Necessary and Sufficient Conditions for State Feedback Equivalence to Negative Imaginary Systems

In this article, we present necessary and sufficient conditions under which a linear time-invariant (LTI) system is state feedback equivalent to a negative imaginary (NI) system. More precisely, we show that a minimal LTI strictly proper system can be rendered NI using full state feedback, if and only if it can be output transformed into a system, which has relative degree less than or equal to two and is weakly minimum phase. We also considered the problems of state feedback equivalence to output strictly negative imaginary systems and strongly strictly negative imaginary systems. Then, we apply the NI state feedback equivalence results to robustly stabilize an uncertain system with strictly negative imaginary uncertainty. A real-world example is provided to illustrate the proposed results, for the purpose of stabilizing an uncertain system.