Adaptive Dynamic Programming for Minimal Energy Control with Guaranteed Convergence Rate of Linear Systems

The traditional linear quadratic optimal control can be summarized as finding the state feedback controller, so that the closed-loop system is stable and the performance index is minimum. And it is well known that the solution of the linear quadratic optimal control problem can be obtained by algebraic Riccati equation (ARE) with the standard assumptions. However, results developed for the traditional linear quadratic optimal control problem cannot be directly applied to solve the problem of minimal energy control with guaranteed convergence rate (MECGCR), because the standard assumptions cannot be satisfied in the MECGCR problem. In this paper, we mainly consider the problem of MECGCR and prove that ARE can be applied to solve the MECGCR problem under some conditions. Furthermore, with the assumption that the system dynamics is unknown, we propose a policy iteration (PI) based adaptive dynamic programming (ADP) algorithm to iteratively solve the ARE using the online information of state and input, without requiring the a priori knowledge of the system matrices. Finally, a numerical example is worked out to show the effectiveness of the proposed approach.