Small Gain Theorem-Based Robustness Analysis of Discrete-Time MJLSs with the Markov Chain on a Borel Space and Its Application to NCSs

TL;DR

This paper develops a small gain theorem for discrete-time Markov jump linear systems with a Borel space Markov chain, providing a finite-dimensional LMI approach for robustness analysis and applying it to networked control systems with delays.

Contribution

It introduces a griding approach to convert infinite-dimensional $H_{}$ analysis problems into finite-dimensional LMIs for robust stability of MJLSs with continuous state spaces.

Findings

Established a small gain theorem for MJLSs on Borel spaces.

Derived a lower bound for the stability radius using LMI techniques.

Validated the approach with two numerical examples.

Abstract

This paper is concerned with the robustness of discrete-time Markov jump linear systems (MJLSs) with the Markov chain on a Borel space. For this general class of MJLSs, a small gain theorem is first established and subsequently applied to derive a lower bound of the stability radius. On this basis, with the aid of the extended bounded real lemma and Schur complements, the robust stability problems for the MJLSs are tackled via linear matrix inequality (LMI) techniques. The novel contribution, primarily founded on the scenario where the state space of the Markov chain is restricted in a continuous set, lies in the formulation of a griding approach. The approach converts the existence problem of solutions of an inequality related to $H_{\infty}$ analysis, which is an infinite-dimensional challenge, into a finite-dimensional LMI feasibility problem. As an application, within the framework of MJLSs, a robustness issue of the sampled-data systems is addressed by using a Markov chain, which is determined by the initial distribution and the stochastic kernel, to model transmission delays existing in networked control systems (NCSs). Finally, the feasibility of the results is verified through two examples.