This paper considers the problem of stabilizing a first-order plants with known time delay using a fractional-order proportional–integral controller [Formula: see text]. Using a generalization of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete analytical characterization of all stabilizing gain values ([Formula: see text]) is provided. The widespread industrial use of fractional PI controllers justifies a timely interest in [Formula: see text] tuning techniques.

This paper presents a new design procedure to tune the fractional order PIλDμ controller that stabilizes a first order plant with time delay. The procedure is based on a suitable version of the Hermite–Biehler Theorem and the Pontryagin Theorem. A Theorem and a Lemma are developed to compute the global stability region of the PIλDμ controller in the (kp,ki,kd) space. Hence, this Theorem and Lemma allow us to develop an algorithm for solving the PIλDμ stabilization problem of the closed loop plant. The proposed approach has been verified by numerical simulation that confirms the effectiveness of the procedure.

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