Phase-root locus and relative stability

A new graphical tool called the phase-root locus is introduced. It is the dual of the conventional root locus, and indicates the motion of closed-loop poles in the s-plane as phase is added to the open-loop transfer function. The root locus/phase-root locus plots are shown to facilitate destabilization diagnosis, which may help determine what part of an unstable physical system requires modification. For example, destabilization may be caused by one closed-loop pole due to a phase-shift at one value of system gain, and by a different pole due to gain variations at another value of system gain. The phase-root locus also allows relative stability information, including phase margin, to be accessed from the s-plane. Thus the s-plane can now be used for robustness analysis/design as well as transient analysis/design. The phase-root locus shows promise as a tool in compensator design as well as in the teaching of classical control theory.