Stability Analysis And Stabilization For Several Switched Singular Systems

Switched systems, which are a class of important hybrid systems, are consisted of several subsystems and a switching law. A switched system can be stable by switching between each subsystem under a switching law. Switched singular (SS) systems whose subsystems are singular systems are a class of switched systems. SS systems have recently become one of the research focuses all over the world because they exist in many practical systems, such as economic systems, biological systems. Typical switched system is composed of a family of subsystems and a switching signal that orchestrates the switching among them. Although each subsystem is very simple, the whole system that consists of switching strategy maybe have very complex dynamic characteristics.In recent years, the research on SS systems has achieved some results. But there is little research on the stability, stochastic control, switching control and observer-based controller design for switched singular systems. Based on the effects of the singular matrices being considered sufficiently, the problems of stability and stabilization are investigated for SS systems by mathematic theory and control theory such as Lyapunov stability theory, stochastic theory. The main contributions of this dissertation are summarized as follows:(1) Using multiple Lyapunov function method, the sufficient conditions of stability for arbitrary switching law of the SS system are obtained and a method to construct a class of multiple Lyapunov functions is presented. However, the problems are difficult to tackle in SS systems since stability, regularity and causality should be considered at the same time and switching between several singular systems makes the problem more complicated. It's difficult to find a common Lyapunov function to satisfy the different subsystems. To solve this problem, based on the existing criteria of stability for discrete-time switched systems and using multiple Lyapunov function method, a sufficient condition of stability for a class of discrete-time SS systems for arbitrary switching laws is proposed. Then a method to construct a class of multiple Lyapunov functions is presented.(2) Using LMI method combined with multiple Lyapunov function method, the problems of stochastic stability for a class of SS systems are studied and a method to design a stochastic controller is presented to make the closed-loop SS systems stable. For SS systems, the problem of controller design is more complicated since the switching property and the singular matrices coexist simultaneously. To solve this problem, the sufficient conditions for a stochastic controller existence are obtained in the form of strict linear matrix inequality. It's the first time that Bernoulli variable is used to design controllers in stabilization problem for SS systems. Based on a linear matrix inequality technology, a sufficient existence condition for such new controller is proposed, which bridges the gain-scheduled (GS) and gain-common (GC) controller design methods. Compared with traditionally GS and GC controllers, the accessible probability of mode is taken into consideration in the presented design method.(3) Using multiple Lyapunov function method,the switching law is designed via dividing the state space for a class of discrete-time SS systems and the sufficient conditions of asymptotic stability of the SS system are obtained. The systematic dynamic would be changed when the systems switched between subsystems, so the design of switching strategy should be considered synthetically when the stability problem is studied. Thus we divide the state space by S-procedure method, and based on the existing criteria of stability for normal switched systems a method to design state feedback controllers is proposed using multiple Lyapunov function method for a class of discrete-time SS systems.(4) The problems of observer-based stabilization for a class of SS systems are studied using LMI method and the observer-based controllers are designed to make the closed-loop SS systems stable.The method estimating the unknown state by designing the observer has practical implications since the state can't be measured directly. So a method to design observer-based controllers by strict LMI is proposed. And a method to obtain the gain matrices by LMI is given for a class of SS systems.(5) The robust H? filtering problem of SS systems being uncertain and partially unknown respectively has been studied. Sufficient existences of switched filters enduring some perturbations are proposed. Based on the presented results, a switched filtering method under such general conditions is developed, where common filter is also designed simultaneously. From the obtained criteria, it is claimed that the presented method makes the designs of common and switched filters into a unified framework. And a sufficient existence of passive filters is given.