A PID controller set of guaranteeing stability and gain and phase margins for time-delay systems

The stabilizing parameter sets and the guaranteed gain-margin (GM) and phase-margin (PM) regions of proportional-integral-derivative (PID) controllers for a class of processes with time-delay are discussed in the paper. The admissible range of stabilizing proportional-gain is first derived by a version of Hermite–Biehler Theorem and the evaluation of some properties of the functions involved in the closed-loop characteristic equation. Then, the stabilizing region in integral-derivative plane, for a fixed proportional-gain, is drawn and identified directly in terms of a graphical stability criterion applicable to time-delay systems. Further, in the stabilizing region, the gain-margin and phase-margin specifications are considered using the same strategy as drawing the stability boundary lines, based on the technique of gain-phase margin tester (GPMT). Illustrating examples are followed in each design step to show the effectiveness of the method.

► Evaluation of some functions involved in the closed-loop characteristic equation.
► Necessary conditions for the explicit stabilizing ranges of Kp values for delay systems of type 0, I and II.
► Determination of stabilizing region in Ki–Kd plane graphically.
► Gain-margin and phase-margin design in the stabilizing region.