The positive real lemma and construction of all realizations of generalized positive rational functions

We here extend the well known positive real lemma (also known as the Kalman–Yakubovich–Popov lemma) to a complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, this approach enables us to characterize systems which can be brought, through a static output feedback, to be generalized positive.