Set Stability Analysis And Feedback Stabilization For Switched Systems

Although increasing attention has been paid to the research of stability theory of switched systems and more and more progress has been reported, most conclusions are obtained based on the assumption that the individual subsystems of switched systems have the origin as a common equilibrium point. Actually, many switched systems converged to an equilibrium set or a manifold eventually other than an equilibrium, such as electrical oscillators, popu-lation dynamics and so on. As pointed out by Rouche, the asymptotic stability of a set is a natural concept available for many practical applications rather than the asymptotic stability of equilibria. Therefore, set stability analysis and design of switched systems is more ap-plicable for practical engineering and is very interesting for further investigations. But, the research of set stability analysis and feedback stabilization of switched systems is still in its beginning stages, and it needs to be further improved.The purpose of this dissertation is to investigate the set stability analysis and feedback stabilization for switched systems. Drawn from the notions and methods of the theory for nonlinear feedback control, we proposed the set stability criteria and the corresponding de-sign techniques of feedback controllers.This thesis is organized as follows:1. The definitions of output-to-V(x) stability, small-time V(x) observability and large-time V(x) observability are proposed to extend the concepts of output-to-state stability and state norm observability. Based on these extended concepts and the boundedness of output integral value, some sufficient conditions are proposed for the stability of the zero-value set of the function V(x) of the switched systems, with are proved via the common Lyapunov function method and multiple Lyapunov function method, respectively. Finally, the relation-ships among the concepts proposed in this paper and the existing notions are discussed in details.2. In order to discuss set stabilization of switched systems by using state feedback, a concept of input-to-V(x) stability is proposed, which extends the notion of input-to-state stability. Based on the input-to-V(x) stability notion, the invariant set state feedback sta-bilizability conditions are proposed for a class of switched systems with Lyapunov stable subsystems. Rigorous proofs are given by using common Lyapunov function and multiple Lyapunov functions respectively. The relationship between input-to-V(x) stability and set input-to-state stability is discussed in details.3. A concept, input/output-to-V(x) stability, is proposed, which extends the concept of input/output-to-state stability. By virtue of this notion and passivity results, some sufficient conditions of invariant set output feedback stabilizability for a class of switched systems are proposed. The results are proved by using common Lyapunov function and multiple Lyapunov functions respectively. The relationship between input/output-to-V(x) stability and input/output-to-state stability is also discussed in details.4. The concepts of state norm observability are further extended in this chapter. Base on these extended notions, the stabilization problem of invariant sets is discussed by using bounded output feedback as well as dynamic output feedback for switched systems with passive subsystems, which are shown by means of multiple Lyapunov functions.5. It is difficult to find a common Lyapunov function for switched systems, and stabil-ity analysis of switched systems by using multiple Lyapunov functions always needs to get the value of Lyapunov functions at switching instants, which violates the classical idea of Lyapunov's methods. As an extension of LaSalle's invariance principle, integral invariance principle can analyze stability by using the boundedness of output integral value and the ob-servability of the system, where the construction of Lyapunov function can be relieved. Here, integral invariance principle is generalized for a class of switched linear systems. Moreover, asymptotic stability is obtained under the observability of switched linear systems.