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We here extend the well known Positive Real Lemma (also known as theKalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positiverational function, when non-minimal realizations are considered. We thenexploit this result to provide an easy construction procedure of all (notnecessarily minimal) state space realizations of generalized positivefunctions. As a by-product, we partition all state space realizations intosubsets: Each is identified with a set of matrices satisfying the same Lyapunovinclusion and thus form a convex invertible cone, cic in short. Moreover, thisapproach enables us to characterize systems which may be brought to begeneralized positive through static output feedback. The formulation throughLyapunov inclusions suggests the introduction of an equivalence class ofrational functions of various dimensions associated with the same systemmatrix.

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