Multivariate root loci: a unified approach to finite and infinite zeros

Abstract

A MacLaurin's series expansion is used to approximate the individual sheets of the characteristic gain function of a linear multivariable system in terms of its state space matrices, at the point at infinity on the frequency plane for small values of characteristic gain. This yields the asymptotic root locus directions for systems with one or more sets of closed loop poles going to infinity for large feedback gains. A bilinear transformation on the frequency variable is used to allow the approximation at any point on the frequency plane. This gives the angles of approach to finite zeros in terms of the state space matrices, and gives a simple way of calculating zeros for square systems. A bilinear transformation on the gain variable is then introduced allowing the characteristic gain function to be approximated at any values of frequency and gain. This enables the directions of departure from the open-loop poles to be determined in terms of the state space matrices of the system.