This paper investigates a novel adaptive fuzzy fractional-order nonsingular terminal sliding mode controller (AFFO-NTSMC) for second-order nonlinear dynamic systems. The technique of fractional calculus and nonsingular terminal sliding mode control (NTSMC) are combined to establish fractional-order NTSMC (FO-NTSMC), in which a new fractional-order (FO) nonsingular terminal sliding mode (NTSM) surface is proposed. Then, a corresponding controller is designed to provide robustness, high performance control, finite time convergence in the presence of uncertainties and external disturbances. Furthermore, a fuzzy system with online adaptive learning algorithm is derived to eliminate the chattering phenomenon in conventional sliding mode control (SMC). The stability of the closed-loop system is rigorously proven. Numerical simulation results are presented to demonstrate the effectiveness of the proposed control method.

This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivative α∈1,2. In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.

No abstract

No abstract

The aim of the paper is to consider the existence and uniqueness of solution of the fractional differential equation with a positive constant coefficient under Hilfer fractional derivative by using the fixed-point theorem. We also prove the bounded and continuous dependence on the initial conditions of solution. Besides, Hyers–Ulam stability and Hyers–Ulam–Rassias stability are discussed. Finally, we provide an example to demonstrate our main results.