Adaptive Neural Optimal Backstepping Control of Uncertain Fractional-Order Chaotic Circuit Systems via Reinforcement Learning

Optimal control has become a hot topic due to its ability to reduce control costs. However, due to the complex form of fractional-order (FO) derivatives, it is difficult to obtain the optimal control solution by solving the FO Hamilton-Jacobi-Belman equation. This article formulates an neural optimal adaptive backstepping control programme for FO chaotic circuit systems with state constraints. To avoid states exceeding constraints during optimal control, a scheme combining a transformation formula with a nonlinear state dependent function is first developed, and then the original system is transformed into an integer-order unconstrained one. To achieve optimal control, a reinforcement learning adaptive backstepping control based on the transformation scheme is introduced, where weight update laws of the reinforcement learning are constructed based on the negative gradient of a positive function rather than the square of Bellman residual, which effectively simplifies the form and design process of the update laws. According to the stability analysis, the formulated programme assures that all signals are bounded and states remain within the specified constraint space. Eventually, a simulation case is displayed to demonstrate the validity of the developed approach.