The first objective of this brief is to discover the solution existence conditions in the methods recently proposed for tuning fractional-order [proportional–integral] (FO-[PI]) controllers. The FO-[PI] controller tuned by these methods can simultaneously ensure the desired phase margin, the desired gain crossover frequency, and the flatness of the phase Bode plot at such a frequency. In this brief, the achievable performance region of these tuning methods is also found in the gain crossover frequency–phase margin plane. Moreover, some results on the shape of this region and the uniqueness of the controller resulting from the considered tuning methods are given. By combining the presented results and those previously obtained on fractional-order [proportional–derivative] tuning methods, a useful result on the fractional-order control system design is derived. In addition, simplified forms for tuning methods previously proposed in the literature are obtained. Also to verify the practical usefulness of outcomes of this brief, some experimental results on velocity control of a dc motor are presented.
['Vahid Badri', 'Mohammad Saleh Tavazoei']
Some Analytical Results on Tuning Fractional-Order [Proportional–Integral] Controllers for Fractional-Order Systems