Research On Finite-Time Stability Analysis And Control Design

This paper mainly deals with various finite-time stability analysis and control design problems. The properties of finite-time stability, finite-time contractive stability and finite-time boundedness are investigated for nonlinear systems, linear time-invariant systems and linear time-varying systems, respectively. And then the state feedback and the output feedback controllers are synthesized in terms of finite-time stability (boundedness). The main contents of this paper are composed of the following three parts:(Ⅰ) Finite-time stability analysis and control design for nonlinear systems.In this part, the problems of finite-time stability analysis and control design are investigated for nonlinear systems. Different from the existing results, the problems studied here are more general: (1) Based on class K_∞functions, the more general definitions are proposed to kinds of finite-time stabilities of nonlinear systems; (2) By Lyapunov functions, the easier and more feasible sufficient conditions are obtained to ensure various finite-time stabilities of nonlinear systems; (3) The Sontag's formula is applied to develop the controller design approaches for finite-time stability, quasi-contractively stability and disturbance rejection of nonlinear systems, respectively.(Ⅱ) Finite-time stability analysis and control design for a class of linear time-invariant systems.In this part, the problems of finite-time stability analysis and control design are dealt with for a class of linear time-invariant systems. In contrast to the previous literature, the systems studied here subject to disturbances generated by an exosystem with fixed reference model. Firstly, based on Lyapunov functions, the sufficient conditions are presented to ensure the finite-time boundedness. Then the design problems are investigated considering both the state feedback and the output feedback cases. In the latter design case, the famous separation principle is applied: First a state feedback controller is designed which guarantees finite-time stability of the closed loop system by assuming that the whole state vector is available; Then a state observer, which tried to retain the properties guaranteed by the state feedback controller is synthesized. The state observer design is based on the classical Luenberger observer. At last, the feasibility analysis is made for the main results of this part and all sufficient conditions are converted into linear matrix inequalities.(Ⅲ) Finite-time stability analysis and control design for a class of linear time-varying systems.In this part, the problems of finite-time stability analysis and control design are considered for a class of linear time-varying systems. Compared with the second part, the systems studied here are extended from time-invariant cases to time-varying ones. So the research methodology is also greatly different: Firstly, based on time-varying Lyapunov functions, the sufficient conditions which are greatly different from the ones brought forward in the second part ensuring finite-time boundedness are presented. Then the design problems are investigated dealing with both the state feedback and the output feedback cases. Different from the second part, the output feedback controller design presented here can be implemented as a single optimizing problem (which means that the controller and state observer are designed at the same time) and the controller structure is more general. So it is less conservable. At last, the feasibility analysis is made for the main results of this part and all sufficient conditions are converted into linear differential matrix inequalities.