Elsevier

Information Sciences

Volume 176, Issue 2, 20 January 2006, Pages 186-200
Information Sciences

Worst case control of uncertain jumping systems with multi-state and input delay information

https://doi.org/10.1016/j.ins.2004.07.019Get rights and content

Abstract

In this paper, the problem of worst case (also called H) Control for a class of uncertain systems with Markovian jump parameters and multiple delays in the state and input is investigated. The jumping parameters are modelled as a continuous-time, discrete-state Markov process and the parametric uncertainties are assumed to be real, time-varying and norm-bounded that appear in the state, input and delayed-state matrices. The time-delay factors are unknowns and time-varying with known bounds. Complete results for instantaneous and delayed state feedback control designs are developed which guarantee the weak-delay dependent stochastic stability with a prescribed H-performance. The solutions are provided in terms of a finite set of coupled linear matrix inequalities (LMIs). Application of the developed theory to a typical example has been presented.

Introduction

Dynamical systems whose structures vary in response to random changes, which may result from abrupt phenomena such as parameter shifting, component and interconnection failures, are frequently occurring in practical situations. Such systems can be modelled by combined continuous and discrete states such as fault-tolerant systems [3], an important class of which is the one with Markovian jumping parameters which has been widely used in jumping systems (JSs). In the linear case, the underlying dynamics are governed by different forms depending on the value of an associated finite-state Markov process. Research into this class of systems and their applications span several decades [1]. For some representative related work on this general topic, we refer the reader to [1], [2], [3], [4], [5], [6], [7], [8], [9], [17], [18], [19], [20], [21], [22], [23] and the references therein. For Markov jumping linear continuous-time systems, the issue of robust stability and control has been investigated in [10], [11], [12] and the counterpart of H-control has been developed in [7]. A recent account on the subject can also be found in [13].

On another research front, dynamical systems with state-delay have been the subject of extensive research during the part two-decades; for a recent coverage on the available results the reader is referred to [14].

The purpose of this paper is to extend the results of [11], [12], [13] further by developing H-control for a class of uncertain Markovian jump parameters with multiple delays in the state and the input. The jumping parameters are treated as continuous-time, discrete-state Markov process and the parametric uncertainties are assumed to be real, time-varying and norm-bounded that appear in the state, input and delayed-state matrices. The time-delay factors are considered unknown and time-varying with know bounds. Complete results for instantaneous and delayed state feedback control designs are developed which guarantee the weak-delay dependent stochastic stability with a prescribed H-performance. The solutions are provided in terms of a finite set of coupled linear matrix inequalities (LMIs).

Notations and Facts: In the sequel, the Euclidean norm is used for vectors. We use Wt, W−1, λ(W) and ∥W∥ to denote, respectively, the transpose of, the inverse of, the eigenvalues of and the induced norm of any square matrix W. We use W > 0 (⩾, <,  0) to denote a symmetric positive definite (positive semidefinite, negative, negative semidefinite) matrix W with λm(W) and λM(W) being the minimum and maximum eigenvalues of W and I to denote the n × n identity matrix. The Lebesgue space L2[0,T] consists of square-integrable functions on the interval [0, T] equipped with the norm ∥ · 2. E[·] stands for mathematical expectation. Sometimes, the arguments of a function will be omitted in the analysis when no confusion can arise.

Fact 1

For any real matrices Σ1, Σ2 and Σ3 with appropriate dimensions and Σ3tΣ3I, it follows thatΣ1Σ3Σ2+Σ2tΣ3tΣ1tα-1Σ1Σ1t+αΣ2tΣ2,α>0

Fact 2

Let Σ1, Σ2, Σ3 and 0 < R = Rt be real constant matrices of compatible dimensions and H(t) be a real matrix function satisfying Ht(t)H(t)  I. Then for any ρ > 0 satisfying ρΣ2tΣ2<R, the following matrix inequality holds:(Σ3+Σ1H(t)Σ2)R-1(Σ3t+Σ2tHt(t)Σ1t)ρ-1Σ1Σ1t+Σ3(R-ρΣ2tΣ2)-1Σ3t

Fact 3 Schur complement

Given constant matrices Ω1, Ω2, Ω3 where Ω1=Ω1tand0<Ω2=Ω2t then Ω1+Ω3tΩ2-1Ω3<0 if and only ifΩ1Ω3tΩ3-Ω2<0or-Ω2Ω3Ω3t-Ω1<0

Section snippets

Problem statement

Given a probability space (Ω,F,P) where Ω is the sample space, F is the algebra of events and P is the probability measure defined on F. Let the random form process {ηt,t[0,T]} be a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in a finite set S={1,2,,s} with generator I=(αij) and transition probability from mode i at time t to mode j at time t + δ, i,jS:pij=Pr(ηt+δ=j|ηt=i)=αijδ+o(δ),ij,1+αijδ+o(δ),i=jwith transition probability rates αij0for

H Control design

Our purpose is to design an H controller of the jumping system ΣJ which will guarantee desirable dynamical behavior. Here we use the instantaneous state feedback control law of the form:u(t)=K(i)x(t),iSThe application of (3.1) in (2.2), (2.3), (2.4) yields the closed-loop system for ηt=iS(ΣJK):x˙(t)=AK(i)x(t)+[A(i)+E(i)Δ(t,i)N(i)]Xτ(t-τ)+[B(i)+H(i)Δa(t,i)M(ηt)]K(i)Xψ(t-ψ)+Γ(i)w(t),t0,x(t)=0,t[-maxjτj,0]z(t)=GK(i)x(t)+G(i)Xτ(t-τ)+F(i)K(i)Xψ(t-ψ)+Φ(i)w(t)whereAK(i)=AKo(i)+B0(i)K(i),GK(i)=G0(i

Example

We consider a pilot-scale three-reach water quality system which can fall into the type (2.2), (2.3), (2.4). Let the Markov process governing the mode switching has generatorJ=-443-3For the two operating conditions (modes), the associated date are:

Mode 1:A0(1)=-0.200-0.1,A1(1)=-0.10-0.1-0.1,A2(1)=0110,A3(1)=-0.90-1-1.1B0(1)=1001,B1(1)=2001,B2(1)=1002,Γ(1)=0.3000.2G0(1)=0.2000.2,G1(1)=0.1000.2,G2(1)=0.2000.1,G3(1)=0.1000.2F0(1)=0.1000.1,f1(1)=0.2000.1,F2(1)=0.1000.2Φ(1)=0.1000.4,M1(1)=M2(1)=0.2

Conclusions

We have investigated the problem of H control for a class of uncertain systems with Markovian jump parameters and multiple-state and input delays. We have designed a state feedback controller such that stochastic stability and a prescribed H-performance are guaranteed. Complete results for weak delay-dependent stochastic stability criteria for the nominal and uncertain time-delay jumping systems have been developed. We have established that the control problem for the system under

Acknowledgments

Professor P. Shi’s work was partially supported by Laboratory of Complex Systems and Intelligence Science, Institute of Automation, Chinese Academy of Sciences, China, while Professor M.S. Mahmoud’s work was supported by the Scientific Research Council of UAE University under grant No. # 02–04–7–11/03. The authors also gratefully acknowledge the very valuable and helpful comments and suggestions by the reviewers, which have improved the quality of the paper.

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