Improved Razumikhin and Krasovskii approaches for discrete-time time-varying time-delay systems

This paper studies stability of discrete-time time-varying time-delay systems. The existing Razumikhin and Krasovskii stability approaches for this class of systems are improved in the sense that the time-shifts of the Razumikhin functions and Krasovskii functionals are allowed to take both negative and positive values. Three kinds of stability concepts, say, uniform stability, uniformly asymptotic stability and uniformly exponential stability, are considered. The improvements of the Razumikhin and Krasovskii approaches are achieved by using the concept of uniformly asymptotically stable (UAS) function, the notion of overshoot associated with the UAS function and an improved comparison lemma. Both delay-dependent and delay-independent stability theorems are obtained for a class of discrete-time linear time-delay systems by using the improved Razumikhin and Krasovskii stability approaches. Numerical examples demonstrate the effectiveness of the proposed methods.