Robust Stability And Stabilization For Singular Systems With Interval Time-Varying Delay

Time delays often occur in many practical and engineering fields and they are often sources of instability and degradation in control systems, so the study of time-delay systems is important both in theory and in practice, and has thus been of great interest to a large number of researchers in the last few decades. However, the range of time-varying delay considered in most exsting literature is from 0 to an upper bound. In practice, a time-varying interval delay is often encounterd, that is, the range of delay varies in an interval for which the lower bound is not restricted to be 0. A typical example of dynamical systems with interval time-varying delay is networked control systems. Since the information of the lower bound of delay must be considered in systems with interval time-varying delay, it is much more complex and difficult to study systems with interval time-varying delays than general time-delay systems and this type of systems has been gradually developed during the last decade. On the other hand, singular system is a kind of dynamic system of more general form and abroad applied background. Up to now, very few papers have investigated the stability and stabilization problem of singular systems with interval time-varying delays, which remains open but challenging.In this dissertation, the background and significance of normal systems with time delays, systems with interval time-varying delays and singular systems are first introduced. The basic characteristics of main methods used to handle delay-dependent stability problem are analyzed in detail, and then the connection among these methods and the limitations of these methods are revealed. Then, the present study situation for systems with interval time-varying delays is introduced from three respects, i.e. selection of Lyapunov-Krasovskii functional (LKF), research methods and research objects, and some open problems in this field are pointed out as well.The robust stability problem for uncertain singular systems with interval time-varying delays is investigated. By using Jensen integral inequality method, constructing the general LKF and introducing some free weighting matrices, a delay-dependent stability criterion for the systems is obtained in terms of linear matrix inequality (LMI). The result is compared with the result by using general integral inequality approach proposed by Zhang xianming, and the conclusion is proved in methematics theory that the results in terms of the two methods are equivalent, but redundant free-weighting matrices are not introduced by using Jensen integral inequality method, so it is more convenient and simpler. Two types of time delay are discussed: continuous but not differential (Case 1) and continuous and differential (Case 2); and two classes of delay-dependent stability conditions are respectively given for the systems to be regular, impulse-free and robustly stable for all admissible uncertainties. Moreover, the relationship between these two classes is also expained.Based on the delay-dependent stability criterion for uncertain singular systems with interval time-varying delays thus derived, four types of state feedback controllers to stabilize the systems are designed, which are memoryless controllers, memorical controllers, resilient (not fragile) controllers related to additive and multiplicative controller gain perturbations. The delay-dependent robust stability conditions are obtained for the corresponding closed-loop systems to be regular, impulse-free and robust stable for all admissible uncertainties. In addition, by adjusting the parameters, the nonlinear terms in the sufficient conditions are transformed into linear ones.The problem of robust stabilization for uncertain singular systems with interval time-varying state and input delays is considered. All the coefficient matrices except the matrix E include norm-bounded uncertainties. Based on the delay-dependent stability criterion for nominal singular systems with interval time-varying delays obtained, by using the idea of generalized quadratic stabilization, the sufficient conditions for the existence of the robust stabilizing memoryless state feedback controllers are given. In the same way, by adjusting the parameters, the conditions in terms of nonlinear LMIs are converted to ones in terms of LMIs. Following that, the special case in which the systems have only the interval time-varying input delays is considered. Compared with the reduction method, the controllers designed by using this method have simpler structures and can be more easily implemented. For uncertain fuzzy singular systems with interval time-varying state delays based on T-S fuzzy model, the Jensen integral inequality method is used to study the delay-dependent stability problem, and the results are extended to systems with time-varying structured uncertainties. Then, based on the stability analysis, memoryless controllers, memorical controllers and resilient (not fragile) controllers related to additive and multiplicative controller gain perturbations are designed. The delay-dependent robust stability conditions in terms of LMIs are obtained for the corresponding closed-loop fuzzy systems to be uniformly regular, uniformly impulse-free and robust stable for all admissible uncertainties.Numerical examples are given to demonstrate the effectiveness of the proposed methods at the end of each chapter in this dissertation.