In this paper, a method is provided for determining all fractional-order (FO) proportional–integral-derivative (PID) controllers that robustly stabilize a single-input and single-output (SISO) transfer function of arbitrary order with a time delay. The parameters of the FO PID controllers are determined 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 stability condition, a multiplicative weight is selected to bound all multiplicative errors of the closed-loop system. In particular, this technique can be used even when the transfer function of a system is not available, as long as the system frequency response can be obtained. An example is given to illustrate the applicability and effectiveness of the method presented.

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