On non-proper negative imaginary systems

This paper studies non-proper negative imaginary systems. First, the concept of negative imaginary transfer functions that may have poles at the origin and infinity is introduced. Then, a generalized lemma is presented to provide a sufficient condition to characterize the non-proper negative imaginary properties of systems. The generalized lemma is given in terms of complex variable ss in the quarter domain of analyticity. Compared to previous results, our result removes the symmetric restriction. Also, a new relationship is established between (lossless) negative imaginary and (lossless) positive real transfer function matrices by using a minor decomposition. The results in this paper give us the possibility to address non-proper and non-symmetric descriptor systems with negative imaginary frequency response. Several examples are presented to illustrate the results.