Robust stability conditions for feedback interconnections of distributed-parameter negative imaginary systems

TL;DR

This paper develops new stability conditions for feedback interconnections of distributed-parameter negative imaginary systems using an IQC framework, accommodating irrational transfer functions and reducing conservatism.

Contribution

It introduces necessary and sufficient IQC-based stability conditions for distributed-parameter NI systems, extending existing results and exploiting frequency properties to improve analysis.

Findings

Derived IQC-based stability conditions for distributed-parameter NI systems.

Generalized stability analysis to irrational transfer functions.

Reduced conservatism by exploiting frequency domain properties.

Abstract

Sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. The IQC framework accommodates distributed-parameter systems with irrational transfer function representations, while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis. The main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of IQCs on positive frequencies that are bounded away from zero and infinity. Two additional sets of IQCs on the DC and instantaneous gains of the systems are shown to be sufficient and necessary for closed-loop stability along a homotopy of systems.