Reinforcement Learning‐Based Sliding Variable Optimal Control for Canonical Nonlinear Systems With Unknown Control Direction

This work tackles how to design optimal tracking controllers for standard nonlinear systems when both the dynamics and the control direction are unknown. Classic backstepping methods often become overly complicated, and typical reinforcement‑learning (RL) approaches have trouble solving the Hamilton‑Jacobi‑Bellman (HJB) equation under these uncertainties. The authors propose a sliding‑variable‑based optimal control (SVBOC) framework. Instead of using recursive backstepping, they build a sliding variable directly from the tracking errors. To approximate the optimal control law, they use a simplified identifier‑critic‑actor RL structure. This setup does not need persistent excitation (PE) conditions; it adapts the network weights by gradient descent on a positive‑definite function. The resulting controller is continuous, so it

avoids the high‑frequency chattering that ordinary sliding‑mode control produces. Lyapunov analysis shows that all closed‑loop signals are semi‑globally uniformly ultimately bounded (SGUUB) and that the tracking errors shrink to a small neighborhood around zero. The framework offers a solid base for advanced control design, but implementing it in practice still requires careful tuning of the neural‑network learning rates and the sliding‑surface parameters.

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