Feedback Stabilization And Performance Optimization Of Switched Systems

A switched system consists of several subsystems and a switching law. The switched system includes both continuous dynamics and discrete dynamics, which make the system dynamics extremely complicated, and the design very difficult. Previous studies on switched systems emphasize on the performance of continuous dynamics or designing switching law to achieve the desired performance, but seldom discuss the performance of switching.Switching is the intrinsic property of the swithed system, which makes the swithed system different from other systems. It is necessary to develop a new theory to cope with switching based on continuous and discrete system theories. To control the switched system, we need to give consideration to switching characteristics and its impact on system performance, while we consider both continuous dynamics and switching law.In order to understand the switched system well and be a convenient discussion for the subsequent development, we introduce basic knowledge about the switched system in the preface. On this basis, the main contents of this dissertation are summarized in the following:First, we study the stabilization for continuous-time switched linear unforced systems based on a control-Lyapunov function approach. The relationship among the stabilization problem, the optimal switching problem for continuous-time switched unforced systems, and the optimal switching problem for the sampling systems is pointed out. An approach is given to construct control-Lyapunov functions. It is concluded that a switched linear unforced system is exponentially stabilizable if and only if there is a control-Lyapunov function. By solving the optimal switching problem, we can design a stabilizing switching law for switched unforced systems.Second, a constructive feedback design method based on canonical decompositions is presented for the stabilization of switched linear control systems. Combined with controllability, we deal with the stabilization problem as follows. On the basis of the controllability criterion, the first step is to transform a controllable multi-input switched system into a controllable single-input one by (nonregular) feedback reduction, the next step is to classify into several cases each with a normal form, and the last step is to verify case by case the stabilizability of the normal forms. In this dissertation, a lemma is given for the quadractic stabilizability. It is pointed out that the ratio of the cases that are quadratically stabilizable to the cases that are non-quadratically stabilizable becomes smaller and smaller with the system dimension increasing. Therefore, it is important to design switching laws and control laws that make the system non-quadratically stabilizable. We discuss the third-order systems as a case study.Third, considering both switching characteristics and its impact on system performance, we investigate the robustly stabilizing switching problem, where we define several switching distance notions between the nominal and perturbed switching laws; propose a constructive algorithm to calculate switching distance; analyse the robustness for the switched system with perturbed switching, and discuss how to design a good switching law against kinds of the system perturbations. The research on switching laws can help us understand the dynamics of the switched system in theory, and improve the switching control design in application.Finally, the conclusions and the prospects of future research are given at the end of this dissertation.