Comparing the stability regions for fractional-order PI controllers and their integer-order approximations

This paper proposes a scheme for computing the stable regions from which the parameters of a fractional-order PI controller can be selected. In particular, we extend a previously investigated method for plotting the stability regions of PI^λ controllers to their integer-order approximations. Next we compare the two stability regions and derive their overlapping sections, which is referred to as the implementable stability region. In order for the final closed-loop system to be stable, the designer must choose the controller parameters from the overlapping section of the two stability regions. The devised scheme we believe can facilitate the design procedure for fractional-order PI controllers. This method can be extended to the general case of a PI^λD^μ controller, where after fixing the fractional-orders of the integrative and derivative actions, the stability regions involve three parameters, (k_p, k_i, k_d), and the implementable stability region would lie at the intersection of three dimensional figures.