Reliable $H_\infty$ Control for Discrete-Time Fuzzy Systems With Infinite-Distributed Delay

Abstract

In this paper, the problem of reliable <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$H_\infty$</tex></formula> control is investigated for discrete-time Takagi–Sugeno (T–S) fuzzy systems with infinite-distributed delay and actuator faults. A discrete-time homogeneous Markov chain is used to represent the stochastic behavior of actuator faults. In terms of a stochastic fuzzy Lyapunov functional, a sufficient condition is proposed to ensure that the resultant closed-loop system is exponentially stable in the mean-square sense with an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$H_\infty$</tex></formula> performance index. Based on the derived condition, the reliable <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$H_\infty$</tex></formula> control problem is solved, and an explicit expression of the desired controller is also given. The case of no failure in the actuator is also considered. A numerical example is given to demonstrate that our results are effective and less conservative.