THE ROLE OF THE CONDITION NUMBER AND THE RELATIVE GAIN ARRAY IN ROBUSTNESS ANALYSIS

This paper studies deviations of open-loop properties in the presence of modeling uncertainties. Our aim is to gain insights into how open-l oop properties and thus potentially closed-loop properties may vary in the face of a diagonally structured uncertainty. We give several esti mates for the worst case deviations of the open-loop transfer function in terms of certain structured singular values and their bounds, and also in terms of certain scaled plant condition numbers, the relative pin array, and the block relative gains. Our analysis shows that the e stimates in terms of the structured singular values and bounds are tig ht in general, so are those in terms of the condition numbers for cert ain cases studied previously in the literature. We show that the worst case deviation will be large when the estimates stated in terms of th e structured singular values, or under certain circumstances in terms of the condition numbers, are large. On the other hand, an example is constructed to show that the relative pin array and block relative pin s may be optimistic measures in assessing these deviations. The develo pments here support and reinforce previous conjectures and results whi ch assert that plants with large condition numbers and/or relative gai ns are potentially difficult to control.