Stability analysis of PID controllers for integral processes with time delay

In this paper, the problems of stabilizing integral plants with time delay using the classical proportional-integral-derivative (PID) controllers and the practical single-parameter (containing only one adjustable parameter) PID controllers are considered, respectively. The complete set of stabilizing parameters of the classical PID controller for the integral plant with time delay and time constant is determined using the extended Hermite-Biehler Theorem applicable to quasi-polynomials. Owning to the difficulty in analyzing the complex closed-loop characteristic equation based on the extended Hermite-Biehler Theorem, a simple method called dual-locus diagram, is adopted to present the stabilizing region of the practical single-parameter PID controller.