Study Of Control Problems For Singular Markovian Jump Systems

Currently, although much effort has been made in control for singular Markovian jump systems, results on this topic are much fewer than those on normal Markovian jump systems. Many unsolve problems need to be studied for this kind of systems. In this dissertation by using the theory of the matrix analysis and the linear matrix inequality (LMI) technique, the problem of control for singular Markovian jump systems is studied, which is motivated by the descriptor system theory, the stochastic system theory and the advanced robust control theory. Main contributions are summarized:(1) The problem of H? fuzzy control for a class of nonlinear time-delay singular Markovian jump systems is discussed. H? fuzzy control for a class of nonlinear time-delay singular Markovian jump systems with partially unknown transition rates is studied first, this class of systems under consideration can be described by Takagi-Sugeno (T-S) fuzzy models, fuzzy state feedback controllers are designed such that closed-loop systems are not only regular, impulse-free and stochastically stable, but also satisfy a prescribed H? performance in terms of LMIs. And this kind of systems with completely known transition rates can be viewed as a special case of the ones we consider here. Then by the method of the augmented matrix, the problem of H? fuzzy control for this kind of systems with completely known transition rates is discussed. When the number of fuzzy rules is relatively large, the quality of the computation for the condition obtained is greatly reduced.(2) The problem of finite-time H? control for singular Markovian jump systems is discussed. Finite-time H?, control for a class of singular Markovian jump systems with partially unknown transition rates based on the state feedback is studied first, sufficient conditions on the stochastic singular finite-time boundedness of singular Markovian jump systems with partially unknown transition rates are obtained, then results are extended to the analysis on the stochastic singular finite-time boundedness of singular Markovian jump systems with completely known transition rates. Then state feedback controllers are designed to ensure the stochastic singular finite-time boundedness and the stochastic singular H? finite-time boundedness of underlying closed-loop systems in forms of strict LMIs. At last, the design of finite-time H? control for singular Markovian jump systems is studied via the static output feedback.(3) By the method of the augmented matrix the problem of robust fuzzy finite-time control for nonlinear singular Markovian jump systems is discussed, this class of systems under consideration can be described by T-S fuzzy models. The designed algorithm for the state feedback controller is provided to guarantee the stochastic singular finite-time boundedness and the stochastic singular H? finite-time boundedness for closed-loop sys-tems of singular Markovian jump systems in terms of strict LMIs with a fixed parameter. Then the method is extended to the designed algorithm for the static output feedback controller of singular Markovian jump systems.(4) The problem of fuzzy stochastic optimal guaranteed cost control for a class of bio-economic singular Markovian jump systems is discussed. By considering the price coefficient to be influenced by a Markovian process, a bio-economic singular Markovian jump model is established, then the model is described by the T-S fuzzy singular Marko-vian jump system. A new design method of fuzzy state feedback controllers is presented to ensure not only the regularity, nonimpulse and stochastic singular finite-time bounded-ness for closed-loop systems of this kind of systems, but also an upper bound achieved for the cost function in forms of strict LMIs. Finally this model is applied to the practical example of the elver breeding.(5) The problem of robust finite-time H? control for a class of uncertain sin-gular stochastic Markovian jump systems with partially unknown transition rates vi-a proportional-differential control law is discussed. Uncertainties are not only in state matrices and input matrices, but also in differential matrices. The stochastic finite-time stability for singular stochastic Markovian jump systems is defined. Based on this def-inition, sufficient conditions for control laws designed are established to guarantee the stochastic finite-time stability for closed-loop systems of singular stochastic Markovian jump systems with partially unknown transition rates and the H? performance satisfied. Numerical examples show the effectiveness of the new definition.(6) The problem of the sliding mode control design for singular stochastic Marko-vian jump systems with uncertainties is studied. A suitable integral sliding function is proposed, a new sufficient condition is developed which not only guarantees the stochas-tic admissibility of the sliding mode dynamics, but is also constructive and determines all the parameter matrices in the integral sliding function. Then a sliding mode control law is synthesized such that the reachability of the specified sliding surface can be ensured. Finally results developed are extended to the sliding mode control design of stochastic Markovian jump systems.