Study On The Stability Of Switched Delay Systems

Switched systems are important class of hybrid systems,which have been widely applied to the engineering and have great significance of theoretical study. Switched systems consist of more than one subsystem and a switching rule indicating the active subsystem at each instant of time. The state trajectory of system is continuous at the time when the model is changed. If time delays are in each subsystem or in the switching rule, it is called switched systems with delays. Founding the criteria of stability of switched systems with delays in the sense of Lyapunov (focusing on asymptotical stability) and Bounded Input and Bounded Output stability (shorting for BIBO stability) is the main consideration of this paper.Firstly, assuming that there exists a Hurwitz linear convex combination in the coefficient matrices of switched linear systems with mixed delays (both discrete and distributed time delays), this paper is to propose the uniform asymptotic stability conditions for this class of switched systems by Lyapunov functional and inequality technique. According to the linear partition of state space, we can construct the stable regions and a state dependent switching rule. By a defined Lyapunov equation and certain inequality, sufficient conditions of stability are given.Secondly, this paper is to discuss the BIBO stability of continuous large scale systems, continuous systems with single delay, mixed delays, multiple mixed delays and discrete systems with single delay. For continuous large scale systems, we use Lyapunov function and inequality to obtain the novel BIBO stabilization criteria expressed in terms of linear matrix inequality (LMI) for continuous multivariable feedback systems. Based on this, the design of state feedback is given by matrix transform. The issue of robust BIBO stabilization for uncertain large scale systems is also addressed. For continuous control systems with single delay, mixed delays, multiple mixed delays, delay-independent and delay-dependent stabilizable criteria for these control systems are presented by theory of Lyapunov functional and Riccati equation, linear matrix inequality technique to guarantee that the bounded input lead to the bounded output. The robust BIBO stability for such systems with perturbation and uncertainty is also discussed. When the norm condition of time-varying uncertain matrices is satisfied, novel robust BIBO stabilization criteria delay-independent and delay-dependent are also established by the introduction of free-weighting matrices and the Schur's lemma. For the discrete systems with time delay, an equivalent transformation to get systems without delay is presented and a control law is designed. At last, delay-independent BIBO stability criteria are given by iterative relation and theory of matrix norm.Lastly, this paper is concerned with the problems of BIBO stabilization of the switched linear systems in presence of time delays under feedback control. This part consists of two cases. Under the assumption that there also exists a Hurwitz linear convex combination in the coefficient matrices of delayed switched linear systems, criteria of BIBO stabilization are established by linear partition of state space and a state-dependent switching rule. Stabilization criteria and the design of state feedback controller are derived and proposed by Lyapunov functional and inequality technique. BIBO stabilization criteria are given in terms of LMI and can be easily solved by LMI Toolbox in Matlab. For switched delay systems in fixed switching regions, the main contribution of the paper is the derivation of sufficient conditions of BIBO stabilization in the form of algebraic Riccati matrix equation based on piecewise quadratic Lyapunov functional. The switching rule is driven by event and state-dependent.