Achievable Performance Region for a Fractional-Order Proportional and Derivative Motion Controller

In recent years, design and tuning of fractional-order controllers for use in motion control have attracted much attention. This paper deals with one of the interesting methods that have been recently proposed for tuning fractional-order proportional and derivative (FOPD) motion controllers. The FOPD controller tuned based on the considered method results in simultaneously meeting the desired phase margin, the desired gain crossover frequency, and the flatness of the phase Bode plot at such a frequency. Since, in this tuning method, the derivative parameter and order are determined in a graphical way, in the first view, it is not clear for which pairs of phase margin and gain crossover frequency an FOPD controller can be found to satisfy the mentioned specifications. In this paper, necessary and sufficient conditions are presented via an analytical approach to check the solution existence of the considered tuning method in both cases of absence and presence of time delay in feedback loop. Furthermore, the uniqueness of the controller parameters obtained from the method is investigated. Moreover, some numerical and experimental examples are presented to confirm the analytical results of this paper.