Gain margins and root locus asymptotic behaviour in multivariable design Part I. The properties of the Markov parameters and the use of high feedback gain

Abstract

From a root locus point of view the ability of a feedback system to tolerate high gains depends on the distribution of the finite zeros (FZ) and the infinite zeros (IZ). The asymptotic directions of the closed loop poles that tend to the IZ with increasing scalar gain are determined by the eigenproperties of a set of parameters called the projected Markov parameters (PMP). Furthermore the points of radiation of the asymptotes called the pivots may also be computed in terms of the elementary state space matrices. The dependence of the order of the IZ and therefore the distribution of the asymptotes on the null structure of the PMP is discussed first and subsequently a design technique which aims at the reduction of the order of the IZ as well as the placement of the corresponding pivots is proposed. The effectiveness of such a design technique in improving the gain margins of feedback systems is illustrated by means of a numerical example.