Robust, positive and exact model reduction via monotone matrices

TL;DR

This paper develops a systematic approach for exact model reduction of positive linear systems using monotone matrix theory, ensuring positivity robustness and providing methods even when minimal realizations are unavailable.

Contribution

It characterizes when positive reductions with non-negative matrices exist and introduces algebraic techniques for constructing positive reductions beyond minimal realizations.

Findings

Characterization of positive reductions with non-negative matrices.

A systematic method for constructing positive reductions.

Robustness of positivity under small perturbations.

Abstract

This work focuses on the problem of exact model reduction of positive linear systems, by leveraging minimal realization theory. While determining the existence of a positive reachable realization remains in general an open problem, we are able to fully characterize the cases in which the new model is obtained with non-negative reduction matrices, and hence positivity of the reduced model is robust with respect to small perturbations of the original system. The characterization is obtained by specializing monotone matrix theory to positive matrices. In addition, we provide a systematic method to construct positive reductions also when minimal ones are not available, by exploiting algebraic techniques.