Padé approximation for time delay systems and a new design method for the fractional-order PI controller

Abstract

Fractional-order PI controllers have been the research topic of many articles thanks to the convenience they bring day by day. This type of controller offers much more useful results compared to integer-order PI. But it makes mathematical operations more complicated. In this study, a simple fractional-order PI controller design method based on the centroid of the convex stability region, which was developed using the stability boundary locus method, is proposed. For this, first, the stability region, which provides all stable controllers, is obtained in the parameter plane of the fractional-order PI controller. Then, by determining the corner and cusp points of this stability region, the convex stability region is obtained. Finally, the centroid of the obtained convex stability region is found, and the controller parameters are calculated. The method used provides significant advantages in terms of calculating the controller parameters without using complex graphical methods, ensuring the stability of the closed-loop system and reducing the computational load considerably. In addition, for the equations used in obtaining the stability region in this study, unlike the studies in the literature, Padé approximations are preferred instead of the time delay term. Especially, it is seen that the results obtained using the second-order Padé approximation almost exactly match the original system. Some numerical examples are given to demonstrate the effectiveness and simplicity of the proposed method.