Connections between discrete- and continuous-time results for positive real and negative imaginary systems

This paper studies the connection between discrete-time negative imaginary and continuous-time negative imaginary functions. Firstly, we analyse differences between two restatements that are claimed to provide necessary and sufficient conditions for systems to be discrete-time positive real. The conclusion is that one is equivalent to the definition of discrete-time positive real while the other one is not. Then, by means of the bilinear transformation, a connection between discrete-time negative imaginary and continuous-time negative imaginary functions is established. A new proof is provided for the discrete-time negative imaginary lemma by applying the connection. Several numerical examples illustrate the developed theory.