Negative Imaginary State Feedback Equivalence for Systems of Relative Degree One and Relative Degree Two

TL;DR

This paper establishes conditions under which linear systems of relative degree one or two can be transformed into negative imaginary systems via state feedback, aiding robust control design.

Contribution

It provides necessary and sufficient conditions for state feedback equivalence to NI systems for systems of relative degree one and two, including a strong strict NI result.

Findings

Systems can be made NI if controllable and weakly minimum phase.

A strong strict NI state feedback equivalence is established.

Application to robust stabilization with NI uncertainties.

Abstract

This paper presents necessary and sufficient conditions under which a linear system of relative degree either one or two is state feedback equivalent to a negative imaginary (NI) system. More precisely, we show for a class of linear time-invariant strictly proper systems, that such a system can be rendered minimal and NI using full state feedback if and only if it is controllable and weakly minimum phase. A strongly strict negative imaginary state feedback equivalence result is also provided. The NI state feedback equivalence result is then applied in a robust stabilization problem for an uncertain system with a strictly negative imaginary uncertainty.