Quantitatively Hyper-Positive Real Functions

TL;DR

This paper introduces a new family of matrix-valued hyper-positive real functions with nested subsets, providing a state-space characterization via a generalized Kalman-Yakubovich-Popov Lemma, and replacing classical linear matrix inclusions with quadratic ones.

Contribution

It defines and analyzes a new class of hyper-positive real functions, establishing their matrix-convexity, inversion closure, and a novel state-space characterization using quadratic matrix inclusions.

Findings

Family of hyper-positive functions is matrix-convex and closed under inversion.

Nested subsets of these functions are characterized.

State-space representation via a generalized KYP Lemma is provided.

Abstract

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.