A Note on the Cross Gramian for Non-Symmetric Systems

TL;DR

This paper introduces an approximation of the cross gramian for non-symmetric, non-square systems using decentralized control concepts, enabling its application in model order reduction.

Contribution

It proposes a novel method to approximate the cross gramian for systems that are neither symmetric nor square, expanding its applicability.

Findings

The non-symmetric cross gramian can be effectively approximated.

Application in model order reduction shows promising results.

Extends the utility of the cross gramian to a broader class of systems.

Abstract

The cross gramian matrix is a tool for model reduction and system identification, but it is only computable for square control systems. For symmetric systems the cross gramian possesses a useful relation to the system's associated Hankel singular values. Yet, many real-life models are neither square nor symmetric. In this work, concepts from decentralized control are used to approximate a cross gramian for non-symmetric and non-square systems. To illustrate this new non-symmetric cross gramian, it is applied in the context of model order reduction.