Approximate Optimized Backstepping Control of Uncertain Fractional-Order Nonlinear Systems Based on Reinforcement Learning

In this article, a feasible reinforcement learning (RL) scheme is proposed for partially unknown fractional-order nonlinear systems (FONSs). First, the fractional Hamilton-Jacobi–Bellman (HJB) equation containing the dynamics of FONSs is proposed by constructing an auxiliary system and equivalent transformation. Then, the optimal solution of FONSs optimal control under a performance constraint is obtained. It is proved that the optimal cost function and optimal control policy can be approximated gradually by the policy iteration. By using the backstepping control, RL, and identifier-actor-critic neural networks (NNs), the unknown dynamics functions are approximated and the approximate optimal controllers are obtained. A Lyapunov function based on optimality error is constructed, then the fractional-order update laws of NNs weights are designed to ensure that the weights converge to the optimum. Thus, the use of the gradient descent algorithm in the context of the fractional-order calculus to train the NNs is avoided. Finally, the error signals are proved to be bounded and the effectiveness of the proposed algorithm is verified by the simulation of two practical examples.