Adaptive tracking control for a class of continuous-time uncertain nonlinear systems using the approximate solution of HJB equation

Abstract

In this paper, an adaptive tracking control scheme is designed for a class of continuous-time uncertain nonlinear systems based on the approximate solution of the Hamilton–Jacobi–Bellman (HJB) equation. Considering matched uncertainties, the tracking control of the continuous-time uncertain nonlinear system can be transformed to the optimal tracking control of the associated nominal system. By building the nominal error system and modifying its cost function, the solution of the relevant HJB equation can be contributed to the adaptive tracking control of the continuous-time uncertain nonlinear system. In view of the complexity on solving the HJB equation, its approximate solution is pursued by the policy iteration algorithm under the adaptive dynamic programming (ADP) framework, where a critic neural network is constructed to approximate the optimal cost function, and an action network is used to directly calculate the approximate optimal control law, which constitutes the tracking control law for the original uncertain system together with the steady control law. The weight convergence of the critic network and the stability of the closed-loop system are provided as the theoretical guarantee based on the Lyapunov theory. Two simulation examples are studied to verify the theoretical results and the effectiveness of the proposed tracking control scheme.