Stabilizing sets of PID controllers for first- order plus double integrating plants with time-delay

This article discusses the problem of determining the parameter sets of stabilizing PID controllers for first-order plus double integrating processes with time-delay. The method adopted in the article is based on a version of Hermite-Biehler theorem valid for quasi-polynomial and a graphical stability criterion for time-delay systems. The admissible range of proportional-gain is first derived. Then, in integral-derivative space, the stabilizing region is drawn and identified directly for a fixed proportional-gain, not to be computed mathematically. An algorithm for obtaining the stabilizing sets of PID controllers is developed also. Case studies illustrate the design procedure and show the shapes of stabilizing regions.