Emgr

Synopsis

emgr - Empirical Gramian Framework (Version 5.99). Empirical gramians can be computed for linear and nonlinear control systems for purposes of model order reduction, uncertainty quantification and system identification. Model reduction using empirical gramians can be applied to the state space, to the parameter space or to both through combined reduction. The emgr framework is a compact open source toolbox for gramian-based model reduction and compatible with OCTAVE and MATLAB.

Features

Application matrix for the empirical gramian framework

emgr encompasses seven types of gramians:

• Empirical Controllability Gramian / Empirical Output Controllability Gramian
• Empirical Observability Gramian / Empirical Average Observability Gramian
• Empirical Cross Gramian / Empirical Non-Symmetric Cross Gramian
• Empirical Linear Cross Gramian / Empirical Non-Symmetric Linear Cross Gramian
• Empirical Sensitivity Gramian (parameter controllability)
• Empirical Identifiability Gramian (parameter observability)
• Empirical Joint Gramian (parameter observability)

emgr overview

Applications:

• Model reduction
  • Proper Orthogonal Decomposition (POD)
  • Modified POD / Approximate Balancing
  • Dominant Subspace Projection Model Reduction (DSPMR)
  • Balanced Proper Orthogonal Decompsition (bPOD)
  • Balanced Truncation
  • Balanced Gains
  • Dynamic Mode Decomposition-Galerkin
• Parameter reduction
• Combined state and parameter reduction
• Decentralized control
• State sensitivity
• Parameter sensitivity
• Parameter identification
• Nonlinearity quantification
• Uncertainty quantification
• System indices
• System norms
• Tau function

Meta Information

Code Meta Data

name:	Empirical Gramian Framework (emgr)
version:	5.99 (2022-04-13)
id:	10.5281/zenodo.6457616 (doi)
author:	Christian Himpe (0000-0003-2194-6754)
topic:	Science, Mathematics, Model Reduction
type:	Toolbox
license:	BSD-2-Clause (open-source)
repository:	https://github.com/gramian/emgr (git)
language:	Matlab
dependencies:	Octave >=5.2, Matlab >=2017b
systems:	Linux, Windows
website:	https://gramian.de
keywords:	Controllability, Observability, Cross Gramian, Model Reduction, Model Order Reduction

Cite

Please cite emgr as:

C. Himpe (2021). emgr - EMpirical GRamian Framework (Version 5.99) [Software]. Available from https://gramian.de . doi:10.5281/zenodo.6457616

 @MISC{emgr599,
  author = {C.~Himpe},
  title = {{emgr - {EMpirical GRamian Framework} (Version~5.99)},
  howpublished = {\url{https://gramian.de}},
  year = {2022},
  doi = {10.5281/zenodo.6457616}
 }

References

• C. Himpe. "emgr - EMpirical GRamian Framework 5.99". Transactions on Mathematical Software, 49(3): 31, 2023.

• C. Himpe. "Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms". Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 141--164, 2021.

• P. Benner, C. Himpe. "Cross-Gramian-Based Dominant Subspaces". Advances in Computational Mathematics 45(5): 91, 2533--2553, 2019.

• S. Grundel, C. Himpe, J. Saak. "On Empirical System Gramians". Proceedings in Applied Mathematics and Mechanics, 19: e201900006, 2019.

• C. Himpe. "emgr - The Empirical Gramian Framework". Algorithms 11(7): 91, 2018.

• C. Himpe, T. Leibner, S. Rave, J. Saak. "Fast Low-Rank Empirical Cross Gramians". Proceedings in Applied Mathematics and Mechanics, 17: 841--842, 2017.

• C. Himpe. "Combined State and Parameter Reduction for Nonlinear Systems with an Application in Neuroscience". Westfälische Wilhelms Universität, Sierke Verlag Göttingen, 2017.

• C. Himpe, M. Ohlberger. "A Note on the Cross Gramian for Non-Symmetric Systems". System Science and Control Engineering 4(1): 199--208, 2016.

• C. Himpe, M. Ohlberger. "The Empirical Cross Gramian for Parametrized Nonlinear Systems". IFAC-PapersOnLine (8th Vienna International Conference on Mathematical Modelling), 48(1): 727--728, 2015.

• C. Himpe, M. Ohlberger. "Model Reduction for Complex Hyperbolic Networks". Proceedings of the European Control Conference: 2739--2743, 2014.

• C. Himpe, M. Ohlberger. "Cross-Gramian Based Combined State and Parameter Reduction for Large-Scale Control Systems". Mathematical Problems in Engineering, 2014: 843869, 2014.

• C. Himpe, M. Ohlberger. "A Unified Software Framework for Empirical Gramians". Journal of Mathematics 2013: 365909, 2013.

Links

• Official website: https://gramian.de

• Github Repo: https://github.com/gramian/emgr

• SWmath: Entry

• Oberwolfach References on Mathematical Software: Entry

• SIAM DSweb Dynamical Systems Software: Entry

Contact

Christian Himpe