Time-dependent Stability Studies

During the last two decades great importance has been attached to delayed systems,for time-delays are often encountered in various practical systems.The existence of the delay may cause the performance of system to drop even instability.Therefore,it has practical meaning and theoretical value to study the delay-dependent stability for systems with delays.This paper is concerned about the delay-dependent stability for systems with time-varying delays and distributed delays.Based on Lyapunov stability theory and linear matrix inequality approach,by constructing a new Lyapunov functional,the new method has been applied to test the positive definiteness of Lyapunov and the negative definiteness of its derivative,which derive the asymptotic stability criterion for improvement.The stability results are less conservative than some existing ones,which is illustrated by listing numerical examples.More specifically,the main contents of this article are as follow:Chapter 1 mainly reviews the research background and significance of systems with time delays.Then the systems with time delays are introduced simply and the current research status of the stability is summarized.And at last,the main content of this article is introduced simply.Chapter 2 mainly introduced the Lyapunov stability theory and linear matrix inequality methods.Moreover,some related lemmas are given,which used in the process of the proof.Chapter 3 discussed the delay-dependent stability of systems with distributed time delays.Firstly,a Lyapunov functional is given and then we propose a new inequality to estimate the derivative of the Lyapunov functional.In addition,we take the Lyapunov functional as a whole to examine its positive definite,rather than restrict each term of it to positive definite as usual.According to Lyapunov stability theory,the less conservative stability criterion is obtained.Finally,the stability results are less conservative than some existing ones,which is illustrated by listing numerical examples.Chapter 4 mainly studied the delay-dependent stability of systems with time-varying delays.A new Lyapunov functional is constructed by introducing the cross terms,which ensures to take the Lyapunov functional as a whole to examine its positive definite,rather than restrict each term of it to positive definite as usual.A new inequality is proposed when estimating the negative definiteness of derivative of the Lyapunov functional.So the stability criterion with less conservative is obtained.Finally,the simulation results are less conservative than some existing ones,which is illustrated by listing numerical examples.Chapter 5 summarizes the main research content of this article,and point out the possibilityof further reduce conservative and indicate the direction for further work.