Robust Control For Markov Jump Systems

Markov jump systems as a special class of hybrid systems with interacting discrete event systems and continuous time dynamic systems are introduced for the strong engineering background. As the development of science and technology, it is quite common for practical systems, such as manufacturing systems, biology and chemistry systems, electric power systems as well as economic systems, having variable structures due to interconnection failures and repair, or environment disturbances, etc. Markovian jump models are often introduced to describe these characters. Due to the complicated structure,the research of jump systems with multi-model switching logic structure cannot be treated as simple combination of traditional control theories for the continuous systems or discrete events systems,respectively. Thus,the study on robust control for Markov jump systems has important theoretic significance and practical value.This dissertation concerns with the problem of robust control for Markov jump systems, constructs some effective performance criteria and obtains a serial of significant results according to Lyapunov stability theory, Linear Matrix inequality(LMI), weak infinitesimal operator, Dynkin formula, Grownwall-Bellman lemma, Shur complement lemma, and projection lemma. This dissertation goes from the easy to the difficult and complicated, first treats with uncertain Markovian jump systems, then a more general and complex case, the problems of robust control, filter design and model reduction related to the Markov jump systems with mode dependent time-delays are intensive researched. The main research work of this dissertation is as follows:a) Under the condition of without disturbance input, for all admissible uncertainties, the stochastic robust stabilization problem for a kind of linear uncertain Markov jump systems is studied, the L₁ performance criterion is established by the introduction of inescapable set and reachable set, and then research the mixed L₁/H_∞control problem. Based on Lyapunov stability theory, the stochastic robust stabilization condition is first given, then the L₁ performance criterion is obtained by choosing appropriate stochastic Lyapunov-Krasovskii function. Upon the proposed performance criterion, a controller design method that guarantees the closed-loop system is stochastically stable and has L₁/H_∞performance constraint is given in terms of a group of LMIs.b) Consider a kind of Markov jump systems with mode-dependent time-varying delays, investigate the L₁ control and filtering problems. Based on the method of approximating the state reachable set with inescapable set to obtain the induced L∞norm, mode-dependent robust L₁ performance criterion is first proposed for mode-dependent time-delay jumping systems. Upon the proposed performance criterion, a sufficient condition for the existence of mode-dependent robust L₁ controller is given in terms of LMIs technology. Furthermore, the problem of robust L₁ filtering is investigated, a sufficient condition for the existence of robust L₁ filter that guarantees the filtering error system is stochastically stable and has L₁ performance constraint is given.c) The markov jump parameters are intyoduced into 2D sysyems. The problems of robust stabilization, robust H_∞control and H_∞model reduction are studied for a class of discrete time 2D jump systems with state delays in Roesser model respectively. At The robust stabilization conditions based on Lyapunov function is proposed firstly, then a sufficient condition for the existence of H_∞controller that guarantees the close-loop system is stochastically stable and certain disturbance inhibition level is obtained, and the controller design problem is cast into a convex optimization problem subjected to LMIs. On the basis of the above, this paper further investigates the H_∞model reduction problem of 2D delayed jump systems. A delay-independent condition of stochastically stability with H_∞performance constraint for 2D delayed jump systems, and then the projection approach is exploited to solve the reduced-order systems. Since the obtained conditions are not expressed in LMI form, the cone complementary linearization method is applied to cast them into convex optimization problems subject to LMI constraints. An explicit parameterization iterative algorithm of the desired reduced-order models is also presented.d) Expand the robust control problem for uncertain time-delay systems with Markov jump parameters to uncertain time-delay neutral systems with Markov jump parameters. The problem of robust H₂/H_∞control is investigated for a class of uncertain neutral jumping systems with discrete and distributed delays. It is desired to design a linear state feedback controller such that the H₂ performance measure is minimized while guaranteeing a prescribed H_∞norm bound on the controlled system. A sufficient condition for the existence of robust H₂/H_∞control law is given in terms of a group of LMIs. Furthermore, the controller design problem is converted into a convex optimization problem subjected to LMI constraints.Besides,simulations are made for main design schemes,and simulation results show the effectiveness of the proposed approaches.Finally,a brief review of this dissertation is given,then some future research areas and open problems in theory and practice are highlighted.