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.
D. Alpay and I. Lewkowicz, "Quantitatively Hyper-positive Real Functions", Linear Algebra and Its Applications, Vol. 623, pp. 316-334, 2021. https://doi.org/10.1016/j.laa.2020.11.014