Control And Filtering For Some Classes Of Markovian Jump Systems

Markovian jump system (or Markovian jump systems) is a special class of hy-brid systems. The changes of subsystems have some statistic property, where the switching system modes belonging to a finite set are driven by a Markov process. Markovian jump system (MJS) can be seen as an extension of deterministic system in format, however, the structure of MJS is more complex, which is very different from deterministic system. In most cases, the criteria of deterministic system can not be applied to MJS directly. Due to the complicated information of MJS, the research method is very different from the traditional system, which is governed by single time or event system. The theory of MJS comes from the fact that it has deep background in theory and application, and it also brings new challenges.This thesis, based on the previous work on MJSs, investigates the problems of delay-dependent stability, H∞control, robust stabilization, H∞filtering and the applications of MJS method in deterministic systems, which also introduces some novel definitions and proposes new ways to deal with some problems. The main contributions of this dissertation are summarized as follows:(1) The stability and H∞control problems for MJSs are studied. On the one hand, the problems of stability and H∞control for MJSs with mode-independent time-varying delays are investigated, where the lower bound of time delay is not 0. Via improved Lyapunov function and Jensen inequality, less conservative results are obtained. On the other hand, by novel Lyapunov function and introducing some slack variables, the stability criteria for MJSs with mode-dependent time delays are established. Moreover, the stabilization problem for MJS with mode-dependent time delays and nonlinearities is concerned. In order to further reduce the conservatism, a new Lyapunov function is constructed, where the bounds of time delays, the time delay states and the information of time delays are included. All the conditions for the desired controllers are given in terms of linear matrix inequalities (LMIs). Numerical examples are used to shown the superiority of the proposed methods.(2) The H∞problem related to discrete-time MJSs with partially unknown transition probabilities is investigated. In contrast with the existing approaches for MJSs with partially unknown transition probabilities where the inaccessible elements separated form the known elements in Lyapunov difference are discarded, another different method to deal with unknown elements in transition matrix is presented. On the basis of the discrete-time MJS transition constraint∑j=1Nλij=1, which is used to estimated the unknown elements, the obtained results not only need to solve less LMIs but also establishe the correlation between known elements and unknown elements.(3) The stabilization problem is considered for networked Markovian jump sys-tems (NMJSs). First of all, for the case where the system mode is accessible with some probability, a partly mode-dependent controller design method is developed. Compared with traditionally mode-independent design method, the results obtained by a partly mode-dependent controller is less conservative, since the distribution property of system modes is concerned. On the other hand, mode-dependent design technique needs the current mode signal to be known exactly, and this ideal as-sumption limits the application of this method. Partly mode-dependent controller can suffer system modes loss with some probability, which is less dependent on signal modes. Under the condition that both network induced delays and packet dropouts occur simultaneously, discrete-time NMJS can be stabilized by a partly mode-dependent controller, whose system states and system modes are transmitted through different channels. In addition, when the plant is a continuous-time NMJS, the existing methods of NMJS fail. However, a partly mode-dependent controller can stabilize an unstable continuous-time NMJS, whose system states and system modes are also transmitted through different channels. Numerical examples are presented to illustrate the usefulness and benefit of the developed results.(4) The H∞filtering problem for MJSs is considered. Sufficient conditions for the existence of the reduced-order H∞filtering are provided in terms of LMIs. Since the established conditions are LMIs with equality constraints, an effective algorithm transferring the non-convex criteria into strict LMIs is proposed. For discrete-time MJSs with partially unknown transition probabilities, a new H∞fil-tering named as partly mode-dependent H∞filter is developed, where the system mode is accessible with some probability. The mode-independent filtering method is to find a common filter for all system modes, no matter what the current sig-nal mode is obtained. The method of partly mode-dependent filtering takes into account the afore-mentioned phenomenon, thus the corresponding solvable solution domain of partly mode-dependent filters is larger and is advantageous for reducing conservatism. Compared with mode-dependent filtering technique, partly mode-dependent filtering method can reduce the dependence on the current mode signal and the burden of data transmission, which is more realistic in practical applications. Numerical examples are given to show the effectiveness of the obtained results.(5) The applications of MJS method in other deterministic systems are stud-ied. Based on the structure uncertainty classification where the uncertainty set is separated into several different subsets according to the maximum singular value of uncertainty, the H∞problem for uncertain deterministic systems is tackled. If the transition rate between two subsets can be observed and modeled as a Markov process, the originally deterministic system is transformed into an MJS. In contrast to common Lyapunov function, the stochastic Lyapunov function modes come from the classification of uncertainties, which is said to be dependent on uncertainties and is less conservative. For singularly perturbed systems (SPSs), an SPS model with distributional probabilities is established, which contains two SPSs, whose singu-larly perturbation parameters may be different. Via introducing Bernoulli variable, the afore-mentioned SPSs are modeled into a new type of SPSs with stochastic parameter matrices. Sufficient condition for mean-square exponential stability sat-isfying H∞performance is established, where the bounds of singularly perturbation parameters are included, however, H∞controllers are constructed, which are inde-pendent of singularly perturbation parameters. Moreover, the obtained results are extended to be Markovian jump singularly perturbed systems (MJSPSs), whose the singularly perturbation parameters and system matrices are driven by two different Markov processes. Numerical examples are used to demonstrate the advantage of the presented results.