Computation of all stabilizing PID controllers for systems with time delay based on singular frequencies

Based on singular frequencies, this paper provides a simple approach to determine the complete set of stabilizing PID parameters for a given but arbitrary system with time delay. The condition for the existence of the stabilizing PID controllers is first presented. Provided that there exists the stabilizing PID controller, the maximal allowable stability range of the proportional gain k_p, integral gain k_i and derivative gain k_d are derived, respectively, all of which are only dependent on the parameters of the controlled plant. Sweeping over any one of k_p, k_i and k_d in their separate allowable ranges, the complete set of stabilizing PID parameters can be obtained by sketching the stability boundary curves over the defined small frequency range and then judging which side of the resulting boundary has less unstable poles. Numerical example is given to illustrate the proposed procedure and the application of the stabilizing results in the PID controller design.