Determination of all stabilizing fractional-order PID controllers that satisfy a weighted sensitivity constraint

This paper presents a method for determining all stabilizing fractional-order (FO) proportional-integral-derivative (PID) controllers that satisfy an H_∞ weighted-sensitivity constraint for a system of integer or non-integer order. All the parameters of 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. In particular, this approach provides all the possible values of the gain parameters of the FO PID controllers that satisfy a given weighted-sensitivity condition even when the transfer function of a system is not available, as long as the frequency response thereof can be obtained. An example is given by way of illustrating the usefulness and effectiveness of the method.