Synthesis of a fractional PI controller for a first-order time delay system

Abstract

This paper considers the problem of stabilizing a first-order plants with known time delay using a fractional-order proportional–integral controller $\text{P}{\text{I}}^{\lambda }$ . Using a generalization of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete analytical characterization of all stabilizing gain values ( $k_p,k_i$ ) is provided. The widespread industrial use of fractional PI controllers justifies a timely interest in $\text{P}{\text{I}}^{\lambda }$ tuning techniques.