New Tools for Finite-Time Fuzzy Control of Fractional-Order Systems: Stability Lemmas, Composite Disturbance Observers, and Singularity-Free Designs

This article presents a series of command-filter-based, finite-time adaptive fuzzy tracking control schemes for fractional-order nonlinear systems (FONSs) subject to input saturation and composite disturbances. A unified design framework is proposed to tackle these issues, combining adaptive fuzzy systems for online uncertainty estimation with novel fractional-order adaptive disturbance observers for robust perturbation approximation. Within this framework, several practical finite-time control schemes are developed, each offering a distinct approach to key design challenges, including finite-time fractional-order filter design, compensation strategies for filter errors, and measures to avoid singularity issues in virtual control law differentiation. The work also includes a review and theoretical comparison with existing methods. The practical finite-time stability of each closed-loop scheme is established via Lyapunov theory, supported by new technical lemmas introduced herein. Finally, numerical simulations are provided to illustrate the effectiveness and relative merits of the proposed control strategies.