Quadratically Parameterized Root Locus Analysis

Abstract

Classical affine root locus applies to controllers that are linear in a parameter k and yield affinely parameterized closed-loop denominator polynomials. This paper presents root locus rules for controllers that are rational in k and yield quadratically parameterized closed-loop denominator polynomials. We show that the quadratic root locus has several advantages relative to the classical affine root locus. Specifically, the quadratic root locus is high-parameter stabilizing for minimum-phase systems that are relative degree 1, 2, or 3. Moreover, the quadratic root locus admits a wider variety of asymptote angles, which provides more controller design flexibility. The quadratic root locus is illustrated on several examples that are difficult to handle using affine root locus, such as high-parameter stabilization of the triple integrator.Index Terms-Affine root locus, quadratic root locus, stabilization.

Determination of vertex polynomials to analyse robust stability of control systems with interval parameters

“…Nowadays, the analysis of the robust quality of control systems with interval parameters is also of great practical importance. There are some well‐known works where the root approach is proposed as the solution to the problem [27, 29, 31, 37–51]. It is worth mentioning that the proposed approach is quite simple and obvious for a designer.…”

Asymptote angles of polynomic root loci

“…Recently, the QRL has attracted the attention of some control researchers. In Wellman and Hoagg (2014), the asymptote rules of the QRL are derived by exploiting the Newton's binomial formula and making an asymptotic approximation of the geometric locus with the big O notation. The inconvenience of that approach is that the proof is very long (five pages), obscure and intricate.…”