Stabilization of first-order systems with time delay using the PID controller

This paper considers the problem of stabilizing a first-order plant with dead-time using a PID controller. Using a version of the Hermite-Biehler theorem applicable to quasi-polynomials, the complete set of stabilizing PID parameters is determined for both open-loop stable and unstable plants. The range of admissible proportional gains is first determined in closed form. For each proportional gain in this range the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid, a triangle or a quadrilateral. For the case of an open-loop unstable plant, a necessary and sufficient condition on the time delay is determined for the existence of stabilizing PID controllers. An example illustrates the applicability of the procedure