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
NOTICE: this is a pre-peer review version of a work that was later accepted for publication in Linear Algebra and Its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Linear Algebra and Its Applications, volume 623, in 2021. https://doi.org/10.1016/j.laa.2020.11.014