Finite-Time Control For Some Classes Of Discrete-Time Switched Systems

The Finite-time Stability(FTS) of a system means that the state of the system does not exceed a certain threshold during a specified time interval when a bound on the initial condition is given. FTS characterizes the transient nature of systems and is a kind of quantitative analysis for systems. In this thesis, FTS analysis and controller design are discussed for some classes of switched systems:discrete-time nonlinear singular switched system, without delays and with time-varying delays, and discrete-time positive Markov jump systems, which can be regarded as a special case of switched systems.Chapter 1 introduces the research background. Firstly, the history and the development of FTS are reviewed. The lack in the finite-time control theory of nonlinear singular switched system and positive Markov jump systems is pointed out, which motives this thesis. Then the main work of this thesis is given. Finally, the notations used in this thesis are listed.In Chapter 2 and Chapter 3, the finite-time control problems under ar-bitrary switching signals are discussed for discrete-time nonlinear singular switched systems are discussed, in cases that without delays and with time-varying delays, where nonlinear item satisfies the Lipschitz-like constraint con-dition. In Chapter 2, the concept of FTS is generalized to discrete-time nonlin-ear singular switched systems and the concept of uniform FTS is introduced. Based on the switched Lyapunov function, and by using the Schur complement and the implicit function theorem, sufficient conditions are developed in the form of linear matrix inequalities (LMIs) which guarantee that discrete-time nonlinear singular switched system is regular, causal, has a unique solution in a neighborhood of the equilibrium point, and is uniformly finite-time stable with respect to (c1, c2, N, R). Then based on the above conditions, an existence con-dition of the finite-time state feedback stabilization controller is proposed and the design method is given. Chapter 3 is concerned with the FTS analysis and state feedback stabilization controller design for a class of discrete-time non-linear singular switched systems with time-varying delay. Sufficient conditions are developed which guarantee that the nonlinear singular switched system with time-varying delays is regular, causal, has a unique solution in a neigh-borhood of the equilibrium point, and is uniformly finite-time stable. Then a delay-dependent condition on the existence of a finite-time state feedback controller is proposed.Chapter 4 studies the stochastic finite-time control problem for discrete-time positive Markov jump systems. For the first time, the necessary and sufficient conditions for the FTS of discrete-time deterministic positive sys-tems is proposed, and the FTS analysis problem is transferred into solving linear programming (Lp) problems. Then the stochastic finite-time stability analysis for positive Markov jump systems is transformed into the finite-time stability analysis for deterministic positive systems. The sufficient conditions of stochastic finite-time stability for positive Markov jump systems are pro-posed, and the design method of the state feedback finite-time controllers is given.