Absolute Stability of Nonlinear Negative Imaginary Systems with Application to Potential Energy Shaping

TL;DR

This paper develops new absolute stability conditions for nonlinear negative imaginary systems, extending linear results and applying to potential energy shaping with static nonlinear feedback.

Contribution

It introduces a novel stability framework for nonlinear NI systems that generalizes prior linear results and connects static nonlinear feedback with storage-function shaping.

Findings

Stability conditions preserve the NI property under certain nonlinear feedbacks.

The framework generalizes existing linear NI stability results.

Application to potential energy shaping demonstrates practical utility.

Abstract

This paper establishes absolute stability conditions for nonlinear negative imaginary (NI) systems interconnected with static nonlinear feedback. We first show that the NI property is preserved when the feedback nonlinearity can be expressed as the gradient of a continuously differentiable function, and the composite storage of the resulting system remains positive definite. This condition provides a direct connection between nonlinear static feedback and storage-function shaping along the measured output channels. Building on this result, conditions are derived for absolute stability of the closed-loop system under mild assumptions. The linear specialization of the results strictly generalizes prior absolute stability results for linear NI systems, allowing coupled nonlinearities not covered by existing slope-restricted or sector-bounded frameworks. Finally, the proposed theory is illustrated through a linear example highlighting this generalization and a nonlinear example that shows the utility of the proposed results in potential energy shaping.