A generalization of the positive real lemma

The positive real lemma (also called the Kalman-Yacubovich-Popov lemma) characterizes the positive realness of the transfer function matrix of a linear dynamic system by algebraic conditions. In the case of a pole-zero cancellation appearing in the transfer function matrix, or in other words, missing the assumptions of controllability and observability, there exist generalized versions, which are discussed and proven applying the Kalman canonical decomposition.