Some new results in the theory of negative imaginary systems with symmetric transfer matrix function

This note represents a first attempt to provide a definition and characterisation of negative imaginary systems for not necessarily rational transfer functions via a sign condition expressed in the entire domain of analyticity, along the same lines of the classic definition of positive real systems. Under the standing assumption of symmetric transfer functions, we then derive a necessary and sufficient condition that characterises negative imaginary transfer functions in terms of a matrix sign condition restricted to the imaginary axis, once again following the same line of argument of the standard positive real case. Using this definition, even transfer functions with a pole at the origin with double multiplicity, as well as with a possibly negative relative degree, can be negative imaginary.