Stability And Robust H₂/H_∞ Control For Discrete-Time Markov Jump Systems

As one of the most basic dynamics models, Markov jump linear systems can be used to represent random failure processes in manufacturing industry, and some investment portfolio models. Recently, some topics related to Markov jump linear systems have attracted more and more attention. This dissertation devotes to coping with stability and stabilizability of discrete-time Markov jump linear stochastic systems with multiplicative noise by means of the operator spectrum. In addition, using Nash game approach, the finite horizon mixed H2/H∞control problem for discrete-time stochastic linear time-varying systems subject to Markov jump parameters and multiplicative noise is settled thoroughly. The main work and contribution of this dissertation are summarized as follows:1. The spectra and unremovable spectra of discrete-time Markov jump linear stochastic time-invariant systems with multiplicative noise are well-defined and investigated. Then a criterion is provided for testing unremovable spectra.2. According to the spectral distribution of an uncontrolled stochastic linear time-invariant system in the complex plane, we distinguish three kinds of stabilities:asymptotical mean square stability-all spectra of the given system belonging to the open left-half complex plane; critical stability-weaker than asymptotical mean square stability and all spectra belonging to the closed left-half complex plane; and essential instability-at least one of the spectra lying in the open right-half complex plane. While dealing with their criteria, two methods are involved: the spectral analysis technique and the generalized Lyapunov equation (GLE) approach, both of which are the most common ways in characterizing system stability.3. A new concept called "D(0,α)-stabilizability" (0< a≤1) is introduced, for which, a necessary and sufficient condition is also proposed via LMI-based approach. A more general regional stability is discussed extensively with some concrete illustrative examples. As one of the applications, the relationship among D(O,α;β)- stability (0≤α<β≤1) of a discrete-time stochastic system, the decay rate of the system state response and the second-order moment Lyapunov exponent is revealed.4. The notions of exact observability and exact detectability for discrete-time Markov jump linear stochastic systems with multiplicative noise are put forward. Stochastic PBH criteria for exact observability and exact detectability are obtained respectively. The relationship among stability, exact observability and the solutions of (GLEs) is also considered. 5. The finite horizon mixed H2/H∞control problem for discrete-time stochastic linear time-varying systems subject to Markov jump parameters and multiplicative noise is handled. Based on four coupled difference matrix-valued recursions (CDMRs), we derive a necessary and sufficient condition for the solvability of H2/H∞control problem and the controller can be designed explicitly according to the solutions of CDMRs. Moreover, a recursive procedure is supplied to solve the CDMRs.