Stability region of fractional-order PI λDμ controller for fractional-order systems with time delay

A simple and effective method to determine the region of fractional-order PI^λD^μ controllers that can stabilize a given fractional-order system with time delay is proposed in this paper. For each known proportional, integral or derivative gain in the PI^λD^μ controllers, the stability region with respect to the other two control gains is derived. Firstly, the boundaries of the fractional-order PI^λD^μ controllers are determined by using the D-decomposition method. Then, an analytical approach is presented to judge which region is the stability one among a lot of areas divided by the resultant boundaries. In comparison with other relevant methods, the main advantage of the proposed method lies in that it can effectively avoid choosing one point from each divided area and finding the stability region of the fractional-order PID controller by testing the system stability corresponding to each chosen point. Moreover, a special phenomenon is revealed: if λ + μ ≠ 2, the boundaries of the stability region in k_i-k_d plane are the curves for a given k value; otherwise, the stability regions in k_i-k_d plane are convex polygons. A numerical example is presented to check the validity of the proposed method. The proposed method can be applied to the fractional-order system free of the detailed model and only the frequency response data of the fractional-order system is required.