Bilinear systems -- A new link to $\mathcal H_2$-norms, relations to stochastic systems and further properties

TL;DR

This paper introduces new theoretical insights into bilinear systems, linking $\

Contribution

It establishes a new connection between output error and $\\mathcal{H}_2$-error, and provides global reachability conditions and stability criteria for bilinear systems.

Findings

New link between output error and $\\mathcal{H}_2$-error.

Global reachability characterization using a Gramian.

Justification of $\\mathcal{H}_2$-optimal model reduction effectiveness.

Abstract

In this paper, we prove several new results that give new insights into bilinear systems. We discuss conditions for asymptotic stability using probabilistic arguments. Moreover, we provide a global characterization of reachability in bilinear systems based on a certain Gramian. Reachability energy estimates using the same Gramian have only been local so far. The main result of this paper, however, is a new link between the output error and the $\mathcal H_2$-error of two bilinear systems. This result has several consequences in the field of model order reduction. It explains why $\mathcal H_2$-optimal model order reduction leads to good approximations in terms of the output error. Moreover, output errors based on the $\mathcal H_2$-norm can now be proved for balancing related model order reduction schemes in this paper. All these new results are based on a Gronwall lemma for matrix differential equations that is established here.