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Analysis And Control Of Discrete-Time Markov And Semi-Markov Stochastic Switching Systems

Posted on:2017-05-16Degree:DoctorType:Dissertation
Country:ChinaCandidate:T YangFull Text:PDF
GTID:1108330503469770Subject:Control Science and Engineering
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The phenomena of stochastic switching, such as the changes of working environ-ment, failures of system components, system time delays, and shift of working points of nonlinear systems, can be readily found in various practical systems. Stochastic switch-ing systems, due to their advantageous capability of modeling these phenomena, have been extensively investigated over the past decades. As an important class of stochastic switching systems, many systematic results on Markov jump systems (MJSs) are avail-able in the literature. However, there still exist some challenging control issues that need to be dealt with, for example, the control issues for the MJSs with asynchronous switch-ing, and the nonlinear MJSs. Besides, the conservatism of some previous results can be further reduced. On the other hand, since semi-Markov jump systems (S-MJSs) relax the Markov property or the non-aftereffect property (Markov property) of transition probabil-ities (TPs), they generalize the scope of stochastic switching systems and thus become a naturally extended research frontier. However, the generality of S-MJSs, i.e., the property that the TPs can depend on the historical knowledge of mode switching sequence, un-avoidably leads to the considerable complexity in studying the S-MJSs, even for the basic stability and stabilization issues. This dissertation research is aimed not only to improve the results on MJSs, but also to systematically investigate the stability and stabilization problems of S-MJSs via the time-varying Lyapunov function approach. Furthermore, the developed theoretical results are verified by some practical examples, including the single-link robot arm, the vehicle suspension system, the pendulum system, the vertical take oil and landing processes of helicopter, the cart-inverted pendulum system, and the population ecological system.Chapter 1 illustrates the motivations on the research of switching systems, especially the stochastic switching systems, and presents the literature review of MJSs and S-MJSs.Chapter 2 studies the time-delaying asynchronous control problem of a class of discrete-time Markov jump linear systems (MJLSs) subject to time delays which occur in both the system state and mode detection. With an extended state space of system mode, the underlying system is able to be remodeled as an MJLS in which the system complexity and the number of system modes are both increased to a certain extent, but remain unchanged when the mode detection delay becomes larger. The TPs for the remod-eled MJLS are also provided based on the multiple-step TPs of the original MJLS. Then, a sufficient condition guaranteeing the stochastic stability of the system is obtained via a modified stochastic Lyapunov function, which leads to the time-delaying asynchronous controller design method. The developed approach effectively solved the problem that the number of modes for the constructed MJLS will grow exponentially as the mode detection delay increases.Chapter 3 investigates the H∞ control and model predictive control problems for the fuzzy MJSs with partly unknown transition probability matrices. First, we study the H∞ control for a class of fuzzy MJSs whose antecedent parts of fuzzy rules are mode-dependent, i.e., the premise variables and/or their fuzzy partitions can be different in different modes. Compared with the fuzzy MJS with mode-independent antecedent parts, the number of fuzzy rules is substantially reduced for the same modeling precision, which in turn effectively relieves the computational burden during the stability analysis and con-troller design. Then, the model predictive control for fuzzy MJSs with input and output constraints is discussed. By introducing some external parameters, the proposed method enjoys less conservatism than the previous results.Chapter 4 is devoted to the state-feedback control and H∞ control problems for S-MJSs in terms of σ-error mean square stability. Firstly, the stability of S-MJLSs is discussed using the semi-Markov kernel. Unlike the previous methods based on the TPs with discrete time as units, our approach circumvents the complex calculation of the up-per bound or approximations of these TPs. Next, the time-varying Lyapunov function approach is proposed, and the comparison of conservatism between the theoretical re-sults based on time-invariant and time-varying Lyapunov function approaches is given in details. Further, the obtained stability criteria are extended to stabilization problems, and the time-varying state-feedback controller design method is proposed. Finally, based on the time-varying Lyapunov function approach and time-varying control scheme, the state-feedback control and H∞ control issues are examined for fuzzy S-MJSs. It should be pointed out that the problem of eliminating the matrix power, which is produced during the analysis with the switching instants being the minimum units, is tackled by the pro-posed approach via introducing some external parameters. In comparison with previous methods, the approach proposed in this chapter not only generalizes the scope of S-MJSs by fully utilizing the probability density function (PDF) of sojourn time, but also reduces the conservatism of the derived results.Chapter 5 is concerned with a class of discrete-time S-MJLSs subject to exponen-tially modulated periodic (EMP) PDF of sojourn time, and the problems of stability and stabilization are addressed in terms of mean square stability. Necessary and sufficient criteria for mean square stability of the general S-MJLSs are first derived, based on which the numerically testable conditions for the systems with the EMP PDF of sojourn time are developed. The adopted Lyapunov function is sojourn-time dependent (STD), by which the existence conditions of STD controller are further obtained. Since the existing studies are typically focused only on the sufficient criteria, it is believed that the present results will contribute to push the current research to a new phase. The idea behind this chapter provides a novel approach to the research of the mean square stability of S-MJLSs.
Keywords/Search Tags:Markov jump systems, semi-Markov jump sysfems, time-varying transition probability, σ-error mean square stability, time-varying Lyapunov function approach, time-varying control scheme
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