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