Non-minimal state-space realizations and the KYP-Lemma

This paper is devoted to discuss the implications of the Kalman-Yakubovich-Popov Lemma, so-called also Positive Real Lemma, when non-minimal state space realizations of transfer matrices of linear continuous systems are used. It is seen that some extra parametrical conditions for positive realness of the transfer matrix to be equivalent to the stability Lemma from a Lyapunov equation with extended constraints appear on the non-minimal parameterizations. Those extra conditions are, in fact, spurious since stable cancellations do not affect to the nature (positive real or not) of the associate transfer matrix.