Computation of stabilizing PID gain regions based on the inverse Nyquist plot

Based on the inverse Nyquist plot, a method is presented to determine the stabilizing PID gain regions for linear systems. It does not depend on the integral and the derivative parameters. The two stable conditions are derived from a generalization of the Hermite-Biehler theorem. Two kinds of key points can be obtained from the inverse Nyquist plot. The PID gain would be divided into several regions by abscissas of these key points. According to the number of intersection of the inverse Nyquist plot and the vertical line, the stabilizing PID gain regions can be determined. Examples are given to show the benefits of the proposed method.