Foundations of not necessarily rational negative imaginary systems theory: Relations between classes of negative imaginary and positive real systems

In this technical note we lay the foundations of a not necessarily rational negative imaginary systems theory and its relations with positive real systems theory. In analogy with the theory of positive real functions, in our general framework negative imaginary systems are defined in terms of a domain of analyticity of the transfer function and of a sign condition that must be satisfied in such domain. In this way, we do not require to restrict the attention to systems with a rational transfer function. In this work, we also define various grades of negative imaginary systems and aim to provide a unitary view of the different notions that have appeared so far in the literature within the framework of positive real and in the more recent theory of negative imaginary systems, and to show how these notions are characterized and linked to each other.