Plotting Residue Loci for Control Systems

This work presents a new graphical method for plotting the residue loci of linear time-invariant control systems as the gain varies. While the classical root locus shows the trajectory of the closed-loop poles, it does not reveal the relative weight of each pole in the system response. The residues of the partial-fraction expansion provide precisely this information, since they determine the magnitude and phase of each modal component. The proposed residue-locus method establishes a set of construction rules analogous to those of Evans’ root locus, allowing the loci to be sketched and interpreted directly in the s-plane. The proposed construction rules are developed analytically and illustrated using representative low-order systems with different pole-zero configurations. For the second-order case with two open-loop poles and no zeros, the residues asymptotically satisfy A1≃−jK/2 and A2≃jK/2, with residue-locus asymptotes at 90∘ and 270∘. The residue-locus method, used together with the root-locus method, provides a unified analytical technique for control-system analysis and design, since it simultaneously describes how the location and the contribution of each closed-loop pole evolve with the gain. The method is intended as a complementary tool for studying transient response, modal dominance, and the relative activation of system modes.