On Positive Real Lemma for Non-minimal Realization Systems

In this paper, we state the positive real lemma and the strictly positive real lemma (KYP lemma) for non-minimal realization systems. First we show the positive real lemma for stabilizable and observable systems under only the constraint on the regularity of the systems, by using the generalized algebraic Riccati equation. Moreover we show that the solution of the Lyapunov equation in the positive real lemma is positive definite. Next we similarly derive the KYP lemma for stabilizable and observable systems with only the above constraint and show that the corresponding solution in the KYP Lemma is positive definite. Finally, as the duals of these problems, we show that the positive real and KYP lemmas for controllable and detectable systems have both positive definite solutions.