Practical Stability Analysis and Observer-Based Joint State and Fault Estimation in k-Caputo Fractional-Order Systems

This manuscript investigates the practical stability of nonlinear dynamical systems governed by the k-Caputo fractional derivative. Using a Lyapunov-based approach, we derive new practical Mittag-Leffler-based stability criteria that provide sufficient conditions ensuring that the trajectories of the considered systems converge to a bounded neighborhood of the equilibrium point. The developed theoretical framework is then applied to the challenging problem of simultaneous state and actuator-fault estimation for nonlinear fractional-order systems. In this setting, we construct an observer capable of jointly reconstructing the system states and the unknown actuator faults, while guaranteeing practical stability through linear matrix inequality (LMI) conditions. The proposed methodology explicitly accounts for nonlinear dynamics through a Lipschitz condition and accommodates bounded actuator faults with bounded fractional derivatives. Numerical examples confirm the convergence of the state and fault-estimation errors and illustrate the practical Mittag-Leffler stability properties of the proposed approach. Overall, the results enrich the stability analysis of fractional-order systems and provide a feasible framework for fault diagnosis in complex nonlinear dynamical settings.