RobustH∞Control for Linear Stochastic Partial Differential Systems with Time Delay

Xisheng Dai,Feiqi Deng,Jianxiang Zhang

doi:10.1155/2014/489408

Abstract

This paper investigates the problems of robust stochastic mean square exponential stabilization and robustH∞for stochastic partial differential time delay systems. Sufficient conditions for the existence of state feedback controllers are proposed, which ensure mean square exponential stability of the resulting closed-loop system and reduce the effect of the disturbance input on the controlled output to a prescribed level ofH∞performance. A linear matrix inequality approach is employed to design the desired state feedback controllers. An illustrative example is provided to show the usefulness of the proposed technique.

Highlights

The H∞ control, since it was first formulated by [1], has been extensively studied in the past years, and a great number of results on this subject have been reported in the literature; see, for example, [2,3,4,5,6,7] and the references therein
This paper investigates the problems of robust stochastic mean square exponential stabilization and robust H∞ for stochastic partial differential time delay systems
We focus on the robust H∞ control problem of linear SPDSs with time delay

Summary

Introduction

The H∞ control, since it was first formulated by [1], has been extensively studied in the past years, and a great number of results on this subject have been reported in the literature; see, for example, [2,3,4,5,6,7] and the references therein. Very recently, [17] studied robust filter for SPDSs by using LMI, which presented an explicit expression for the robust H∞ filter Motivated by these facts, our main purpose in this paper is to examine stochastic exponential stabilization and robust H∞ control for linear SPDSs with time delay under Dirichlet boundary and Robin boundary conditions, respectively. We consider the problem of stochastic exponential stabilization for which a state feedback controller is designed such that the resulting closed-loop system is mean-square exponentially stable. The problem of robust H∞ control is addressed for which a state feedback controller is designed, for which is the resulting closed-loop system mean-square exponentially stable, and is a prescribed H∞ performance. The Rnf such that E ∫0t ∫O f(x, s)Tf(x, s)dx dt := ∫Ω ∫0t ∫O f(x, s, ω)Tf(x, s, ω)dx dt Pdω < ∞, where E represents the mathematical expectation and {f(x, t), t ∈ [0, T]} is stochastic process at the space location x ∈ O and a function of three arguments, that is, f(x, t, ω), x ∈ O, t ∈ [0, T], ω ∈ Ω

Problem Statement and Preliminaries

Mean Square Exponential Stabilization

An Illustrative Example

Conclusions