A Frequency-Domain Approach to Nonlinear Negative Imaginary Systems Analysis

TL;DR

This paper extends the theory of negative imaginary systems to nonlinear systems using a frequency-domain approach, providing stability conditions and demonstrating effectiveness through examples and simulations.

Contribution

It introduces a novel frequency-domain framework for nonlinear NI systems and develops stability criteria based on integral quadratic constraints.

Findings

Established a finite-frequency characterization of nonlinear NI systems

Derived a feedback stability condition for nonlinear NI systems

Validated the approach with examples and simulations

Abstract

In this study, we extend the theory of negative imaginary (NI) systems to a nonlinear framework using a frequency-domain approach. The extended notion is completely characterized via a finite-frequency integration over a "kernel function" on energy-bounded input and output signal pairs. The notion is closely related to and carefully contrasted with the well-studied extension of negative imaginariness -- the theory of counterclockwise dynamics. A condition for feedback stability of the proposed nonlinear NI systems is then developed based on the technique of integral quadratic constraints. Examples and simulations on feedback interconnections of typical nonlinear systems are provided to demonstrate the effectiveness.