On positive real lemma for stabilizable and detectable systems

In this paper, we state the positive real lemma for stabilizable and detectable systems. First, without assumptions of controllability and observability, we show the positive real lemma for stable systems under only some constraint with respect to invariant zeros. Moreover we show that the solution of the Lyapunov equation in the positive real lemma is positive definite. Next we show the positive real and strictly positive real lemmas for stabilizable and detectable systems under only that constraint.