Adaptive dynamic programming-based optimal control for nonlinear state constrained systems with input delay

This paper investigates the problem of adaptive optimal tracking control for full-state constrained strict-feedback nonlinear systems with input delay. To facilitate the study, a novel control approach is developed by combining the backstepping design technique and adaptive dynamic programming (ADP) theory. At first, an intermediate variable is introduced to approximate the input delay using Pade approximation. Then, barrier Lyapunov functions are incorporated into the backstepping procedure to handle the state constraints. Moreover, neural networks are employed to approximate unknown functions in the presence of uncertainties. Based on this, an adaptive backstepping feedforward controller is developed, which converts the tracking task into an equivalent regulation problem for the affine form nonlinear system. To obtain the optimal control of the affine form nonlinear system, a critic network is constructed within the ADP framework to approximate the solution of Hamilton-Jacobi-Bellman equation, and online learning is utilized to obtain the optimal feedback control. The resulting controller consists of feedforward and feedback parts. Meanwhile, all signals in the closed-loop system are guaranteed to be uniformly ultimately bounded. Finally, the effectiveness of the proposed control scheme is illustrated through a numerical example.