Exponential Stability Analysis Of Discrete Time-delay Systems

It is well known that time delays occur in many practical systems,such as network control systems,neural networks and gene regulatory networks.At the same time,time delay is a typical destructive factor of system stability.For this reason,the stability analysis of time-delay systems becomes one of the important issues in the control theory.This paper gives the necessary conditions as well as necessary and sufficient conditions of exponential stability for discrete time-delay systems,and sufficient conditions of robust exponential stability for a class of uncertain discrete time-delay systems.The main contents are as follows:1.The necessary conditions of exponential stability for discrete time-delay systems.The fundamental matrix of discrete time-delay systems is firstly defined,and thereby the concept of Lyapunov matrix is proposed.The Lyapunov-Krasovskii(L-K)functional based on Lyapunov matrix is introduced and it's quadratic lower bounds are given under the premise that the system is exponentially stable.Then,some properties of Lyapunov matrices are given.By introducing appropriate functionals,the necessary conditions of exponential stability are obtained.As applications of Lyapunov matrix the robust stability analysis of a class of uncertain for the system under consideration linear discrete time delay systems is presented.Finally,the validity of the proposed methods are verified by numerical examples.2.The necessary and sufficient conditions of exponential stability for discrete time-delay systems.We first construct a boundary value problem of matrix difference equations,which is a generalization of the Lyapunov matrix equation for delay-free linear discrete-time systems.Also,the existence,uniqueness and properties of solution to the boundary value problem are investigated.In addition,for linear discrete time-delay systems,we introduce a new concept-Lyapunov matrix which can be viewed as a generalization of the unique positive definite solution of delay-free Lyapunov matrix equation.It should be noted that the Lyapunov matrix can be represented by the solution of the boundary value problem mentioned above.Then,a necessary and sufficient condition which guarantees the existence and uniqueness of Lyapunov matrix is investigated.Furthermore,by constructing Lyapunov matrix-based complete-type L-K functional,several necessary and sufficient conditions under which the considered systems are exponentially stable are proposed.Finally,two numerical examples are presented to show the advantages of the proven theoretic results.