Splitting Root-Locus Plot into Algebraic Plane Curves

TL;DR

This paper introduces a method to decompose the Root Locus plot of a rational transfer function into algebraic plane curves using algebraic geometry, and constructs a dual curve representation called the 'Algebraic Dual Root Locus.'

Contribution

It presents a novel algebraic geometry approach to split Root Locus plots into individual curves and introduces the concept of the dual Root Locus using duality principles.

Findings

Decomposition of Root Locus into algebraic plane curves.

Construction of the Algebraic Dual Root Locus.

Application of primary decomposition in algebraic geometry.

Abstract

In this paper we show how to split the Root Locus plot for an irreducible rational transfer function into several individual algebraic plane curves, like lines, circles, conics, etc. To achieve this goal we use results of a previous paper of the author to represent the Root Locus as an algebraic variety generated by an ideal over a polynomial ring, and whose primary decomposion allow us to isolate the planes curves that composes the Root Locus. As a by-product, using the concept of duality in projective algebraic geometry, we show how to obtain the dual curve of each plane curve that composes the Root Locus and unite them to obtain what we denominate the "Algebraic Dual Root Locus".