Quantitatively hyper-positive real functions

Hyper-positive real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of functions, related through nested inclusions, introduced. Structurally, this family turns out to be matrix-convex and closed under inversion. A state-space characterization these a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, passive systems, here substituted by Quadratic Inclusions.