Adaptive optimal control of unknown nonlinear systems with different time scales

The adaptive optimal control of unknown nonlinear system with different time scales is considered in this paper. The commonly used singular perturbation theory (SPT) to solve this problem is based on the accurately reduced system model, which is extremely difficult to be obtained in practical application. To overcome this difficulty, an adaptive dynamic programing-based optimal control algorithm with the simplified actor–critic–identifier structure is developed. A different time scales dynamic neural network (DTSDNN) identifier with a novel updating law derived from a properly designed Lyapunov function is proposed to estimate the unknown system dynamics. Furthermore, the critic NN with an improved adaptive law considering the NN weight estimation error information is designed, which can achieve faster convergent speed compared with the commonly used gradient method. Lyapunov approach is used to guarantee exponential convergence to a bounded region in the neighborhood of the optimal control and uniformly ultimately bounded (UUB) stability of the closed-loop system. Two examples are provided to illustrate the effectiveness and applicability of the developed approach.