Necessary and Sufficient Conditions for Perfect Command Following and Disturbance Rejection in Fractional Order Systems

The aim of this paper is to present a modified explanation of the classic internal model principle for certain class of finite-dimensional, time-invariant, deterministic fractional-order systems commonly known as fractional systems of commensurate order. The necessary and sufficient conditions for perfect command tracking and disturbance rejection are provided. The difficulty of applying the classic internal model principle to fractional-order systems is due to the difference between integer-order and fractional-order systems from the zero-pole cancellation point of view. The notion of zero-pole cancellation is discussed for the systems under consideration in a well posed mathematical framework. It is also shown that fractional elements can be used for command tracking and disturbance rejection purposes which provides more flexibility for controller design applications. Two illustrative examples confirm the applicability of the proposed theorems.