The intersection of the root-loci of multivariable systems with the imaginary axis

A scalar polynomial expression in the scalar feedback gain, that is used to evaluate the root-loci of a linear multivariable system, is derived. It is shown that the set of values of the scalar gain for which the root-loci intersect the imaginary axis (excluding the origin) is a subset of these polynomial roots. A linear equation in the feedback gain, whose solution yields the gain values for which the root-loci pass through the origin, is found. Together with the polynomial expression, this equation provides an easy way of calculating the values of the feedback gain for which one or more of the closed-loop poles moves from one half plane to another.