An Online Gradient-Based Adaptive Fractional-Order Sliding Mode Framework for Chaos Synchronisation of Brushless DC Motor Drives

In applications such as electric vehicles, aerospace systems, and precision servo applications, brushless direct current (BLDC) motor drives may experience chaotic dynamics of current, torque, and speed under uncertain operating and loading conditions. These dynamics can degrade tracking performance and impose mechanical stress on the system. Fractional-order sliding mode control (FOSMC) performs well for chaos suppression and synchronisation, but existing methods often use fixed, manually tuned, or offline-optimised gains that cannot cope with time-varying disturbances or parameter drift. In this work, the study propose a fractional-order sliding mode controller (FOSMC) whose sliding-surface and reaching-law parameters are tuned online by a deterministic, gradient-based adaptation law master–slave chaos synchronisation of BLDC drives. The controller uses an online gradient-based gain adaptation law structurally inspired by, but not identical to, actor-parameter updates used in deep reinforcement learning (DRL) algorithms such as deep deterministic policy gradient (DDPG) and twin delayed DDPG (TD3); no actor–critic network, replay buffer, or policy-gradient training loop is implemented in this study. The adaptation mechanism repeatedly updates the sliding-surface parameters and reaching law during operation. A fractional-order Lyapunov analysis proves asymptotic convergence of the synchronisation error. The study compared the proposed controller with PID, conventional integer-order sliding mode control (SMC), and fixed-gain FOSMC, and evaluated it in MATLAB/Simulink under nominal, load-disturbance, and parameter-mismatch conditions using 10 independent random seeds for each test condition. The proposed adaptive FOSMC (AG-FOSMC) achieved a mean tracking RMSE of 0.00190 ± 0.00012, compared with 0.00246 ± 0.00006 for fixed-gain FOSMC and 0.00455 ± 0.00006 for conventional SMC. The PID controller did not converge under the tested uncertainty. In addition, the proposed method reduced the total variation of the control signals by around 44% and 78% compared with the control signals with fixed-gain FOSMC and conventional SMC, respectively. The variation in RMSE under ±20% changes in the electrical time-constant parameter, 15% un-modelled parameter mismatch, and measurement noise confirmed robust performance against parameter drift without manual retuning. The findings suggest that lightweight online adaptation can improve tracking precision and robustness, reduce chattering in chaotic BLDC drive systems, and provide a foundation for future implementation of fully trained deep-RL agents.