The positive real lemma and construction of all realizations of generalized positive rational functions

TL;DR

This paper extends the Positive Real Lemma to generalized positive rational functions with non-minimal realizations, providing a systematic way to construct all such realizations and analyze their properties.

Contribution

It generalizes the Positive Real Lemma to complex matrix-valued functions and introduces a method to construct all realizations, including non-minimal ones, using Lyapunov inclusions.

Findings

Provides a construction procedure for all realizations of generalized positive functions.

Partitions realizations into convex sets based on Lyapunov inclusions.

Characterizes systems that can be made generalized positive via static output feedback.

Abstract

We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, we partition all state space realizations into subsets: Each is identified with a set of matrices satisfying the same Lyapunov inclusion and thus form a convex invertible cone, cic in short. Moreover, this approach enables us to characterize systems which may be brought to be generalized positive through static output feedback. The formulation through Lyapunov inclusions suggests the introduction of an equivalence class of rational functions of various dimensions associated with the same system matrix.