Foundations of negative imaginary systems theory and relations with positive real systems

TL;DR

This paper develops a broad, domain-based negative imaginary systems theory that includes discrete-time systems, linking it with positive real systems and passivity, and provides stability analysis for their interconnections.

Contribution

It introduces a non-rational, domain-based framework for negative imaginary systems, including the first study of discrete-time cases, and unifies existing concepts with positive real systems.

Findings

First discrete-time negative imaginary systems studied in literature

Unified framework linking positive real and negative imaginary systems

Stability analysis for discrete-time system interconnections

Abstract

In this paper we lay the foundations of a not necessarily rational negative imaginary systems theory and its relations with positive real systems theory and, hence, with passivity. In analogy with the theory of positive real functions, in our general framework, negative imaginary systems are defined in terms of a domain of analyticity of the transfer function and of a sign condition that must be satisfied in such domain. In this way, on the one hand, our theory does not require to restrict the attention to systems with rational transfer function and, on the other hand | just by suitably selecting the domain of analyticity to be either the right half complex plane or the complement of the unit disc in the complex plane | we particularize our theory to both continuous-time and to discrete-time systems. Indeed, to the best of our knowledge, this is first time that discrete-time negative imaginary systems are studied in the literature. In this work, we also aim to provide a unitary view of the different notions that have appeared so far in the literature within the framework of positive real and in the more recent theory of negative imaginary systems, and to show how these notions are characterized and linked to each other. A stability analysis result for the interconnection of discrete-time systems is also derived.