On the Kalman-Yakubovich-Popov lemma for positive systems

The classical Kalman-Yakubovich-Popov lemma gives conditions for solvability of a certain inequality in terms of a symmetric matrix. The lemma has numerous applications in systems theory and control. Recently, it has been shown that for positive systems, important versions of the lemma can equivalently be stated in terms of a diagonal matrix rather than a general symmetric one. This paper generalizes these results and a new proof is given.