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 K_p, integral gain K_i, and derivative gain K_d. In this paper, they will be plotted on the (K_p, K_i), (K_p, K_d), and (K_i, K_d) 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.