Suboptimal Output Tracking and Regulation of a Class of Nonlinear Systems

Abstract

This paper proposes an approximate optimal approach for output tracking and regulation of a class of nonlinear systems. This class includes systems which are decomposable into a linear part and a nonlinear part. It is shown that the Hamilton–Jacobi–Bellman equation associated with optimal control of such systems, under an assumption on form of its solution, can be approximated as an algebraic equation. Decomposition of this algebraic equation into a linear part and a nonlinear part and their subsequent solutions together form an approximate value function. This value function is found to be nearly optimal locally around the origin. Thus, a locally stabilizing controller with the proposed value function as feedback gain is proposed. This approach achieves output tracking and regulation by converting these into a stabilization problem through internal model principle resulting in a near-optimal tracking and regulation. This proposed algebraic equation can be used for estimation too, i.e., it plays the same role in nonlinear settings as that played by an algebraic Riccati equation in linear quadratic regulation and linear quadratic estimation. Finally, the proposed approach can compensate integral windup effects due to actuator saturation through modification of filtering characteristics of internal model. Simulation results for an undamped spring-mass system are given for efficacy of the approach.