Proportional-Integral Controller for Stabilization of Second-Order Delay Processes

This paper considers the problem of determining the complete stabilizing set of proportional-integral (PI) controllers for a second-order process with time delay by employing a version of the Hermite-Biehler theorem applicable to quasipolynomials. With the poles of open-loop system being complex, we first provide the result to find the admissible range of the proportional gain. Then by choosing a fixed proportional gain in this range, we can ascertain the complete region of integral gain which can stabilize the second-order delay process. Similarly, the result for the case of open-loop real poles is also obtained. It is mentioned that the condition to obtain the parameter set for stabilizing the given plant is sufficient and necessary.