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 ink 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.
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Wellman et al. (2014) studied this question.