Stability Analysis Of Linear Time-varying Delay System

In control theory,the past state of the system affect the current state to a certain extent.In other words,the change of the system is related not only to the current state,but also to the past state.This kind of system is called time-delay system.In some cases,time-delay can even cause system instability.Therefore,the problem of time-delay has always been a hot topic in the field of control.Since 1970 s,Lyapunov functional has been introduced into the stability analysis of time-varying delay systems.The Lyapunov method has become an effective tool for the analysis of time-delay systems.At present,there is still no uniform method for constructing LyapunovKrasovskii functional(abbreviated L-K functional).In the construction of L-K functional,integral items are generally included,and nowadays,it has been gradually expanded from single integral and double integral to triple integral and quadruple integral.In this paper,the stability of continuous time-varying delay systems is studied.By using Lyapunov's second method,the Lyapunov function with quadruple integral term is selected,and the stability condition of time-varying delay system with less conservatism is obtained by means of integral inequality and delay partition method.According to Lyapunov's second method,we need to derive the functional,and in order to make the stability criterion be expressed in the form of linear matrix inequality(LMI),we need to deal with some integral terms generated by the derivation of Lyapunov function.The processing method includes identical deformation or in the process of magnification,various forms of integral inequalities are adopted,including Jensen integral inequality,Wirtinger integral inequality,Bessel-Legendre integral inequality(abbreviated as B-L integral inequality),and so on.Combined with the integral inequality method,the degree of magnification of the cross-term integral affects the conservatism of the criterion.In this paper,the following two integral inequalities are used to reduce the conservatism of the stability criterion of the system:Wirtinger integral inequality is used to amplify the integral term in the functional derivative.Since the Wirtinger integral inequality is closer to the truth value than the Jensen integral inequality,the final functional derivative is also closer to the truth value,which makes its stability condition less conservative.The stability condition of a time-varying delay system with less conservative property is obtained by applying the Wirtinger integral inequality.In the process of studying the method of integral inequality,it is observed that the conservatism of B-L integral inequality is less than that of Wirtinger integralinequality.B-L integral inequality is used to deal with some cross-term integrals.The obtained stability criterion is improved and a new stability criterion is obtained.Finally,the superiority of the new stability criterion is proved by the same numerical example.Through the above two integral inequalities,two less conservative stability conditions are obtained.