Finite-time Boundedness Analysis And H_? Control For Several Classes Of Jumping Systems

Markov jump systems are a special kind of hybrid systems.The switching between the modes of the systems obeys Markov process.Markov jump systems have strong modeling ability for the actual systems whose the structures or/and parameters have random jumps.In recent years,Markov jump systems have attracted widespread attention and have been widely used in communication systems,network control systems,and industrial systems and so on.The transition rates are the key to the study of Markov jump systems.Currently,the investiga-tion of time-invariant transition rate matrix has been very mature.It can be divided into four categories:completely known,the bounded uncertain,partially unknown,generally uncertain transition rate matrix,respectively.However,for practical projects,time-varying is inevitable,so research on time-varying transition rates(i.e.nonhomogeneous Markov chains)are particu-larly important,and the research results for nonhomogeneous Markov jump systems can also enrich and improve the control theory.On the other hand,in control theory,Lyapunov stability stays for a long time in infinite time domain.However,due to the needs of practical problems,researchers are showing more and more interest in the state trajectories of dynamical systems over finite time intervals,that is,finite time stability 'has become one of the hottest research issues today.On the other hand,in actual operation,the existence of external disturbance is a source of instability and often causes undesirable performance,even makes system out of con-trol.In order to reject the instability of systems caused by disturbance,the state feedback H?control problem has been extensively studied.Therefore,the purposes of this paper are to summarize the existing research results and further discuss the problems of the finite-time boundedness and finite-time control for sev-eral Markov jump systems.It mainly includes four aspects,the problems of robust finite time H? control for uncertain Markov jump systems and uncertain Markov jump time-varying de-lay systems,the problems of stability analysis and controller synthesis for nonhomogeneous Markov jump systems,the problems of finite time H? control for nonhomogeneous Markov jump systems.The first two issues mainly discuss generally uncertain transition rates.The last two focus on the time-varying transition rates are constrained by polytopes.The details are as follows:1.The first chapter mainly introduces the basic concepts,research status and research significance of Markov jump systems,and introduces the stability,finite-time stability,transi-tion rate matrix,linear matrix inequalities,H? control,time delay and other basic concepts,respectively.2.The second chapter is concerned with the problems of robust finite-time H? control for a class of uncertain continuous-time Markov jump systems with generally uncertain transition rate matrix.By constructing Lyapunov functional,sufficient conditions ensuring the finite-time boundedness is developed for given systems for the first time.Broaden some existing conclu-sions and further design robust finite-time Hx state feedback controller,which guarantee Hx finite-time bounded for the closed-loop systems.Finally,one numerical example exemplifies the effectiveness of the proposed control methods3.The third chapter considers the problems of robust finite-time control for a class of uncertain Markov time-delay jump systems with generally uncertain transition rate matrix.Based on the results of former chapter.By constructing Lyapunov functional and introducing some free-weight matrices,sufficient conditions ensuring the finite-time boundedness is devel-oped for the given systems for the first time.Improve and broaden some existing conclusions and further design robust finite-time H? state feedback controllers,which guarantee the Hx finite-time bounded for the closed-loop system.Finally,two numerical examples exemplify the effectiveness and less conservativeness of the proposed control methods.4.The fourth chapter focuses on the problems of stability analysis and controller syn-thesis for a class of nonhomogeneous Markov jump systems with both state and input delays The time-varying transition rates are described as a polytope set.By constructing a Lyapnouv-Krasovskii functional and introducing some appropriately slack matrices,a stochastic stability condition is proposed in term of infinite matrix inequalities.Improve and broaden some existing conclusions and a new stochastic stability condition is further derived in form of a finite set of LMIs.Then,in terms of linear matrix inequalities techniques,the sufficient condition on the existence of state-feedback controller is presented and proved.Finally,two numerical exam-ples are given to illustrate the effectiveness of the theoretical results and the proposed design approaches.5.The fifth chapter mainly discusses the problems of finite-time H? control for a class of nonhomogeneous Markov jump systems,based on the discussion of the previous three chapters The time-varying transition rates are assumed to be polytope type.By constructing parameter-dependent Lyapunov functional,the sufficient condition of parameter-dependent finite-time boundedness is established and then according to projection lemma and the methods used in the former chapter to linearize the nonlinear parameter-dependent infinite dimension inequalities so that we can get H? disturbance level and finite-time H? controllers.Finally,two numerical examples show the effectiveness of the proposed methods.6.The last chapter summarizes the work done in the full text and points out the next research direction.