Periodic Averaging of Discrete-Time Systems: A Time-Delay Approach

This article is concerned with the stability of discrete-time systems with fast-varying coefficients that may be uncertain. Recently, a constructive time-delay approach to averaging was proposed for continuous-time systems. In the present article, we develop, for the first time, this approach to discrete-time case. We first transform the system to a time-delay system with the delay being the period of averaging, which can be regarded as a perturbation of the classical averaged system. The stability of the original system can be guaranteed by the resulting time-delay system. Then under assumption of the classical averaged system being exponentially stable, we derive sufficient stability conditions for the resulting time-delay system, and find a quantitative upper bound on the small parameter that ensures the exponential stability. Moreover, we extend our method to input-to-state stability (ISS) analysis of the perturbed systems. Finally, we apply the approach to the practical stability of discrete-time switched affine systems, where an explicit ultimate bound in terms of the switching period is presented. Two numerical examples are given to illustrate the efficiency of results.