On lossless negative imaginary systems

The paper is concerned with the notion of lossless negative imaginary systems and their stabilization using strictly negative imaginary controllers through positive feedback. Firstly, the concept of lossless negative imaginary transfer functions is introduced and some properties of such transfer functions are studied. Secondly, a Lossless Negative Imaginary Lemma is given which establishes conditions on matrices appearing in a minimal state-space realization that are necessary and sufficient for a transfer function to be lossless negative imaginary. Thirdly, a necessary and sufficient condition is provided for the stabilization of a lossless negative imaginary system by a strictly negative imaginary controller. Finally, a flexible structure example is presented to illustrate the theory.