The Bounded Real Lemma for Internally Positive Systems and H-Infinity Structured Static State Feedback

We consider the bounded real lemma for internally positive linear time-invariant systems. We show that the
$H_{\infty}$
norm of such systems can be evaluated by checking the existence of a certain diagonal quadratic storage function. Taking advantage of this fact, the problem of designing a structured static state feedback controller achieving internal stability, contractiveness, and internal positivity in closed loop becomes convex and tractable.