Dynamics and control of initialized fractional-order systems

Due to the importance of historical effects in fractional-order systems, this paper presents a general fractional-order system and control theory that includes the time-varying initialization response. Previous studies have not properly accounted for these historical effects. The initialization response, along with the forced response, for fractional-order systems is determined. The scalar fractional-order impulse response is determined, and is a generalization of the exponential function. Stability properties of fractional-order systems are presented in the complex w-plane, which is a transformation of the s-plane. Time responses are discussed with respect to pole positions in the complex w-plane and frequency response behavior is included. A fractional-order vector space representation, which is a generalization of the state space concept, is presented including the initialization response. Control methods for vector representations of initialized fractional-order systems are shown. Finally, the fractional-order differintegral is generalized to continuous order-distributions which have the possibility of including all fractional orders in a transfer function.