Asian Journal of Control, Vol. 6, No. 1, pp. 1-8, March 2004 1 CONTROL OF INTERCONNECTED JUMPING SYSTEMS: AN H ∞ APPROACH Magdi S. Mahmoud, Peng Shi, and Abdulla Ismail ABSTRACT This paper investigates, by using an H ∞ approach, the problems of sto- chastic stability and control for a class of interconnected systems with Mark- ovian jumping parameters. Both cases of finite- and infinite-horizon are studied. It is shown that the problems under consideration can be solved if a set of coupled differential or algebraic Riccati equations are solvable. KeyWords: H ∞ -control, interconnected systems, Markovian jump parame- ters, stochastic stability and stabilization. I. INTRODUCTION Problems associated with decentralized control and stabilization of interconnected systems are receiving considerable interests [1-8] where most of the effort are focused on dealing with the interaction patterns and per- forming the control analysis and design on the subsystem level. Most of the time, dynamical systems are subject to frequent unpredictable structural changes. It turns out that these systems can be conveniently modeled as piece-wise deterministic systems, where the underlying dynamics are represented by different forms depending on the value of an associated Markov chain process. An important class of such systems is the jump linear sys- tems. Research into this class of systems and their appli- cations span several decades. For some representative prior work on this general topic, we refer the reader to [9-15,18-25] and the references therein. To date the problems of stability analysis and con- trol design of linear interconnected systems with Mark- ovian jump parameters, to the best of the authors’ knowledge, has not yet been fully investigated. Manuscript received November 19, 2002; accepted April 3, 2003. Magdi S. Mahmoud and Abdulla Ismail are with College of Engineering, UAE University, PO Box 17555 Al-Ain, United Arab Emirates. Peng Shi is wiith School of Technology, University of Gla- morgan, Pontypridd, Wales, UK, CF37 1DL. He was with Weapons Systems Division, Defense Science and Technology Organization, Department of Defence, PO Box 1500, Edin- burgh 5111 SA, Australia. This work was partially supported by the UAE Scientific Research Council under grant # 03/11-7-04-02. In this paper, we are aiming to initiate the study and develop criteria of stochastic stability and stabiliza- tion of a class of linear interconnected systems with Markovian jump parameters. The jumping parameters are treated as continuous-time, discrete-state Markov process. The notion of stochastic decentralized stability is firstly introduced. Here the purpose is to design de- centralized state feedback controller such that stochastic stability and a prescribed H ∞ -performance are guaran- teed. Both cases of finite-horizon and infinite-horizon are treated. We demonstrated that the decentralized con- trol problem for interconnected Markovian jump systems can be essentially solved in terms of the solutions of a finite set of coupled differential (or algebraic) Riccati equations. Notations and Facts. In the sequel, the Euclidean norm is used for vectors. We use W t , 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 ma- trix W. We use W > 0 (≥, <, ≤ 0) to denote a symmetric positive definite (positive semidefinite, negative, nega- tive semidefinite matrix W with σ M (W) being the maxi- mum singular value of W and I to denote the n × n iden- tity matrix. The Lebesgue space L 2 [0, ∞) consists of square-integrable functions on the interval [0, ∞). E[⋅] stands for mathematical expectation. 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 de- fined on F. Fact 1. ([18]) For any real matrices Σ 1 , Σ 2 and Σ 3 with appropriate dimensions and 3 3 , t I ΣΣ≤ it follows that 1 1 3 2 2 3 1 1 1 2 2 , 0. t t t t t α α α − ΣΣΣ+ΣΣΣ≤ ΣΣ+ΣΣ∀>