Averaging-Based Stability of Discrete-Time Delayed Systems via a Novel Delay-Free Transformation

Abstract

In this article, we study, for the first time, the stability of linear delayed discrete-time systems with small parameter $\varepsilon > 0$ and rapidly varying coefficients. Recently, an efficient constructive approach to averaging-based stability via a novel delay-free transformation was introduced for continuous-time systems. Our paper extends this approach to discrete-time systems. We start by introducing a discrete-time change of variables that leads to a perturbed averaged system. By employing Lyapunov analysis, we derive linear matrix inequalities (LMIs) for finding the maximum values of the small parameter $\varepsilon > 0$ and delay (either constant or time-varying) that guarantee exponential stability of the original system. We show that differently from the continuous-time, in the discrete-time, given any bounded delay, there exists a small enough $\varepsilon$ such that our LMIs are feasible (i.e., the system is exponentially stable). Numerical examples illustrate the efficiency of the proposed approach.