A geometric description of the set of stabilizing PID controllers

Abstract This article presents a geometric description of the set of stabilizing PID controllers. The three-dimensional parameter space is partitioned by the stability crossing surface into regions such that the number of characteristic roots on the right half-plane (RHP) remains constant within each region. The set of stabilizing PID parameters consists those regions with no RHP root. The stability crossing surface is a ruled surface, and it is completely determined by a curve known as the discriminant. The discriminant may be divided into sectors between its cusps, and each sector corresponds to a positive patch and negative patch. The stability crossing surface is composed of these patches.