Extreme Value Condition of Stabilizing PID Region

Abstract

This paper considers the problem of stabilizing linear time-invariant plants by a PID controller. If one of the tuning parameters reaches the extreme value, the closed-loop system contains a double pole on the positive imaginary axis or the integral gain is equal to zero. Using this property, the admissible ranges of the proportional gain, the integral gain and the derivative gain are derived respectively. If the proportional gain is fixed, the stability region in the plane with respect to the integral gain and the derivative gain is determined by plotting the stability boundary locus. The effectiveness of the method presented is illustrated a numerical example.