An Extended Kalman-Yakubovich-Popov Lemma for Positive Systems∗

An extended Kalman-Yakubovich-Popov Lemma for positive systems is proved, which generalizes earlier versions in several respects: Non-strict inequalities are treated. Matrix assumptions are less restrictive. Moreover, a new equivalence is introduced in terms of linear programming rather than semi-definite programming. As a complement, we also prove that a symmetric Metzler matrix with m non-zero 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 non-zero entries.