Stabilising regions of PID controllers for nth-order all pole plants with dead-time

A new synthesis procedure of PID (proportional-integral-derivative) controllers for nth-order all pole plants with dead-time is developed via a parameter space approach. A necessary and sufficient condition-based graphical stability criterion is applied to determine the complete stabilising regions of PID controllers. First, the admissible stabilising range of the proportional-gain is derived in terms of a version of the Hermite-Beihler theorem applicable to quasi-polynomials. Then, the boundaries of the stabilising region in the integral-derivative plane are drawn directly, not calculated mathematically, for a Axed value of proportional-gain in the stabilising range. An algorithm for determining the stabilising parameter set of the PID controller is also proposed. Finally, numerical examples are given to show the shapes of the stabilising regions of the PID controllers for different order plants with dead-time.