Root locus for a controller class that yields quadratic gain parameterization

We present rules for constructing the root locus for a linear feedback system, where the closed-loop denominator polynomial is quadratic in the root-locus parameter k. These quadratic root-locus rules apply to a class of controllers that are rational functions of k. We provide controller design examples to demonstrate the quadratic root locus. For example, we use quadratic root locus to high-gain stabilize the triple integrator; this is not possible with classical root locus.