Study On Stability And Stabilization For Several Classes Of Continuous-time And Discrete-time Impulsive Systems

Impulse is a common phenomenon in nature and all kinds of industrial systems. The e?ect of impulses on the stability of the system usually can not be neglected. This e?ect may be positive, also may be negative. Thus the stability analysis of impulsive systems is more complex than general systems. The impulses are known as stabilizing impulses if they make unstable systems stable. Destabilizing impulses make the system lost the original stability. ”Neutral”-type impulses do not change the stability of the system.From the point of view of control theory, stabilizing impulses are more attractive because they can be used to design impulsive controllers. Compared with the continuous control methods, impulse control has advantages like shorter response time, less information to be transferred, more easily to be realized and so on, and has drawn more and more attention of control scholars. In this paper, based on the existing research results, by using the theory of stability, we study the problem of the stability and stabilization for several classes of continuous-time and discrete-time impulsive systems. The main work and research results of this dissertation are outlined as follows:1. The problem of stability and stabilization of impulsive neural networks is studied. The considered networks are represented as the interconnection of two lower order subnetworks by using decomposition techniques. A time-varying Lyapunov function is introduced to study this hybrid interconnection structure and a su?cient condition of stability is obtained which relies on both the lower bound and the upper bound of impulsive intervals. The obtained condition does not restrain the stability of both continuous and discrete dynamics. Thus it can be applied to deal with not only the case where at least one dynamics is stable but also the case where both continuous and discrete dynamics are unstable. Moreover, based on linear matrix inequalities, a su?cient condition for the existence of reduced-order impulsive controllers is developed.2. The problem of stability and stabilization of impulsive neural networks with discrete time-varying delay and unbounded continuously distributed delay is studied.First, the impulse-time-dependent Lyapunov function and functional are applied to catch hybrid dynamic characteristics of networks and two novel stability criteria are obtained.The ?rst one demands the discrete time-varying delay is bounded, while the second one demands not only the discrete time-varying delay is bounded but also its derivative is less than one. The ?rst one obviously has a wider range of application, while the e?ect of the second one is better than the ?rst when the discrete delay is constant. The two criteria are still applicable where both continuous and discrete dynamics of the considered networks are unstable. Moreover, two design conditions for linear state feedback impulsive controllers are presented in terms of linear matrix inequalities.3. The problem of stability and stabilization of discrete-time linear delayed systems with nearly-periodic impulses is studied. Nearly-periodic impulse is a class of impulse which is close to a nominal periodic impulse, the error between them is a uncertain term whose range is known. In order to eliminate the in?uence of time delay, system is ?rst transformed to a equivalent discrete-time impulsive linear time invariant system by increment-dimensional method, then a linear di?erence inclusion is introduced to describe the nature of solution of system in impulsive instants. A su?cient condition of stability of systems is obtained by constructing a discrete time-varying Lyapunov function which depends on impulses. The obtained result is still applicable when both the two discrete-time subsystems of the considered system are unstable. Based on linear matrix inequalities, design criteria for reduced-order and full-order impulsive controller are addressed.4. The problem of stability and stabilization of discrete-time switched linear systems with time-varying delays and nearly-periodic impulses under arbitrary switching is studied. A equivalent discrete-time switched linear time-varying system with nearly-periodic impulses is obtained by increasing the dimension of the considered system. Based on linear di?erence inclusion method, the problem of stability of the considered system is transformed to that of the linear di?erence inclusion which describes system states in impulsive instants. A discrete-time time-varying Lyapunov function depended on switching, time delay and impulse is introduced and a stability criterion is derived. Whether or not discrete-time subsystems in every switched subsystems are stable, the criterion is e?ective. Based on the obtained stability result, we consider the design problem of impulsive controller and delayed impulsive controller. Su?cient conditions for the existence of reduced-order and full-order impulsive controllers and reduced-order and full-order delayed impulsive controllers are developed.