Stabilization of time-delay systems using the PID controller

This work considers the problem of stabilizing a given linear time-invariant plant with dead time using a fixed order controller. Specific controller structures considered are the constant gain (P), proportional-integral (PI) and proportional-integral-derivative (PID). Using a version of the Hermite-Biehler Theorem applicable to quasipolynomials, a complete analytical characterization of all stabilizing gain values is provided as a closed form solution for a system described by a first or second-order model with time delay. A similar approach is then used to tackle the problem of stabilizing a first-order plant with time delay using a PI controller, and once again the complete stabilizing set is determined. Based on these results, the problem of stabilizing a first-order plant with dead time using a PID controller is analyzed and solved 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 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. Several examples illustrate the applicability of the procedure.; Since PID controllers remain popular among industrial practitioners, the results presented in this work should find widespread applications, specially in the development of computationally efficient tools for PID controller design and analysis. This line of research also provides an instrument to understand the relationship between the time delay exhibited by a system and its stabilization using a fixed order controller.