Simplified Optimized Backstepping Control for a Class of Nonlinear Strict-Feedback Systems With Unknown Dynamic Functions

In this article, a control scheme based on optimized backstepping (OB) technique is developed for a class of nonlinear strict-feedback systems with unknown dynamic functions. Reinforcement learning (RL) is employed for achieving the optimized control, and it is designed on the basis of the neural-network (NN) approximations under identifier-critic-actor architecture, where the identifier, critic, and actor are utilized for estimating the unknown dynamic, evaluating the system performance, and implementing the control action, respectively. OB control is to design all virtual controls and the actual control of backstepping to be the optimized solutions of corresponding subsystems. If the control is developed by employing the existing RL-based optimal control methods, it will become very intricate because their critic and actor updating laws are derived by carrying out gradient descent algorithm to the square of Bellman residual error, which is equal to the approximation of the Hamilton-Jacobi-Bellman (HJB) equation that contains multiple nonlinear terms. In order to effectively accomplish the optimized control, a simplified RL algorithm is designed by deriving the updating laws from the negative gradient of a simple positive function, which is generated from the partial derivative of the HJB equation. Meanwhile, the design can also release the condition of persistence excitation, which is required in most existing optimal controls. Finally, effectiveness is demonstrated by both theory and simulation.