An Approximate Optimal Control Approach for Robust Stabilization of a Class of Discrete-Time Nonlinear Systems With Uncertainties

In this correspondence paper, the robust stabilization of a class of discrete-time nonlinear systems with uncertainties is investigated by using an approximate optimal control approach. The robust control problem is transformed into an optimal control problem under some proper restrictions on the bound of the uncertainties. For the purpose of dealing with the transformed optimal control, the discrete-time generalized Hamilton-Jacobi-Bellman equation is introduced and then solved using the successive approximation method with neural network implementation. In addition, a numerical simulation is included to illustrate the effectiveness of the robust control strategy.

I. Introduction

It is known that the model uncertainties must be considered during the controller design process since they may cause deterioration of the control systems. In general, we say a controller is robust if it works even if the actual system deviates from its nominal model based on which the controller is designed. In fact, the robustness of control systems has been attended and studied by control scientists for many years. Robust control has become an important topic of modern control theory [1]–[3]. Lin et al. [3] pointed out that under proper restricted conditions, the robust control problem can be converted into an optimal control problem. Though the detailed operation procedure was not given, it provided an alternative method to deal with the robust stabilization problem. Thus, optimal control methods can be employed to design robust controllers. In fact, the research on linear optimal control has matured during the last several decades. However, when studying the nonlinear optimal control problem, we have to solve the Hamilton–Jacobi–Bellman (HJB) equation, which is often a difficult task. Therefore, some indirect methods have been proposed in order to overcome the difficulty in solving the nonlinear HJB equation. In [4], a recursive method was employed to deal with the optimal control problem for continuous-time nonlinear systems by successively solving the generalized HJB (GHJB) equation. The GHJB equation, which gives the cost of an arbitrary control law, can be used to improve the performance of this control and to approximate the HJB equation successively as well. In [5], adaptive (or approximate) dynamic programming (ADP) was presented to solve the optimal control problem, mainly for discrete-time nonlinear systems, based on function approximation structures such as neural networks. In recent years, the research on optimal control based on GHJB formulation and the ADP approach has acquired much attention from scholars [6]–[10]. Specifically, ADP has become one of the key directions for future research in intelligent control and understanding intelligence.