Stability And Robust Control For Switched Descriptor Systems

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.