On relative gain array and condition number

Abstract

Deviations of open-loop properties in the presence of modeling uncertainties are studied. The aim is to gain insights into how open-loop properties and, thus, potentially closed-loop properties may vary in the face of a diagonally structured uncertainty. Several estimates are given for the worst-case deviations of the open-loop transfer function in terms of the scaled plant condition numbers, the relative gain array, and the block relative gains. The worst-case deviation will be largely exactly when the estimates stated in terms of the condition number are large. An example is constructed to show that the relative gain array and block relative gains may be optimistic measures in assessing these deviations. The developments here support previous conjectures and results which assert that plants with large condition numbers and/or relative gains are potentially difficult to control. >