Quantitatively hyper-positive real rational functions III

TL;DR

This paper introduces a new family of matrix-valued hyper-positive real rational functions, explores their properties, and provides a state-space characterization using quadratic matrix inclusions, advancing the understanding of absolute stability in control systems.

Contribution

It defines nested subsets of hyper-positive functions, proves their matrix-convexity and closure under inversion, and extends classical LMI techniques to quadratic matrix inclusions for stability analysis.

Findings

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

State-space characterization via a quadratic matrix inclusion version of KYP lemma.

Provides a framework for analyzing absolute stability using quadratic matrix inclusions.

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.