Dynamic Analysis Of Impulsive Switched Systems

In this paper,we consider the stability properties about impulsive switched systems where ik∈{1.2,…,p}, fik-1∈C[R+×Rn,Rn], Ik∈C[R+×Rn,Rn], I0=x0,0<t0<t1<…<tk<… are impulse times, and lim(?) tk=∞, x’(t)=fik (t,x)is the system which is switched to the ik subsystem in the interval [tk,tk+1). When the system (I) is switched to the same subsystem, it subjects to the same impulsive effect. And we give some applications for the theorem.Switched system is one kind of very important hybrid dynamical systems consisting of a family of continuous-time or discrete-time subsystems and a rule that orchestrates switching between them. Switched system has such properties:Even if every subsystem is not stable, by constructing suitable switching rules can also ensure the stability of the system. On the contrary, even if every subsystem is stable, system can also be unstable if the choices of switching law is not appropriate. At present, switched systems have a wide range of applications in the disk of computer control system, aerospace transporta-tion control, radio communications, chemical process control, mechanical control and power systems and other fields. In recent years, because the switch system can improve system performance and meet the needs of the rapid development of intelligent control, the academia show great interests to it, and they have got some good results, especially in the area of stability.But in the process of actual switching, switching impulse often exists in the system, in this case, due to the presence of impulse, system can produce complex dynamic behav-ior. Literature puts forward a new model that is the impulsive and switched system. This model is under the control of PWM (Pulse width modulated), the switching control of machine tool working model can more accurately describe the actual situation in the instantaneous impulse switched system. Most of the time, impulse and switching are going inevitable simultaneously, and how to control these factors, making the production and things develop in accordance with the established goals are a problem worthy to be discussed. In recent years, more and more scientific researchers have been researching for this system. In view of the practical significance and theoretical value, this paper showed different methods to study stability of impulsive and switched systems. It is specifically divided into two chapters.In the first chapter, what we researched is the nonlinear impulsive switched system including stable subsystems and unstable subsystems. We mainly use multiple Lyapunov functions and Dwell time approach to study the stability of the system. We will show the stability of the whole system if the selection of dwell time is large enough and the total dwell time of the stable system and unstable system satisfy a certain proportion. Finally, examples verify the validity of the theorem are given.In the second chapter, firstly, we mainly study impulsive and switched system with delay. We mainly use multiple Lyapunov functions, dwell time method and delay dif-ferential inequality to study the exponential stability of impulsive and switched system with delay, and got corresponding results. Finally, examples are given to illustrate the application of the theorems.