Conditions for preserving negative imaginary properties in feedback interconnections and an application to multi-agent systems

Abstract

We derived necessary and sufficient conditions for a feedback interconnection of negative imaginary systems to have negative imaginary properties. We first consider two negative imaginary systems (or strictly negative imaginary systems) which are interconnected via positive feedback. Then necessary and sufficient conditions for determining whether the feedback interconnection has negative imaginary properties are presented. These conditions are specialised to a dc loop gain that is easy to check under some assumptions. Together with stability results in the literature, these conditions could be applied to check the stability of a string of several coupled negative imaginary subsystems. Such an example is given to illustrate the usefulness of our proposed results.