Asymptotic Stability And H_∞Control For Generalized Systems With Time-Delay

Generalized systems with time-delay are also named singular systems withtime-delay, differential difference algebraic equations with time-delay or differentialalgebraic systems with time-delay. It is essentially systems with time-delay coupledby differential (difference) equations with time-delay and algebraic equations.However, in many practical problems, the state change rate not only depends on thecurrent state, but also depends on the past history state, which performance in themathematical model is the systems with time-delay. Because time delays, as the mainfactors, can seriously affect the system stability, the researches on systems withtime-delay and generalized systems with time-delay have important significance bothin theory and in practice. However, the range of time delay for generalized systemsconsidered mainly in present research works is constant time delay. In practice, a timedelay is often encountered, that is, the range of delay varies in an interval. Since thechanges of the time delay must be considered in systems, it is much more complexand difficult to study systems with interval time-varying delay. On the other hand,switched system is a class of hybrid system which composed of several subsystemsand a switching law that specifies the active subsystems. If all of the subsystems aregeneralized systems, in that case they are called generalized switched systems.Generalized switched systems with interval time-varying delay attract the interest ofseveral scholars more and more due to their extensive applications in engineering andgreat significance of theoretical study.Stability is one of the fundamental problems in control theory. Now, there havebeen many significant results on stability problems of generalized systems, butinvestigation on stability problem of nonlinear generalized systems or generalized systems with interval time-delay is premature. In addition, the theory of H_∞controloriginated early in1980s. Since it has remedied the shortage of control theory inpractice application and the wide applicability of the model itself, it has been attachedimportant and developed into one of the most important aspect for the control theory.The research of H_∞control for normal systems has been developed sufficiently, allkinds of methods to design H_∞controller are put forward one after another, but theresearch on generalized systems, especially generalized systems with intervaltime-delay, has still a lot of space for improvement. In recent years, the researches onthe stability and H_∞performance analysis are one of the hotpots science problemsof the current control theory for control community’s researchers. The aboveproblems are mainly studied in this paper. Aiming at these problems, several classesof generalized systems with time-delay or interval time-varying delay are studied.Some related conclusions are obtained about system stability and H_∞controllerdesign method. The main contents are as follows:In chapter1, the historical development and broad research situation aboutstability and H_∞control for generalized systems with time-delay are reviewed.In chapter2, the fundamental knowledge of singular systems and linear matrixinequalities is introduced. Some definitions and lemmas which will be used later inthis paper are presented.In chapter3, the problem of asymptotic stability is considered for a class ofnonlinear generalized systems with time-delay. It is considered that in actual systemsthere inevitably exists some nonlinearity. By constructing Riccati matrix equations,using Lyapunov function method and related inequalities in this paper, someasymptotic stability criterions are obtained for a class of nonlinear generalizedsystems with time-delay. Based on the results, there criterions are extended to a classof function with double variables in the nonlinear term. The given variables satisfyLipschitz condition, respectively. Then the new asymptotic stability criterions areobtained for the discussed nonlinear generalized system with time-delay. At last, by using the MATLAB software, the numerical example is provided to demonstrate therationality of the asymptotic stability criterion for the generalized systems withtime-delay discussed. It lays some theoretical base on the study of asymptomaticstability for nonlinear generalized systems with time-delay.In chapter4，the problems of stability, H_∞control and H_∞memoryless statefeedback controller design for a class of discrete switched generalized system withinterval time-varying delay are investigated. Based on linear matrix inequalities,common Lyapunov function method and Lyapunov-Krasovskii functional method,stability and H_∞performance analysis are discussed for the system. According tothe study of discrete switched generalized system with interval time-varying delaywithout control, the design method of H_∞memoryless state feedback controller isfinally obtained for discrete switched generalized system with control and intervaltime-varying delay in any switching law. Finally, a numerical example is given todemonstrate the feasibility and the effectiveness of the proposed method.In chapter5, a conclusion is given about the research works in this paper, at thesame time the perspective on the furthermore research is also proposed.