New forms of Riccati equations and the further results of the optimal control for linear discrete-time systems

Abstract

For linear discrete-time systems, the traditional finite horizon optimal controller is proved to render the closed-loop systems asymptotically stable under some assumptions in literature. In this paper, a new form of finite horizon discrete-time Riccati equation is proposed. It is proved that the new form of finite horizon discrete-time Riccati equation is equivalent to the other three old ones. Based on this new form of finite horizon discrete-time Riccati equation, the finite horizon optimal controller of linear discrete time systems is proved to render the closed-loop system exponentially stable without any assumptions. At the same time, a new form of infinite horizon discrete-time Riccati equation is proposed when the discrete system is controllable or stabilizable. It is proved that the new form of infinite horizon discrete-time Riccati equation is equivalent to the other three old ones too. Based on this new form of infinite horizon discrete-time Riccati equation, the infinite horizon optimal controller of linear discrete-time systems is proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally an unstable batch reactor and an unstable inverted pendulum are used to verify the theory results of this paper.