Optimal Feedback Controller Design For Linear Svstems Based On Finite-time Stability Theory

The finite-time control method can stabilize system in finite time and provide a fast convergence speed for closed-loop system,which has strong theoretical research and practical engineering significance.In this paper,considering quadratic performance index,the finite-time optimal state feedback control and the finite-time optimal tracking control are studied respectively by using theory of finite-time stability and optimization methods.The main research works are as follows.(1)The finite-time optimal state feedback control for linear system is studied.Firstly,combining the finite-time Lyapunov stability theorem,minimum principle and the Hamilton-Jacobi-Bellman equation,a design method of nonlinear finite-time optimal state feedback controller is proposed based on quadratic performance index.The designed feedback controller can not only guarantee the finite-time stability of closed-loop system,but also minimize the performance index function.Then,a specific finite-time V-function which satisfies sufficient conditions is constructed for given system.The controller containing this function enables the system to have a fast convergence speed and can converge to the equilibrium point in finite time.Numerical simulations are presented to illustrate the effectiveness of the state feedback controller.(2)The finite-time optimal tracking control based on augmented state for linear systems is studied.By using of the finite-time stability theorem and the augmented state method,a design method of the finite-time optimal tracking controller is proposed,and sufficient conditions involving the finite-time V-function are given.Then,a finite-time V-function which satisfies the conditions is constructed for given system.The corresponding finite-time tracking controller which enables the tracking error to converge to zero in finite time,having a fast response speed and high tracking accuracy.Numerical simulations are presented to illustrate the effectiveness of the tracking controller.(3)The finite-time optimal tracking control based on the augmented error system for linear system is studied.Firstly,an augmented error system is constructed by using the common methods of preview control.Considering the quadratic performance index,a design method of tracking controller is proposed by using the conclusions of finite-time optimal state feedback control.Then,a finite-time V-function that satisfies the conditions is constructed for given system.The controller with this function makes the augmented error system converge in finite time which has a fast convergence speed.At the same time,tracking error of the original control system can converge in finite time.Numerical simulations are presented to illustrate the effectiveness of the controller.