Conditions for the Stabilization of Linear Systems Using Classical PID Controllers

This paper addresses the stabilization of single-input single-output linear time-invariant delay-free systems by means of classical PID controllers. The considered plants contain one or two unstable poles together with additional stable dynamics. Rather than addressing controller tuning or performance optimization, the problem is formulated from a structural stabilizability perspective, namely, whether a stabilizing controller of a prescribed classical structure exists. Necessary and sufficient conditions for the existence of stabilizing P, PI, PD, and PID controllers are established explicitly in terms of the plant pole locations. Some of these conditions are obtained as delay-free specializations of previously reported results for time-delay systems, whereas additional results are derived specifically to complete the characterization of the delay-free case. The resulting conditions allow stabilizability to be determined without numerical searches or iterative tuning procedures. The analysis also identifies classes of plants for which P and PI controllers cannot achieve stabilization while derivative action enables stabilization through PD or PID controllers. A Nyquist-based interpretation is provided to clarify this effect, and illustrative examples are presented to support the theoretical results.