Partial-fraction Expansion of Lossless Negative Imaginary Property and A Generalized Lossless Negative Imaginary Lemma

TL;DR

This paper develops a partial-fraction expansion for lossless negative imaginary systems, introduces a generalized lemma allowing poles at zero, and explores relationships with lossless positive real systems, supported by numerical examples.

Contribution

It presents a generalized lossless negative imaginary lemma with poles at zero and establishes new relationships with lossless positive real systems.

Findings

Necessary and sufficient condition for non-proper lossless negative imaginary systems

New relationships between lossless positive real and lossless negative imaginary systems

Generalized lemma with minimal state-space realization allowing poles at zero

Abstract

This paper studies a partial-fraction expansion for lossless negative imaginary systems and presents a generalized lossless negative imaginary lemma by allowing poles at zero. First, a necessary and sufficient condition for a system to be non-proper lossless negative imaginary is developed, and a minor partial-fraction expansion of lossless negative imaginary property is studied. Second, according to the minor decomposition properties, two different and new relationships between lossless positive real and lossless negative imaginary systems are established. Third, according to one of the relationships, a generalized lossless negative imaginary lemma in terms of a minimal state-space realization is derived by allowing poles at zero. Some important properties of lossless negative imaginary systems are also studied in this paper, and three numerical examples are provided to illustrate the developed theory