A Pseudo S-Plane Mapping of Z-Plane Root Locus

In this paper, inspired by the geometric inversion transformation, a novel transformation of the z-plane root locus to a pseudo s-plane is proposed. In the z-plane, the stability of a discrete closed-loop system (a sampled-data control system) requires that all the system poles lie within the unit circle. In root locus analysis, the unit circle region seems congested, compared to the stability region of a continuous system, which is the left half of the s-plane. In the case of fast sampling, the poles of a discrete system can really be in a small neighborhood, thus making the control implementation difficult. The geometric transformation developed in this work helps widen or enlarge the space for the system poles and preserves most of the features of z-plane root loci, including marginal stability and root loci branching off at vertical angles. The usefulness of the new transformation in design of discrete control systems is demonstrated in a numerical example.