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Discrete-time Markov jump linear systems. (English) Zbl 1081.93001

Probability and Its Applications. London: Springer (ISBN 1-85233-761-3/hbk). x, 280 p. (2005).
The following class of dynamic systems is considered \[ {\mathcal G} = \begin{cases} x(k+1)= A_{\theta (k)} x(k) + B_{\theta(k)} u(k) + G_{\theta (k)} w(k), \\ y(k) = L_{\theta (k)} x(k) + H_{\theta (k)} w(k),\\ z(k)= C_{\theta (k)} x(k) + D_{\theta (k)} u(k),\\ x(0)= x_0, \quad \theta (0) = \theta_0,\end{cases} \] where \(x(k)\) represents the state variable of the system, \(u(k)\) the control variable, \(w(k)\) the noise sequence acting on the system, \(y(k)\) the measurable variable available to the controller, and \(z(k)\) the output of the system. All matrices have appropriate dimensions and \(\theta (k)\) stands for the state of a Markov chain taking value in a finite set \(N = \{ 1, \ldots, N\}, V =\{ v_1, \ldots, v_N\}\) and the transition probability matrix of the Markov chain \(P = [p_{ij}].\) The book consist of eight chapters.
The first chapter presents the class of Markov Jump Linear Systems (MJLS) via some application-oriented examples with motivating remarks and an outline of the problems. Chapter 2 provide the bare essential of background.
In Chapter 3 (Stability) the authors consider just the first equation of \({\mathcal G}\) with \(w(k)\) either a sequence of independent identically distributed second order random variables or an \(l_2\)-sequence of random variables. The control law \(u(k)\) and output \(y(k)\) are considered in the case when a feedback control is based on an estimate \({\hat \theta} (k)\) instead of \(\theta (k).\) The key concept of mean square stability for MJLS is presented. The chapter is concluded analysing the almost sure stability of MJLS.
Chapter 4 (Optimal control) deals with the quadratic and \(H_2\)-Optimal Control Problems. The authors obtain a solution for the finite horizon and \(H_2\)-control problems through, respectively, a set of control coupled difference and algebraic Riccati equations. In Chapter 5 (Filtering) the filtering problem is considered. The solution is derived through a difference and algebraic Riccati-like equation. The case with uncertainties on the parameters of the system is analyzed by using convex optimization. In Chapter 6 (Quadratic Optimal Control with Partial Information) the linear quadratic optimal control problem for the case in which the controller has only access to the output variable \(y(k)\) besides the jump variable \(\theta (k)\) is considered. A Markov jump linear controller is designed from two sets coupled difference (for the finite horizon case) and algebraic (for the \(H_2\)-control case) Riccati equation, one associated with the control problem and the other one associated with the filtering problem.
Chapter 7 (The \(H_\infty\)-Control) deals with an \(H_\infty\)-like theory for MJLS following an approach based on a worst-case design problem. Chapter 8 (Design Techniques and Examples) presents and discusses some applications of the theoretical results obtained in this book, It also presents design-oriented techniques based on convex optimization for control problems of MJLS.
This book is theoretically oriented, although some illustrative examples are included. The book is primarily intended for students and practitioners of control theory.

MSC:

93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49K45 Optimality conditions for problems involving randomness
93E20 Optimal stochastic control
93E11 Filtering in stochastic control theory
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