Dual Heuristic Programming for Optimal Control of Continuous-Time Nonlinear Systems Using Single Echo State Network

This article presents an improved online adaptive dynamic programming (ADP) algorithm to solve the optimal control problem of continuous-time nonlinear systems with infinite horizon cost. The Hamilton-Jacobi-Bellman (HJB) equation is iteratively approximated by a novel critic-only structure which is constructed using the single echo state network (ESN). Inspired by the dual heuristic programming (DHP) technique, ESN is designed to approximate the costate function, then to derive the optimal controller. As the ESN is characterized by the echo state property (ESP), it is proved that the ESN can successfully approximate the solution to the HJB equation. Besides, to eliminate the requirement for the initial admissible control, a new weight tuning law is designed by adding an alternative condition. The stability of the closed-loop optimal control system and the convergence of the out weights of the ESN are guaranteed by using the Lyapunov theorem in the sense of uniformly ultimately bounded (UUB). Two simulation examples, including linear system and nonlinear system, are given to illustrate the availability and effectiveness of the proposed approach by comparing it with the polynomial neural-network scheme.