A Note on the Cross Gramian for Non-Symmetric Systems

By Himpe Christian and Ohlberger Mario
Working Paper (2015)

Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions

Working Paper (2019)

Hochs Peter and Roberts A.J.

New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian

Advances in Computational Mathematics (2019)

Rydel Marek

On Empirical System Gramians

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Grundel Sara, Himpe Christian, and Saak Jens

Cross-Gramian-Based Dominant Subspaces

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Benner Peter and Himpe Christian

On Reduced Input-Output Dynamic Mode Decomposition

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Himpe Christian, Benner Peter, and Mitchell Tim

Fast Low-Rank Empirical Cross Gramians

Working Paper (2017)

Himpe Christian, Leibner Tobias, Rave Stephan, and Saak Jens

emgr - The Empirical Gramian Framework

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Himpe Christian

Cross-Gramian-Based Model Reduction: A Comparison

Working Paper (2016)

Himpe Christian and Ohlberger Mario

Parareal in Time Intermediate Targets Methods for Optimal Control Problems

International Series of Numerical Mathematics (2013)

Mohamed Kamel Riahi, Maday Yvon, and Julien salomon

Abstract

The cross Gramian matrix is a tool for model reduction and system identification, but it is only applicable to 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.

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Please cite the paper as follows:

Himpe C. and Ohlberger M. (2015) A Note on the Cross Gramian for Non-Symmetric Systems. Working Paper.

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