Analysis And Synthesis Of Semi-markovian Jump Systems

When subject to random changes, most physical systems would observe changes in their structures. These changes may result from abrupt phenomena such as component and interconnection failures, parameters shifting, tracking, and the time required to measure some of the variables at different stages. Systems with this character may be modeled as hybrid ones, that is, a discrete random variable called the mode or regime is appended to the continuous state variable. The mode describes the random jumps of the system parameters and the occurrence of discontinuities. With a strong theoretical signaficance and application value, such a system model is useful particularly since it allows the decision maker to cope adequately with the discrete events that disrupt or change the normal operation of a system significantly, by using the knowledge of their occurrence and the statistical information on the rate at which these events take place. The Markovian stochastic process, with its powerful modeling capability in application areas such as the aerospace industry, industrial processes, biomedical industry and socioeconomics, has proved to be of vital importance as a typical class of hybrid dynamical system. However,MJS have many limitations in applications, since the jump time of a Markov chain is, in general, exponentially distributed, and the results obtained for the MJS are intrinsically conservative due to constant transition rates. Compared with the MJS, semi-Markovian jump systems(S-MJS) are characterized by a fixed matrix of transition probabilities and a matrix of sojourn time probability density functions. Due to their relaxed conditions on the probability distributions, S-MJS have much broader applications than the conventional MJS. Thus, this area of research is significant because of both its theoretical and practical values.The emphasis in this thesis is the studies of stochastic stability analysis, robust control, filtering and fault detection for S-MJS, with special attentions on some complex dynamic systems such as delay systems, singular systems, neutral-type stochastic systems,fuzzy systems and neural networks etc. The presentation of this thesis is divided into two parts: Part one considers the S-MJS with phase-type sojourn time distribution. By using a supplementary variable technique and a plant transformation, phase-type S-MJS have been transformed into its associated Markovian jump systems, and the control and estimation problems are then solved for the associated Markovian jump systems. Part two focuses on the S-MJS with sojourn time governed by non-exponential distribution and the filtering problems have been solved for S-MJS with partially unknown, entirely unknown,and polytype uncertain transition probabilities. The main contents are as follows:Chapter 1 first introduced the research background and significance of Markovian systems, as well as the current research status Markovian jump systems, so as to provide a basis of reference for further research of semi-Markovian jump systems. Secondly, the main differences between semi-Markovian jump systems and Markovian jump systems have been provided, followed by a description of the advantages of the semi-Markovian jump systems and its broad application prospects. Additionally, we have mentioned several problems that are yet to be solved, methods that require refinements and the main research contents of this dissertation.Chapter 2 investigated the problem of stochastic stability for a class of semiMarkovian systems with mode-dependent time-variant delays. By Lyapunov function approach, together with a piecewise analysis method, a su?cient condition is proposed to guarantee the stochastic stability of the underlying systems. As more time-delay information is used, our results are much less conservative than some existing ones in literature.Finally, two examples are given to show the effectiveness and advantages of the proposed techniques.Chapter 3 is concerned with the constrained regulation problem for a class of singular semi-Markovian jump systems. A semi-Markovian system of this kind has been transformed into its associated Markovian system via a supplementary variable technique and a plant transformation. Motivated by recent develops in positively invariant set, necessary and su?cient conditions for the existence of full rank solutions for a class of nonlinear equations are derived, and a new algorithm that provides a solution to the constrained regulation problem is presented. The correctness and effectiveness of the proposed approaches are illustrated by a simulation example.Chapter 4 considers the state estimation and sliding mode control problems for phase-type semi-Markovian jump systems. Using a supplementary variable technique and a plant transformation, a finite phase-type semi-Markov process has been transformed into a finite Markov chain, which is called its associated Markov chain. As a result, a phase-type semi-Markovian jump systems can be equivalently expressed as its associated Markovian systems. A sliding surface is then constructed and a sliding mode controller is synthesized to ensure that the associated Markovian jump systems satisfy the reaching condition. Further, an observer-based sliding mode control problem is investigated. Su?-cient conditions are established for the solvability of the desired observer. Two numerical examples are presented to show the effectiveness of the proposed design techniques.Chapter 5 is concerned with the quantized control design problem for a class of semiMarkovian jump systems with repeated scalar nonlinearities. A semi-Markovian system of this kind has been transformed into an associated Markovian system via a supplementary variable technique and a plant transformation. A su?cient condition for associated Markovian jump systems is developed. This condition guarantees that the corresponding closed-loop systems are stochastically stable and have a prescribed H∞ performance. The existence conditions for full- and reduced-order dynamic output feedback controllers are proposed, and the cone complementarity linearization procedure is employed to cast the controller design problem into a sequential minimization one, which can be solved e?-ciently with existing optimization techniques. Finally, an application to cognitive radio systems demonstrates the e?ciency of the new design method developed.Chapter 6 is concerned with the fault detection problem for semi-Markovian jump systems. Firstly, based on the Takagi-Sugeno fuzzy approach, a fault detection filter is constructed such that the prescribed performance requirement can be guaranteed. The existence conditions for full- and reduced-order fault detection filters are provided, and the cone complementarity linearization procedure is employed to cast the filter design problem into a sequential minimization one, which can be solved e?ciently with existing optimization techniques. A numerical example and a practical application are given to to illustrate the effectiveness of the proposed technique. Secondly, the fault detection filtering problem is concerned for a class of switched systems with semi-Markovian parameters and known sojourn probability. An event-driven control strategy is developed to reduce the frequency of transmission, and a su?cient condition for sojourn probability dependent switched systems is presented. A fault detection filter is designed such that the corresponding filtering error system is stochastically stable and have a prescribed performance. In addition, the cone complementarity linearization procedure is employed to cast the filter design problem into a sequential minimization one, which can be solved efficiently with existing optimization techniques. Finally, a numerical example is provided to verify the effectiveness of the proposed design scheme.Chapter 7 is concerned the problem of exponential passive filtering is investigated for a class of stochastic neutral-type neural networks with both semi-Markovian jump pa- rameters and mixed time delays. Our aim is to estimate the state by a Luenberger-type observer such that the filter error dynamics are exponentially mean-square stable with an expected decay rate and an attenuation level. Su?cient conditions for existence of passive filters are obtained and a cone complementarity linearization procedure is employed to transform a nonconvex feasibility problem into a sequential minimization problem,which can be readily solved by existing optimization techniques. Numerical examples are given to demonstrate the effectiveness of the proposed techniques.