Making Nonlinear Systems Negative Imaginary via State Feedback

TL;DR

This paper introduces a state feedback method to stabilize nonlinear systems by making them nonlinear negative imaginary (NI) or output strictly NI, with conditions for stability and handling NI uncertainties.

Contribution

It provides a novel state feedback stabilization approach for nonlinear systems of relative degree up to two, rendering them nonlinear NI or OSNI systems with stability guarantees.

Findings

Conditions for nonlinear NI and OSNI system transformation

Existence of stabilizing state feedback under certain conditions

Extension to systems with nonlinear NI uncertainties

Abstract

This paper provides a state feedback stabilization approach for nonlinear systems of relative degree less than or equal to two by rendering them nonlinear negative imaginary (NI) systems. Conditions are provided under which a nonlinear system can be made a nonlinear NI system or a nonlinear output strictly negative imaginary (OSNI) system. Roughly speaking, an affine nonlinear system that has a normal form with relative degree less than or equal to two, after possible output transformation, can be rendered nonlinear NI and nonlinear OSNI. In addition, if the internal dynamics of the normal form are input-to-state stable, then there exists a state feedback input that stabilizes the system. This stabilization result is then extended to achieve stability for systems with a nonlinear NI uncertainty.