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

Abdelhamid Djari, Abdelaziz Aouiche, Riadh Djabri, Hanane Djellab, Mohamad A. Alawad and Yazeed Alkhrijah ()
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Abdelhamid Djari: Department of Electrical Engineering, Echahid Cheikh Larbi Tebessi University, Tebessa 12002, Algeria
Abdelaziz Aouiche: LSSS Laboratory, Department of Electronics and Communications, Echahid Cheikh Larbi Tebessi University, Tebessa 12002, Algeria
Riadh Djabri: Department of Electrical Engineering, Echahid Cheikh Larbi Tebessi University, Tebessa 12002, Algeria
Hanane Djellab: LSSS Laboratory, Department of Electronics and Communications, Echahid Cheikh Larbi Tebessi University, Tebessa 12002, Algeria
Mohamad A. Alawad: Department of Electrical Engineering, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
Yazeed Alkhrijah: Department of Electrical Engineering, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia

Mathematics, 2025, vol. 13, issue 19, 1-22

Abstract: 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.

Keywords: chaotic systems; fractional-order systems; control and stabilization; backstepping control; Mittag–Leffler principle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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