Determination of all stabilizing fractional-order PID controllers

A new method for finding all fractional-order (FO) proportional-integral-derivative (PID) controllers that stabilize a given system of integer or non-integer order is proposed. The stability boundaries of such FO PID controllers are calculated in the frequency domain and are given in terms of the proportional gain K_p, integral gain Kh and derivative gain Kd. In this paper, they will be plotted on the (K_p, K_i), (K_p, K_d), and (K_i, K_d) planes. A key advantage of this approach is that it provides the stability boundaries even when the transfer function of a system is not available, as long as the frequency response of the system can be obtained. Moreover, the method does not require complicated mathematical calculations. An example is presented to illustrate the effectiveness of this method. The results are compared with those of conventional integer-order (IO) PID controllers for a sample example.