Fractional integro-differential sliding mode control of a class of distributed-order nonlinear systems

Distributed-order systems arise as a natural generalization of fractional- and integer-order systems, and these are commonly associated with slow and ultra-slow dynamics, which motivates designing robust schemes to enforce fast stabilization. This paper proposes a robust sliding mode controller that induces a stable motion in a finite time, relying on a dynamic extension to produce an integer-order reaching phase. Thus, a continuous fractional sliding mode controller is designed to compensate not necessarily integer-order differentiable disturbances. The theoretical results demonstrate that both the disturbance is exactly compensated and the pseudo-state converges asymptotically to the origin. A numerical simulation is carried out to demonstrate the reliability and efficacy of the proposed method, where a comparison to a conventional sliding mode scheme reveals a superior performance for the proposed fractional sliding mode approach.