Making nonlinear systems negative imaginary via state feedback✩

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 input-affine nonlinear system can be rendered nonlinear NI and nonlinear OSNI if it has a normal form with relative degree less than or equal to two after possible output transformation. 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. A numerical example is provided to illustrate the process of stabilizing an uncertain system using the proposed results. & COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

This paper proposes a state feedback stabilization approach for nonlinear systems with relative degree less than or equal to two by transforming them into nonlinear negative imaginary (NI) systems. The paper provides conditions under which a nonlinear system can be made a nonlinear NI system or a nonlinear output strictly negative imaginary (OSNI) system. It is shown that if an input-affine nonlinear system has a normal form with relative degree less than or equal to two after output transformation, it can be rendered nonlinear NI and nonlinear OSNI. Additionally, if the internal dynamics of the normal form are input-to-state stable, there exists a state feedback input that can stabilize the system. The paper also extends this stabilization result to systems with nonlinear NI uncertainty, and presents a numerical example to illustrate the process of stabilizing an uncertain system using the proposed approach.