Dynamics Analysis On Linear Switched System

Recently, a novel family of hybrid systems called switched system has attractedmore and more attention from many researchers. A general switched system consists ofseveral subsystems and a switching rule that orchestrates the switching among them.Due to the fact that a wide range of applications in economical, biological, physical andengineering systems can be modeled mathematically by certain switched systems fromthe real world, therefore, switched systems have been investigated extensively. Since Z.Sun (2002) resolved the problem of the controllability of linear switched systems, thedynamics analysis of linear switched systems has reached new levels. Undoubtedly,Dynamic analysis is the foundation problem of the control synthesis for switchedsystems and should be investigated deeply enough.Compared with the results of previous literatures for ordinary differential dynamicsystems, not only do we concentrate on each subsystem, but also on the switchingsequence during investigation of switched systems. As is known, switching amongdifferent subsystems may produce complex dynamical phenomena such as chaos andmultiple limit cycles. Consequently, we cannot analyse the dynamics of switchedsystems by virtue of those approaches which are used to investigate dynamicalproperties of ordinary differential equations. Although the switched systems have beeninvestigated extensively and quite a number of fundamental concepts and powerful toolshave been developed from various disciplines, obstacle remains, in which manysignificant problems are yet not explored or understood clearly. Hence, it is worthwhileto eliminate the problems involved in these hybrid systems.In this dissertation, controllability, observability and reachability of linear switchedsystems with continuous-time and discrete-time subsystems have been studied andrelated algebraic and geometrical criteria have been established. Meanwhile, somesubspace algorithms have been designed to search the reachable and unobservablesubspace. In addition, a category of linear switched impulsive systems have beenresearched deeply. Correspondingly, geometrical criteria of controllability andreachability have been established. Furthermore, in terms of invariance theory,reachable and controllable subspace algorithms have been obtained. By employing theapproaches which were adopted extensively in the investigation of switched systems,we have inspected linear impulsive systems and stability criteria. More specifically, the main research results of this dissertation are listed as follows:1Controllability and observability of time-varying linear switched systems withcontinuous-time and discrete-time subsystems have been investigated. Meanwhilesufficient and necessary conditions for controllability and observability have beenestablished, which are formulated in the form of matrix rank. Besides, some simple andpractical sufficient conditions have been deduced;②Combined with geometrical control theory, geometrical criteria of reachabilityand observability for time-invariant linear switched systems with continuous-time anddiscrete-time subsystems have been established. And two simple subspace algorithmshave been introduced to obtain reachable and observable subspaces. Moreover, we havepointed out that the reachable set of switched systems may not be a subspace ifdiscrete-time subsystems are irreversible;③Reachability and controllabitly of a special novel family of linear switchedimpulsive system have been investigated. By virtue of invariance, we have verified thatthe reachable and controllable subspaces of switched systems can be obtained by thedevised subspace algorithm;④By employing multiple Lyapunov function and average dwell time, which areextensively applied to the stability analysis of switched system, we have investigatedthe stability of linear impulsive system and obtained some significant criteria;⑤Typical numerical examples have been given to illustrate the effectiveness ofthe theoretical results.