Fractional-Order Backstepping Approach Based on the Mittag–Leffler Criterion for Controlling Non-Commensurate Fractional-Order Chaotic Systems Under Uncertainties and External Disturbances

Chaotic systems appear in a wide range of natural and engineering contexts, making the design of reliable and flexible control strategies a crucial challenge. This work proposes a robust control scheme based on the Fractional-Order Backstepping Control (FOBC) method for the stabilization of non-commensurate fractional-order chaotic systems subject to bounded uncertainties and external disturbances. The method is developed through a rigorous stability analysis grounded in the Mittag–Leffler function, enabling the step-by-step stabilization of each subsystem. By incorporating fractional-order derivatives into carefully selected Lyapunov candidate functions, the proposed controller ensures global system stability. The performance of the FOBC approach is validated on fractional-order versions of the Duffing–Holmes system and the Rayleigh oscillator, with the results compared against those of a fractional-order PID (FOPID) controller. Numerical evaluations demonstrate the superior performance of the proposed strategy: the error dynamics converge rapidly to zero, the system exhibits strong robustness by restoring state variables to equilibrium quickly after disturbances, and the method achieves low energy dissipation with a high error convergence speed. These quantitative indices confirm the efficiency of FOBC over existing methods. The integration of fractional-order dynamics within the backstepping framework offers a powerful, robust, and resilient approach to stabilizing complex chaotic systems in the presence of uncertainties and external perturbations.