40 research outputs found
emgr - The Empirical Gramian Framework
System Gramian matrices are a well-known encoding for properties of
input-output systems such as controllability, observability or minimality.
These so-called system Gramians were developed in linear system theory for
applications such as model order reduction of control systems. Empirical
Gramian are an extension to the system Gramians for parametric and nonlinear
systems as well as a data-driven method of computation. The empirical Gramian
framework - emgr - implements the empirical Gramians in a uniform and
configurable manner, with applications such as Gramian-based (nonlinear) model
reduction, decentralized control, sensitivity analysis, parameter
identification and combined state and parameter reduction
Model Reduction for Complex Hyperbolic Networks
We recently introduced the joint gramian for combined state and parameter
reduction [C. Himpe and M. Ohlberger. Cross-Gramian Based Combined State and
Parameter Reduction for Large-Scale Control Systems. arXiv:1302.0634, 2013],
which is applied in this work to reduce a parametrized linear time-varying
control system modeling a hyperbolic network. The reduction encompasses the
dimension of nodes and parameters of the underlying control system. Networks
with a hyperbolic structure have many applications as models for large-scale
systems. A prominent example is the brain, for which a network structure of the
various regions is often assumed to model propagation of information. Networks
with many nodes, and parametrized, uncertain or even unknown connectivity
require many and individually computationally costly simulations. The presented
model order reduction enables vast simulations of surrogate networks exhibiting
almost the same dynamics with a small error compared to full order model.Comment: preprin
On Reduced Input-Output Dynamic Mode Decomposition
The identification of reduced-order models from high-dimensional data is a
challenging task, and even more so if the identified system should not only be
suitable for a certain data set, but generally approximate the input-output
behavior of the data source. In this work, we consider the input-output dynamic
mode decomposition method for system identification. We compare excitation
approaches for the data-driven identification process and describe an
optimization-based stabilization strategy for the identified systems
A Unified Software Framework for Empirical Gramians
A common approach in model reduction is balanced truncation, which is based on Gramian matrices classifying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical Gramians not only extended this concept to nonlinear systems but also provided a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical Gramians. The Gramian types will be discussed and applied in a model reduction framework for multiple-input multiple-output systems
Model Order Reduction for Gas and Energy Networks
To counter the volatile nature of renewable energy sources, gas networks take
a vital role. But, to ensure fulfillment of contracts under these
circumstances, a vast number of possible scenarios, incorporating uncertain
supply and demand, has to be simulated ahead of time. This many-query gas
network simulation task can be accelerated by model reduction, yet,
large-scale, nonlinear, parametric, hyperbolic partial differential(-algebraic)
equation systems, modeling natural gas transport, are a challenging application
for model order reduction algorithms.
For this industrial application, we bring together the scientific computing
topics of: mathematical modeling of gas transport networks, numerical
simulation of hyperbolic partial differential equation, and parametric model
reduction for nonlinear systems. This research resulted in the "morgen" (Model
Order Reduction for Gas and Energy Networks) software platform, which enables
modular testing of various combinations of models, solvers, and model reduction
methods. In this work we present the theoretical background on systemic
modeling and structured, data-driven, system-theoretic model reduction for gas
networks, as well as the implementation of "morgen" and associated numerical
experiments testing model reduction adapted to gas network models
A note on the cross Gramian for non-symmetric systems
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