Synthesis of PID controllers for high-order plants with time-delay

This note deals with the problem of the stabilizing parameter set of PID controllers for high-order all pole plants with time-delay. The time constants of the plants are assumed to be arbitrary, either positive or negative. First of all, the stabilizing range of the proportional gain is given in terms of a version of the Hermite–Beihler Theorem and the properties of the imaginary part of the closed-loop characteristic quasi-polynomial. Then, based on a graphical stability criterion for time-delay systems, the stabilizing region in integral-derivative space is drawn and identified for a fixed admissible proportional gain. An algorithm for determining the stabilizing parameter set of PID controllers is also proposed. Finally, case studies are provided to illustrate the shapes of the stabilizing regions.