Emgr

Synopsis

emgr - Empirical Gramian Framework (Version 5.7). 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 Observability Gramian
• Empirical Cross Gramian (including an Empirical Non-Symmetric Cross Gramian)
• Empirical Linear Cross Gramian
• Empirical Sensitivity Gramian (parameter controllability)
• Empirical Identifiability Gramian (parameter observability)
• Empirical Joint Gramian (parameter observability)

emgr overview

applicable to:

• Linear + Nonlinear Control Systems
• First + Second Order Control Systems
• Parametrized | Parametric Systems
• Time Invariant + Varying Systems
• Discretized PDEs

and with sample code for:

• Balanced Truncation and balancing related methods
• Combined State and Parameter Reduction
• Parameter Identification + Sensitivity Analysis
• Parameter Reduction + Robust Reduction
• Optimal Sensor + Actuator Placement
• Decentralized Control
• Nonlinearity Quantification

References

• 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". In: Proceedings in Applied Mathematics and Mechanics, vol. 17(1): 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". Proceedings of the 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 2014: 2739--2743, 2014.

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

• Meta Information: INI code-metadata

• Oberwolfach References on Mathematical Software: Entry

Contact

Christian Himpe