Optimal Control of Nonlinear Continuous-Time Systems in Strict-Feedback Form

This paper proposes a novel optimal tracking control scheme for nonlinear continuous-time systems in strict-feedback form with uncertain dynamics. The optimal tracking problem is transformed into an equivalent optimal regulation problem through a feedforward adaptive control input that is generated by modifying the standard backstepping technique. Subsequently, a neural network-based optimal control scheme is introduced to estimate the cost, or value function, over an infinite horizon for the resulting nonlinear continuous-time systems in affine form when the internal dynamics are unknown. The estimated cost function is then used to obtain the optimal feedback control input; therefore, the overall optimal control input for the nonlinear continuous-time system in strict-feedback form includes the feedforward plus the optimal feedback terms. It is shown that the estimated cost function minimizes the Hamilton-Jacobi-Bellman estimation error in a forward-in-time manner without using any value or policy iterations. Finally, optimal output feedback control is introduced through the design of a suitable observer. Lyapunov theory is utilized to show the overall stability of the proposed schemes without requiring an initial admissible controller. Simulation examples are provided to validate the theoretical results.

I. Introduction

The stabilization of different classes of nonlinear systems in the presence of uncertainties is thoroughly studied in [1]–[4]. The adaptive neural network (NN) controller is proposed when the dynamics of the nonlinear system have parameter uncertainties [4], [5]. However, optimality is preferred over stability for such nonlinear systems [4], [6]. Therefore, optimal adaptive control schemes are being investigated in the recent years.