Research On The Stability Of Switched Systems With Unstable Subsystems

Abstract As one of the most important hybrid systems, switched systems are often used for modeling various control problems and some complex processes in nature. In spite of their apparent simplicity, switched systems display a very complicated dynamical behaviors because of the multiple subsystems and various possible switching signals. In the last decade, the investigations of stability for switched systems have attracted consid-erable attention in Systems Engineering, Computer Sciences and Mathematics commu-nities. This dissertation is devoted to research the switched systems with unstable sub-systems, which is theoretically challenging and of fundamental importance to numerous applications. This paper is organized as follows.In Chapter1, we introduce some research background and status on the dissertation.Chapter2studies the problem of robust stability for switched linear systems with all the subsystems being unstable. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, suffi-cient conditions of exponential stability for both deterministic and uncertain switched lin-ear systems are presented by using the invariant subspace theory and average dwell time method. Under the given switching signals, the activation time ratio between two sets of unstable subsystems is required to be located between two constants which are computed using the desired stability degree of the switched system. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law.Chapter3discusses some stability properties of discrete-time switched linear sys-tems (DSLSs) with unstable subsystems. First, under certain hypothesis, the necessary condition of stability for DSLSs is obtained. Second, using the average dwell time (ADT) strategy, we discuss the sufficient condition of exponential stability for switched linear systems with two assumptions. Finally, two examples are presented to show the effec-tiveness of the proposed approach. It is worth pointing out that there are some results for DSLSs obtained by using ADT technique. Those results are based on the switching signal is arbitrary, which usually is used to deal with continuous-time switching law. However, the switching signal of DSLSs is not arbitrar}'because the state of those systems depends only on discrete-time points not on all time points. In order to solve this contradiction, this chapter proposes a novel concept of discrete-time switching signal (DTSS) and develops the ADT technique for DTSS.In Chapter4, the classical Lyapunov theorems for nonlinear autonomous systems are extended. In particular, our results relax the requirement of V along the system trajec-tories being non-increasing, and thus Lyapunov function is a special case of generalized Lyapunov function. In particular, stability theorems of nonlinear systems are present-ed by replacing "V along the system trajectories is non-increasing" with "V along the system trajectories may increase its value during some proper time intervals". Different from the multiple Lyapunov function method, it only needs to construct a Lyapunov-like function instead of a collection of Lyapunov-like functions. These results are applied to the research for stability of switched systems with unstable subsystems. Two numerical examples are presented to demonstrate the proposed approach.Chapter5investigates the relationship between reset positive systems and switched positive systems. Firstly, as an extension of reset control systems, a novel concept of reset positive systems (RPS) is proposed. Secondly, the relationship between RPS and switched positive systems (SPS) is established. These relationship provides us an effective approach to transfer the stability of reset control system to investigate the stability of the discrete-time switched system which may be solved with many tools. Thirdly, the sufficient and necessary condition of those RPS asymptotically stable for any reset is obtained, while the sufficient and necessary condition of those RPS stabilizable is also presented. Last but not least, some stability criteria for RPS are given.Chapter6concerns the problem of stability for switched systems with extended av-erage dwell time in both continuous-time and discrete-time cases. By introducing three novel concepts of closed-chain, r-open-chain and quasi-cyclic switching signals, the sta-bility and stabilization conditions of the switched systems with ADT or mode-dependent ADT (MDADT) switching are obtained, we develops and enriches the existing results of stability under constrained switching. On the other hand, the chapter provides a solution to the open problem of how to obtain a tighter bound on ADT and MDADT.