Residues in Partial Fraction Decomposition Applied to Pole Sensitivity Analysis and Root Locus Construction

TL;DR

The paper introduces a new interpretation of residues from partial fraction decomposition, showing that the pole-velocity $ rac{dp_j}{dK} $ in a linearly controlled feedback system is given by the negative residue $-ar{k}_j$ of the closed-loop transfer function $G_0(s)$ at the pole $p_j$, with infinite velocity for multiple poles. This insight underpins three contributions: (1) a pole-sensitivity framework distinguishing how poles move with respect to the control gain $K$, (2) a novel, more efficient root-locus construction algorithm based on difference equations and a stabilizing term, and (3) a generalized pole-sensitivity method to system parameters using residues of auxiliary transfer functions $G_i(s,{f h})$ derived from the characteristic polynomial $ riangle(s,{f h})$. The proposed methods are validated conceptually and numerically, including a DC motor example, and shown to outperform MATLAB's rlocus in execution time while preserving output fidelity. Collectively, the work provides analytical pole-trajectory insights and practical tools for faster control-system design and analysis.

Abstract

The applications of the partial fraction decomposition in control and systems engineering are several. In this letter, we propose a new interpretation of residues in the partial fraction decomposition, which is employed for the following purposes: to address the pole sensitivity problem, namely to study the speed of variation of the system poles when the control parameter changes and when the system is subject to parameters variations, as well as to propose a new algorithm for the construction of the root locus. The new algorithm is proven to be more efficient in terms of execution time than the dedicated MATLAB function, while providing the same output results.

Figures (6)

• Figure 1: Linearly controlled feedback system.
• Figure 2: Root locus and poles velocity vectors $\frac{d p_j}{dK}$ of the feedback system associated with $G(s)$ in \ref{['num_es_1']}.
• Figure 3: Root loci of the feedback system associated with $G(s)$ in \ref{['stabil_term_comp']} when $K$ ranges from $0$ to $10$ computed using Algorithm \ref{['Prop_3_bis']} with and without the stabilizing term $\Delta(s,k_i)$ in \ref{['d_p_K_dK_sol_bis_new_bis']}.
• Figure 4: Comparison between the average execution times of Algorithm \ref{['Prop_3_bis']} and rlocus for the construction of the root locus.
• Figure 5: Power-Oriented Graphs block scheme of a DC electric motor.