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A Koopman Operator Approach for Computing and Balancing Gramians for Discrete Time Nonlinear Systems [article]

Enoch Yeung, Zhiyuan Liu, Nathan O. Hodas
2017 arXiv   pre-print
We introduce the Koopman operator as a canonical representation of the system and apply a lifting technique to compute gramians in the space of full-state observables.  ...  In this paper, we consider the problem of quantifying controllability and observability of a nonlinear discrete time dynamical system.  ...  In this work we use Koopman operators to construct controllability and observability gramians for a class of discrete time nonlinear systems with exogenous inputs.  ... 
arXiv:1709.08712v1 fatcat:ls5o3m5abrfehodh6a5a5bepyq

Decomposition of Nonlinear Dynamical Systems Using Koopman Gramians [article]

Zhiyuan Liu, Soumya Kundu, Lijun Chen, Enoch Yeung
2017 arXiv   pre-print
In this paper we propose a new Koopman operator approach to the decomposition of nonlinear dynamical systems using Koopman Gramians.  ...  We introduce the notion of an input-Koopman operator, and show how input-Koopman operators can be used to cast a nonlinear system into the classical state-space form, and identify conditions under which  ...  In this paper we adopt a similar Koopman operator approach for system decomposition as Raak et al.  ... 
arXiv:1710.01719v1 fatcat:ghapjdmjgvbljo3yahdymphx5e

Model Order Reduction for Gas and Energy Networks [article]

Christian Himpe, Sara Grundel, Peter Benner
2021 arXiv   pre-print
and parametric model reduction for nonlinear systems.  ...  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 system identification method is based on the Koopman operator, which is an infinite dimensional, but linear operator, mapping (a transformation, or observable of the) state x k at discrete-time step  ... 
arXiv:2011.12099v3 fatcat:fl275ki6ljalblu7ycvkbi64du

Linearly-Recurrent Autoencoder Networks for Learning Dynamics [article]

Samuel E. Otto, Clarence W. Rowley
2019 arXiv   pre-print
This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator.  ...  Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems.  ...  We would also like to thank William Eggert for his invaluable help and collaboration on the initial iterations of the LRAN code.  ... 
arXiv:1712.01378v2 fatcat:oxmdv7vmpzbyhng72o2phf5y54

Model order reduction for gas and energy networks

Christian Himpe, Sara Grundel, Peter Benner
2021 Journal of Mathematics in Industry  
networks, numerical simulation of hyperbolic partial differential equation, and parametric model reduction for nonlinear systems.  ...  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.  ...  Funding This work is supported by the German Federal Ministry for Economic Affairs and Energy, in the joint project: "MathEnergy -Mathematical Key Technologies for Evolving Energy Grids", sub-project:  ... 
doi:10.1186/s13362-021-00109-4 fatcat:3khgfoifkfdd3nelc4s65bzo2e

Including inputs and control within equation-free architectures for complex systems

Joshua L. Proctor, Steven L. Brunton, J. Nathan Kutz
2016 The European Physical Journal Special Topics  
The increasing ubiquity of complex systems that require control is a challenge for existing methodologies in characterization and controller design when the system is high-dimensional, nonlinear, and without  ...  Further, we discuss the link between these equation-based methods and recently developed equation-free methods such as the Dynamic Mode Decomposition and Koopman operator theory.  ...  JLP would like to thank Bill and Melinda Gates for their active support of the Institute for Disease Modeling and their sponsorship through the Global Good Fund.  ... 
doi:10.1140/epjst/e2016-60057-9 fatcat:mbt6slhmfzaxllagsffiqcfiwm

Reduced-order models for flow control: balanced models and Koopman modes [chapter]

Clarence W. Rowley, Igor Mezić, Shervin Bagheri, Philipp Schlatter, Dan S. Henningson
2009 IUTAM Bookseries  
We then describe a new method for analyzing nonlinear flows based on spectral analysis of the Koopman operator, a linear operator defined for any nonlinear dynamical system.  ...  ] , and discuss strengths and weaknesses of each approach.  ...  Acknowledgments This work was supported by the National Science Foundation, award CMS-0347239, and the Air Force Office of Scientific Research, awards FA9550-05-1-0369 and FA9550-07-1-0127.  ... 
doi:10.1007/978-90-481-3723-7_6 fatcat:4yrpl2quzrfihluwam4aydu6dq

emgr—The Empirical Gramian Framework

Christian Himpe
2018 Algorithms  
Empirical Gramian are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation.  ...  System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality.  ...  , for a hyperbolic SISO system, and third for a nonlinear SIMO system.  ... 
doi:10.3390/a11070091 fatcat:f3kvq4e5bzhrresk64qqks7fbi

Computational Hydrodynamic Stability and Flow Control Based on Spectral Analysis of Linear Operators

Shervin Bagheri
2012 Archives of Computational Methods in Engineering  
Spectral decomposition of the linearized Navier-Stokes operator, the Koopman operator, the spatial correlation operator and the Hankel operator provide a means to gain physical insight into the dynamics  ...  The theory and the algorithms are exemplified on flow over a flat plate and a jet in crossflow, as prototypes for the laminar-turbulent transition and three-dimensional vortex shedding.  ...  The time-discrete spectrum and the time-continuous spectrum of the Koopman operator are shown in Fig. 15 . The two spectra are related to each other via a linear transformation (see Sect. 4.1).  ... 
doi:10.1007/s11831-012-9074-0 fatcat:wesb356mgverdag2i6nx7yvyte

Modal Analysis of Fluid Flows: An Overview [article]

Kunihiko Taira, Steven L. Brunton, Scott T. M. Dawson, Clarence W. Rowley, Tim Colonius, Beverley J. McKeon, Oliver T. Schmidt, Stanislav Gordeyev, Vassilios Theofilis, Lawrence S. Ukeiley
2017 arXiv   pre-print
This step typically starts with a modal decomposition of an experimental or numerical dataset of the flow field, or of an operator relevant to the system.  ...  The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (Balanced POD), dynamic mode decomposition (DMD), Koopman  ...  We gratefully acknowledge the contributions made by DRS to this document and for his continued support of much of this work.  ... 
arXiv:1702.01453v2 fatcat:3s6lovtg3bdzbnoodqy24t4iv4

Data-driven approximations of dynamical systems operators for control [article]

Eurika Kaiser, J. Nathan Kutz, Steven L. Brunton
2019 arXiv   pre-print
The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics.  ...  Koopman operator.  ...  Acknowledgements EK gratefully acknowledges support by the "Washington Research Foundation Fund for Innovation in Data-Intensive Discovery" and a Data Science Environments project award from the Gordon  ... 
arXiv:1902.10239v1 fatcat:23bw5qdalbenddnelslxtmes7m

Modern Koopman Theory for Dynamical Systems [article]

Steven L. Brunton, Marko Budišić, Eurika Kaiser, J. Nathan Kutz
2021 arXiv   pre-print
This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems  ...  The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated  ...  We also thank Shervin Bagheri, Bing Brunton, Bethany Lusch, Ryan Mohr, Frank Noe, Josh Proctor, Clancy Rowley, and Peter Schmid for many fruitful discussions on DMD, Koopman theory, and control.  ... 
arXiv:2102.12086v2 fatcat:2oylyx25dbctvkjfnirfcgjuqu

Construction of energy-stable projection-based reduced order models

Irina Kalashnikova, Matthew F. Barone, Srinivasan Arunajatesan, Bart G. van Bloemen Waanders
2014 Applied Mathematics and Computation  
The key idea is to apply to the PDEs a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables.  ...  An approach for building energy-stable Galerkin reduced order models (ROMs) for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection  ...  Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04  ... 
doi:10.1016/j.amc.2014.10.073 fatcat:hcivo43d2bdhhmmxtetdhinfja

Generative stochastic modeling of strongly nonlinear flows with non-Gaussian statistics [article]

Hassan Arbabi, Themistoklis Sapsis
2022 arXiv   pre-print
As such, this framework represents the chaotic time series as the evolution of a stochastic system observed through the lens of a nonlinear map.  ...  Here, we propose a data-driven framework to model stationary chaotic dynamical systems through nonlinear transformations and a set of decoupled stochastic differential equations (SDEs).  ...  Boyko Dodov and AIR Worldwide for providing the reanalysis climate data, as well as Prof. Christian Lessig who performed the projection of the data to a spherical wavelet basis. We also thank Profs.  ... 
arXiv:1908.08941v5 fatcat:z6ionhdarbdldm4qg7rbku7rnm

On reduced input-output dynamic mode decomposition

Peter Benner, Christian Himpe, Tim Mitchell
2018 Advances in Computational Mathematics  
We compare excitation approaches for the data-driven identification process and describe an optimization-based stabilization strategy for the identified systems.  ...  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  ...  The authors are grateful for the helpful feedback and comments provided by the two anonymous referees.  ... 
doi:10.1007/s10444-018-9592-x fatcat:nazxklco3zbc5llfcuszashcdi
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