Discrete-time negative imaginary systems

In this paper we introduce the notion of a discrete-time negative imaginary system and we investigate its relations with discrete-time positive real system theory. In the framework presented here, discrete-time negative imaginary systems are defined in terms of a sign condition that must be satisfied in a domain of analyticity of the transfer function, in analogy with the case of discrete-time positive real functions, as well as analogously to the continuous-time case. This means in particular that we do not need to restrict our notions and definitions to systems with rational transfer functions. We also provide a discrete-time counterpart of the different notions that have appeared so far in the literature within the framework of strictly positive real and in the more recent theory of strictly negative imaginary systems, and to show how these notions are characterized and linked to each other. Stability analysis results for the feedback interconnection of discrete-time negative imaginary systems are also derived.