Stabilizing parameters of PID controllers for second-order systems with time- delay

This article focuses on the stabilization, using PID controllers, of second-order systems with a pair of complex poles plus time-delay. The parameters constraint of the controlled plant for the existence of a stabilizing PID controller is first provided. And the stabilizing range of proportional-gain is derived in terms of a version of Hermite-Biehler Theorem applicable to quasi- polynomial. Then, based on a graphical stability criterion for time-delay systems, the stabilizing region in integral-derivative space is drawn and identified, and an algorithm for finding the stabilizing parameter set of PID controller is also developed. Finally, examples are given to illustrate the design steps and the shapes of stabilizing regions.