Determination of all stabilizing fractional-order PID controllers that satisfy a robust performance constraint

This paper presents a method for finding all stabilizing fractional-order (FO) proportional-integral-derivative (PID) controllers that satisfy a robust performance constraint for a system of integer or non-integer order with a time delay. All the solutions to such FO PID controllers are calculated in the frequency domain and are given in terms of the proportional gain Kp, integral gain Ki, and derivative gain Kd. In this paper, they will be plotted on the (Kp, Ki), (Kp, Kd), and (Ki, Kd) planes. For a robust performance constraint, a weight is selected to bound all multiplicative errors resulting from parametric uncertainty and a sensitivity function weight is selected to set performance specifications. This approach provides all the possible gain parameter values of FO PID controllers that satisfy a robust performance condition. An example is given to illustrate the usefulness and effectiveness of the method.