On the Kalman-Yakubovich-Popov Lemma for Positive Systems

An extended Kalman-Yakubovich-Popov (KYP) Lemma for positive systems is derived. The main difference compared to earlier versions is that non-strict inequalities are treated. Matrix assumptions are also less restrictive. Moreover, a new equivalence is introduced in terms of linear programming rather than semi-definite programming. As a complement to the KYP lemma, it is also proved that a symmetric Metzler matrix with $m$ nonzero entries above the diagonal is negative semi-definite if and only if it can be written as a sum of $m$ negative semi-definite matrices, each of which has only four nonzero entries. This is useful in the context large-scale optimization.