Necessary and sufficient conditions for lossless negative imaginary systems

This paper studies some useful properties of lossless negative imaginary transfer matrices for both continuous-time and discrete-time systems. Necessary and sufficient conditions are established for lossless negative imaginary systems both in frequency domain and state-space realization. Meanwhile, a minor decomposition method for lossless negative imaginary systems is proposed in terms of a partial-fraction expansion. This method is important for developing non-proper lossless negative imaginary theory in this paper by allowing poles at the origin and infinity. Two new relationships between lossless positive real and lossless negative imaginary systems are consequently established. According to these new established relationships, a generalized continuous-time lossless negative imaginary lemma and a different version of discrete-time lossless negative imaginary lemma are developed in terms of a minimal state-space realization. Several examples are provided to illustrate the main results.