Based on the average dwell time method and multiple Lyapunov method, sufficient conditionsare given to guarantee continuous switched descriptor systems finite time stable and finitetime bounded. Based on the switched Lyapunov function approach, sufficient conditions arederived, under which, discrete switched descriptor systems are regular, causal, uniformfinite-time bounded and uniform finite-time stable under arbitrary switching law.
3) The input-to-state stability problem is discussed for a class of nonlinear switcheddescriptor systems. The problem is divided into two cases. For the first case, each subsystemis input-to-state stable, and for the second case, some subsystems are not input-to-state stable,even unstable. Based on the dwell time approach and the Gronwall-Bellman inequality, thesufficient conditions are derived to guarantee the systems to be input-to-state stable from thedefinition of input-to-state stability. Compared with existing methods, it is not necessary toconstruct an input-state stability control Lyapunov function, and not required to design thespecific structure of the control input. Through designing the appropriate switching rule, thewhole system can be input-to-state stable. It can be seen that the difficulty of controllers'design is reduced to some extent.
4) The problem of guaranteed cost control and robust H-infinity control for switcheddescriptor systems is considered. First of all, by using the common Lyapunov function methodcombined with a convex combination technology, and multiple Lyapunov function method,the design method of optimal guaranteed cost controllers is given for a class of switcheddescriptor systems with time-delay. Compared with existing results, the methods used in thisdissertation are less conservative. Then, by the multiple Lyapunov function method, the statefeedback controllers and static output feedback controllers are designed respectively todiscuss the H-infinity control of the switched descriptor systems. The specific design methodsof controllers and switching law are obtained. Compared to the existed results, the model ofthe study is more general, and the methods adopted in this dissertation are less conservative.
引文
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