Stability Analysis Of Impulsive Switched Systems

In this paper,we consider the stability properties about impulsive switched systemswhere ik∈{1,2,, p}, fi+k1∈C[R×Rn, Rn], Iik∈C[R+×Rn, Rn],, Ii0≡x0,0<t0<t1<<tk<are impulse times, and limtk, x′(t)=fhe iksubsystem in the inâ†'te∞k=∞ik(t, x)is the systemwhich is switched to trval (tk, tk+1]. When the system (I) isswitched to the same subsystem, it subjects to the same impulsive efect. And we givesome applications for the theorem.Switched system is one kind of very important hybrid dynamical systems consist-ing of a family of contains-time subsystems and a rule that orchestrates the switchingbetween them. It has widespread applications in emerging fields such as neural net-works, secure communication, the control of intelligent, large power systems,aerospaceand other fields, these systems typical examples including computer hard disk, drive sys-tem, restricted robot control system, auto steering system, the plane’s steering controlsystem, the television channel switching system,etc. In recent years, there has been agrowing interest in the study of switched systems. Many achievements have been gainedin the aspects of stability of switched systems[37]. In addition, impulse is a transientphenomena of abrupt change, the phenomena of impulsive widely exist in the practicalproblems of many fields in the modern technology, many of them can be described byimpulsive diferential systems. In many cases,pulse, switch are together proceed. Theproblem that how to control these factors which makes the production or the develop-ment of things according to the established goal is a worthy of discussion. In recentyears, impulsive switched systems has attracted more and more attention from scholarsand researchers.However, the results of the stability of impulsive switched systems arestill few[818]. In this paper, the system is in[10]the first study, but only to study ex-ponential stability of impulsive switched systems. In view of the practical significanceand theoretical value,this paper studies the stability in terms of two measures by thevariational Lyapunov method and exponential stability. The paper is divided into two chapters.In chapter1, using the method of variational Lyapunov functions and comparisonprinciple, we discuss the stability in terms of two measures. In section4of this chapter,we established a new comparison principle[1920]by comparing with a scalar diferentialsystem.We get several stability criteria in terms of two measure for system(I). And letgik≡0, We can also get direct result of stability in terms of two measures. At last,applications of the main results are shown by several examples.Literature[3]proposed a cogitation: if each subsystem is stable, as long as timelength between any two consecutive switching subsystems is large enough, the stabilityof system can realize. But the requirements is too harsh. Literature[6]give the notion ofdwell time.Dwell time is the time length between any two consecutive switching subsys-tems.if the dwell time is proper large, we can get the exponential stability of the switchedsystems. Literature[7]proposed the notion of the average dwell time,"Average"thiswords reflected: the activation time with one or two subsystems may less than Ï„, but theaverage dwell time with each subsystems must more than Ï„.In1982, when Laypunov is expounding the x1=x2==xn=0stability withrespect to the variables,he said:"we can study more general problem: stability with re-spect to some of these variables, for example,In the case of x1, x2,, xm(m <n)."Theresearch on the stability with respect to part of variable is actual requirement. People areonly interested in partial variable,or because of the technical difculty, partial variablecan not be controlled, so the study of the stability with respect to part of the variablesbecome very important. In recent years, the results of the stability with respect to partof the variables of impulsive switched systems are still few.In chapter2section3, firstly, we exploit the thought of the literature[5]and inves-tigate the exponential stability of impulsive switched systems by a lyapunov functionmethod.Secondly, we exploit the thought of the literature[7]and study the exponentialstability of impulsive switched systems by multiple lyapunov function method and thedwell time techniques.Finally some examples which show the efectiveness of the resultsare worked out, and numerical simulations are performed to both illustrate and verifythe stability analysis. In section4of this chapter, we study the exponential stabilitywith respect to part of the variables of impulsive switched systems. Several numericalexamples are worked out to illustrate our results.