Bilinear Transformation for Discrete-Time Positive Real and Negative Imaginary Systems

This paper studies the connection between discrete-time and continuous-time negative imaginary systems. First, we analyze differences between two statements that are claimed to provide equivalent conditions for systems to be discrete-time positive real. Our conclusion is that one is equivalent to the definition of discrete-time positive real transfer matrices, the other is not. Second, by means of the bilinear transformation, a connection between discrete-time and continuous-time negative imaginary transfer matrices is established. Third, the concept of discrete-time lossless negative imaginary systems is introduced, and a discrete-time lossless negative imaginary lemma is developed to characterize the lossless negative imaginary properties in terms of minimal state-space realization. Some properties of discrete-time lossless negative imaginary transfer matrices are also studied. Several numerical examples illustrate the developed theory.