Properties of positive feedback interconnected negative imaginary systems

Abstract

This paper addresses the problem of finding a set of ‘if and only if’ criteria for a positive feedback interconnection of two Negative Imaginary (NI) systems to retain the NI property. First, we have derived a state-space characterisation for NI systems that does not impose the minimality restriction. Then, two more essential pre-requisite technical lemmas are provided to prove the main results of this paper, which deal with interconnected system properties. Under some technical assumptions, the paper shows that a positive feedback interconnection of two NI systems with possible pole(s) at s = 0 preserves the NI property. Subsequently, the result is specialised when the NI systems do not have any pole(s) at s = 0 and then to SNI systems. The proposed results have potential applications in decentralised control of large vehicle platoons. Furthermore, the results may be utilised in simplifying complex physical networks and in physical network analysis and synthesis.