Exponential Stability and Robust H∞ Control for Discrete-Time Time-Delay Infinite Markov Jump Systems

Yueying Liu,Ting Hou

doi:10.1155/2018/3676083

Abstract

In this paper, exponential stability and robust H∞ control problem are investigated for a class of discrete-time time-delay stochastic systems with infinite Markov jump and multiplicative noises. The jumping parameters are modeled as an infinite-state Markov chain. By using a novel Lyapunov-Krasovskii functional, a new sufficient condition in terms of matrix inequalities is derived to guarantee the mean square exponential stability of the equilibrium point. Then some sufficient conditions for the existence of feedback controller are presented to guarantee that the resulting closed-loop system has mean square exponential stability for the zero exogenous disturbance and satisfies a prescribed H∞ performance level. Numerical simulations are exploited to validate the applicability of developed theoretical results.

Highlights

During the past decades, Markov jump systems have been the subject of a great deal of research since they have been used extensively both in theory and in applications
[18, 23] investigated the exponential stability and infinite horizon H2/H∞ control problem for discrete-time infinite Markov jump systems with multiplicative noises, respectively, but they neglected the effects of time-delay
Stability will be analyzed, and a sufficient condition is obtained for system (1) with u(t) = 0 and V(t) = 0 to have mean square exponential stability

Summary

Introduction

Markov jump systems have been the subject of a great deal of research since they have been used extensively both in theory and in applications. The work in [18] demonstrated the inequivalence between stochastic stability and mean square exponential stability in discrete-time case With this motivation, infinite Markov jump systems have stirred widespread research interests. [18, 23] investigated the exponential stability and infinite horizon H2/H∞ control problem for discrete-time infinite Markov jump systems with multiplicative noises, respectively, but they neglected the effects of time-delay. Stability and control for time-delay stochastic systems with infinite Markov jump and multiplicative noises have not received enough attention despite their importance in practical applications, which motivates us for the present research. We aim to address the exponential stability and H∞ control problem for a class of discrete-time time-delay stochastic systems with infinite Markov jumps and multiplicative noises in this paper. D fl {1, 2, . . .}. l2(Z+; Rm) fl {ς ∈ Rm | ς is Ftmeasurable, and (∑∞ t=0 E‖y(t)‖2)1/2 < ∞}

Preliminaries

Main Results

Illustrative Example

Conclusions