Worst case control of uncertain jumping systems with multi-state and input delay information
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 -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 -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 -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 consists of square-integrable functions on the interval [0, T] equipped with the norm ∥ · ∥2. 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 , it follows that 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 , the following matrix inequality holds: Fact 3 Schur complement Given constant matrices Ω1, Ω2, Ω3 where then if and only if
Section snippets
Problem statement
Given a probability space where Ω is the sample space, is the algebra of events and P is the probability measure defined on . Let the random form process be a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in a finite set with generator and transition probability from mode i at time t to mode j at time t + δ, :with transition probability rates
Control design
Our purpose is to design an controller of the jumping system ΣJ which will guarantee desirable dynamical behavior. Here we use the instantaneous state feedback control law of the form:The application of (3.1) in (2.2), (2.3), (2.4) yields the closed-loop system for where
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 generatorFor the two operating conditions (modes), the associated date are:
Mode 1:
Conclusions
We have investigated the problem of 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 -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.
References (23)
Markov Models and Optimization
(1992)Stochastic Stability and Control
(1967)- et al.
Control of a hybrid conditionally linear Gaussian processes
J. Optimiz. Theory Appl.
(1992) - et al.
An optimal stochastic production planning problem with randomly fluctuating demand
SIAM J. Control Optim.
(1987) - et al.
Controllability, stabilizability and continuous-time Markovian jump linear-quadratic control
IEEE Trans. Automat. Contr.
(1990) - et al.
Stochastic stability properties of jump linear systems
IEEE Trans. Automat. Contr.
(1992) Stability and control for linear systems with jump Markov perturbations
Stochastic Anal. Appl.
(1995)- et al.
Analysis design of controllers in systems with random attributes, Part I
Automat. Rem. Contr.
(1961) Feedback control of a class of linear systems with jump parameters
IEEE Trans. Automat. Contr.
(1969)- et al.
control for linear systems with Markovian jumping parameters
Contr-Theor Adv. Technol.
(1993)
Deterministic and Stochastic Time-Delay Systems
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