Dynamic Mode Decomposition for Compressive System Identification

Bai, Zhe; Kaiser, Eurika; Proctor, Joshua L.; Kutz, J. Nathan; Brunton, Steven L.

doi:10.2514/1.j057870

Title: Dynamic Mode Decomposition for Compressive System Identification

Journal Article · Tue Mar 19 00:00:00 EDT 2019 · AIAA Journal

DOI:https://doi.org/10.2514/1.j057870· OSTI ID:1582371

Bai, Zhe ^[1]; Kaiser, Eurika ^[1]; Proctor, Joshua L. ^[2]; Kutz, J. Nathan ^[1]; Brunton, Steven L. ^[1]

Univ. of Washington, Seattle, WA (United States)
Inst. of Disease Modeling, Bellevue, WA (United States)

Dynamic mode decomposition has emerged as a leading technique to identify spatiotemporal coherent structures from high-dimensional data, benefiting from a strong connection to nonlinear dynamical systems via the Koopman operator. In this work, we integrate and unify two recent innovations that extend DMD to systems with actuation [Proctor et al., 2016] and systems with heavily subsampled measurements [Brunton et al., 2015]. When combined, these methods yield a novel framework for compressive system identification [code is publicly available at: https://github.com/zhbai/cDMDc]. It is possible to identify a low-order model from limited input-output data and reconstruct the associated full-state dynamic modes with compressed sensing, adding interpretability to the state of the reduced-order model. Moreover, when full-state data is available, it is possible to dramatically accelerate downstream computations by first compressing the data. We demonstrate this unified framework on two model systems, investigating the effects of sensor noise, different types of measurements (e.g., point sensors, Gaussian random projections, etc.), compression ratios, and different choices of actuation (e.g., localized, broadband, etc.). In the first example, we explore this architecture on a test system with known low-rank dynamics and an artificially inflated state dimension. The second example consists of a real-world engineering application given by the fluid flow past a pitching airfoil at low Reynolds number. This example provides a challenging and realistic test-case for the proposed method, and results demonstrate that the dominant coherent structures are well characterized despite actuation and heavily subsampled data.

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Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC); US Air Force Office of Scientific Research (AFOSR); US Army Research Office (ARO); G&B Moore Foundation; A. P. Sloan Foundation

Grant/Contract Number:: AC02-05CH11231; FA9550-16-1-0650; W911NF-17-1-0118; 2013-10-29; 3835

OSTI ID:: 1582371

Journal Information:: AIAA Journal, Vol. 58, Issue 2; ISSN 0001-1452

Publisher:: AIAACopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 38 works

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References (86)

Adaptive algorithms for sparse system identification Kalouptsidis, Nicholas; Mileounis, Gerasimos; Babadi, Behtash Signal Processing, Vol. 91, Issue 8 https://doi.org/10.1016/j.sigpro.2011.02.013	journal	August 2011
Identification of Sparse Linear Operators Heckel, Reinhard; Bolcskei, Helmut IEEE Transactions on Information Theory, Vol. 59, Issue 12 https://doi.org/10.1109/TIT.2013.2280599	journal	December 2013
CoSaMP: iterative signal recovery from incomplete and inaccurate samples Needell, Deanna; Tropp, Joel A. Communications of the ACM, Vol. 53, Issue 12 https://doi.org/10.1145/1859204.1859229	journal	December 2010
Dynamic mode decomposition of numerical and experimental data Schmid, Peter J. Journal of Fluid Mechanics, Vol. 656 https://doi.org/10.1017/S0022112010001217	journal	July 2010
Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum Bagheri, Shervin Physics of Fluids, Vol. 26, Issue 9 https://doi.org/10.1063/1.4895898	journal	September 2014
Chaos as an intermittently forced linear system Brunton, Steven L.; Brunton, Bingni W.; Proctor, Joshua L. Nature Communications, Vol. 8, Issue 1 https://doi.org/10.1038/s41467-017-00030-8	journal	May 2017
Principal component analysis in linear systems: Controllability, observability, and model reduction Moore, B. IEEE Transactions on Automatic Control, Vol. 26, Issue 1 https://doi.org/10.1109/TAC.1981.1102568	journal	February 1981
Dynamic Mode Decomposition with Control Proctor, Joshua L.; Brunton, Steven L.; Kutz, J. Nathan SIAM Journal on Applied Dynamical Systems, Vol. 15, Issue 1 https://doi.org/10.1137/15M1013857	journal	January 2016
Compressive Sensing [Lecture Notes] Baraniuk, Richard G. IEEE Signal Processing Magazine, Vol. 24, Issue 4, p. 118-121 https://doi.org/10.1109/MSP.2007.4286571	journal	August 2007
Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations Sharma, Ati S.; Mezić, Igor; McKeon, Beverley J. Physical Review Fluids, Vol. 1, Issue 3 https://doi.org/10.1103/PhysRevFluids.1.032402	journal	July 2016
Compressed sensing Donoho, D. L. IEEE Transactions on Information Theory, Vol. 52, Issue 4 https://doi.org/10.1109/TIT.2006.871582	journal	April 2006
Low-Dimensional Approach for Reconstruction of Airfoil Data via Compressive Sensing Bai, Zhe; Wimalajeewa, Thakshila; Berger, Zachary AIAA Journal, Vol. 53, Issue 4 https://doi.org/10.2514/1.J053287	journal	April 2015
Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition Rowley, C. W. International Journal of Bifurcation and Chaos, Vol. 15, Issue 03 https://doi.org/10.1142/S0218127405012429	journal	March 2005
Online Sparse System Identification and Signal Reconstruction Using Projections Onto Weighted $\ell_{1}$ Balls Kopsinis, Yannis; Slavakis, Konstantinos; Theodoridis, Sergios IEEE Transactions on Signal Processing, Vol. 59, Issue 3 https://doi.org/10.1109/TSP.2010.2090874	journal	March 2011
Spectral analysis of fluid flows using sub-Nyquist-rate PIV data Tu, Jonathan H.; Rowley, Clarence W.; Kutz, J. Nathan Experiments in Fluids, Vol. 55, Issue 9 https://doi.org/10.1007/s00348-014-1805-6	journal	September 2014
The immersed boundary method: A projection approach Taira, Kunihiko; Colonius, Tim Journal of Computational Physics, Vol. 225, Issue 2 https://doi.org/10.1016/j.jcp.2007.03.005	journal	August 2007
Generalizing Koopman Theory to Allow for Inputs and Control Proctor, Joshua L.; Brunton, Steven L.; Kutz, J. Nathan SIAM Journal on Applied Dynamical Systems, Vol. 17, Issue 1 https://doi.org/10.1137/16M1062296	journal	January 2018
Constrained sparse Galerkin regression Loiseau, Jean-Christophe; Brunton, Steven L. Journal of Fluid Mechanics, Vol. 838 https://doi.org/10.1017/jfm.2017.823	journal	January 2018
Koopman-mode decomposition of the cylinder wake Bagheri, Shervin Journal of Fluid Mechanics, Vol. 726 https://doi.org/10.1017/jfm.2013.249	journal	June 2013
Randomized dynamic mode decomposition for nonintrusive reduced order modelling: RANDOMIZED DYNAMIC MODE DECOMPOSITION FOR NIROM Bistrian, Diana Alina; Navon, Ionel Michael International Journal for Numerical Methods in Engineering, Vol. 112, Issue 1 https://doi.org/10.1002/nme.5499	journal	February 2017
Dynamic mode decomposition using a Kalman filter for parameter estimation Nonomura, Taku; Shibata, Hisaichi; Takaki, Ryoji AIP Advances, Vol. 8, Issue 10 https://doi.org/10.1063/1.5031816	journal	October 2018
Parameter-Varying Aerodynamics Models for Aggressive Pitching-Response Prediction Hemati, Maziar S.; Dawson, Scott T. M.; Rowley, Clarence W. AIAA Journal, Vol. 55, Issue 3 https://doi.org/10.2514/1.J055193	journal	March 2017
Reduced-order unsteady aerodynamic models at low Reynolds numbers Brunton, Steven L.; Rowley, Clarence W.; Williams, David R. Journal of Fluid Mechanics, Vol. 724 https://doi.org/10.1017/jfm.2013.163	journal	April 2013
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems Benner, Peter; Gugercin, Serkan; Willcox, Karen SIAM Review, Vol. 57, Issue 4 https://doi.org/10.1137/130932715	journal	January 2015
Variable Projection Methods for an Optimized Dynamic Mode Decomposition Askham, Travis; Kutz, J. Nathan SIAM Journal on Applied Dynamical Systems, Vol. 17, Issue 1 https://doi.org/10.1137/M1124176	journal	January 2018
State-space model identification and feedback control of unsteady aerodynamic forces Brunton, Steven L.; Dawson, Scott T. M.; Rowley, Clarence W. Journal of Fluids and Structures, Vol. 50 https://doi.org/10.1016/j.jfluidstructs.2014.06.026	journal	October 2014
N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems Van Overschee, Peter; De Moor, Bart Automatica, Vol. 30, Issue 1 https://doi.org/10.1016/0005-1098(94)90230-5	journal	January 1994
Subspace-based methods for the identification of linear time-invariant systems Viberg, Mats Automatica, Vol. 31, Issue 12 https://doi.org/10.1016/0005-1098(95)00107-5	journal	December 1995
A dynamic mode decomposition approach for large and arbitrarily sampled systems Guéniat, Florimond; Mathelin, Lionel; Pastur, Luc R. Physics of Fluids, Vol. 27, Issue 2 https://doi.org/10.1063/1.4908073	journal	February 2015
Cluster-based reduced-order modelling of a mixing layer Kaiser, Eurika; Noack, Bernd R.; Cordier, Laurent Journal of Fluid Mechanics, Vol. 754 https://doi.org/10.1017/jfm.2014.355	journal	August 2014
System identification using chaos with application to equalization of a chaotic modulation system Leung, H. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 45, Issue 3 https://doi.org/10.1109/81.662721	journal	March 1998
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control Korda, Milan; Mezić, Igor Automatica, Vol. 93 https://doi.org/10.1016/j.automatica.2018.03.046	journal	July 2018
An overview of subspace identification Qin, S. Joe Computers & Chemical Engineering, Vol. 30, Issue 10-12 https://doi.org/10.1016/j.compchemeng.2006.05.045	journal	September 2006
Compressed sensing and dynamic mode decomposition L. Brunton, Steven; L. Proctor, Joshua; H. Tu, Jonathan Journal of Computational Dynamics, Vol. 2, Issue 2 https://doi.org/10.3934/jcd.2015002	journal	January 2015
Closed-Loop Turbulence Control: Progress and Challenges Brunton, Steven L.; Noack, Bernd R. Applied Mechanics Reviews, Vol. 67, Issue 5 https://doi.org/10.1115/1.4031175	journal	August 2015
Extracting spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition Brunton, Bingni W.; Johnson, Lise A.; Ojemann, Jeffrey G. Journal of Neuroscience Methods, Vol. 258 https://doi.org/10.1016/j.jneumeth.2015.10.010	journal	January 2016
Sparsity enabled cluster reduced-order models for control Kaiser, Eurika; Morzyński, Marek; Daviller, Guillaume Journal of Computational Physics, Vol. 352 https://doi.org/10.1016/j.jcp.2017.09.057	journal	January 2018
Sparse Sensing and DMD-Based Identification of Flow Regimes and Bifurcations in Complex Flows Kramer, Boris; Grover, Piyush; Boufounos, Petros SIAM Journal on Applied Dynamical Systems, Vol. 16, Issue 2 https://doi.org/10.1137/15M104565X	journal	January 2017
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information Candes, E.J.; Romberg, J.; Tao, T. IEEE Transactions on Information Theory, Vol. 52, Issue 2, p. 489-509 https://doi.org/10.1109/TIT.2005.862083	journal	February 2006
Linear observer synthesis for nonlinear systems using Koopman Operator framework Surana, Amit; Banaszuk, Andrzej IFAC-PapersOnLine, Vol. 49, Issue 18 https://doi.org/10.1016/j.ifacol.2016.10.250	journal	January 2016
Sparse Sensor Placement Optimization for Classification Brunton, B. W.; Brunton, S. L.; Proctor, J. L. SIAM Journal on Applied Mathematics, Vol. 76, Issue 5 https://doi.org/10.1137/15M1036713	journal	January 2016
Dynamic Mode Decomposition Kutz, J. Nathan; Brunton, Steven L.; Brunton, Bingni W. Society for Industrial and Applied Mathematics https://doi.org/10.1137/1.9781611974508	book	January 2016
Sparsity-promoting dynamic mode decomposition Jovanović, Mihailo R.; Schmid, Peter J.; Nichols, Joseph W. Physics of Fluids, Vol. 26, Issue 2 https://doi.org/10.1063/1.4863670	journal	February 2014
On dynamic mode decomposition: Theory and applications Kutz, J. Nathan; Brunton, Steven L.; Luchtenburg, Dirk M. Journal of Computational Dynamics, Vol. 1, Issue 2 https://doi.org/10.3934/jcd.2014.1.391	journal	December 2014
Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition Dawson, Scott T. M.; Hemati, Maziar S.; Williams, Matthew O. Experiments in Fluids, Vol. 57, Issue 3 https://doi.org/10.1007/s00348-016-2127-7	journal	February 2016
An eigensystem realization algorithm for modal parameter identification and model reduction Juang, Jer-Nan; Pappa, Richard S. Journal of Guidance, Control, and Dynamics, Vol. 8, Issue 5 https://doi.org/10.2514/3.20031	journal	September 1985
Spectral analysis of nonlinear flows Rowley, Clarence W.; MeziĆ, Igor; Bagheri, Shervin Journal of Fluid Mechanics, Vol. 641 https://doi.org/10.1017/S0022112009992059	journal	November 2009
Analysis of Fluid Flows via Spectral Properties of the Koopman Operator Mezić, Igor Annual Review of Fluid Mechanics, Vol. 45, Issue 1 https://doi.org/10.1146/annurev-fluid-011212-140652	journal	January 2013
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator Arbabi, Hassan; Mezić, Igor SIAM Journal on Applied Dynamical Systems, Vol. 16, Issue 4 https://doi.org/10.1137/17M1125236	journal	January 2017
Résumé of the AIAA FDTC Low Reynolds Number Discussion Group's Canonical Cases Ol, Michael; Altman, Aaron; Eldredge, Jeff 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition https://doi.org/10.2514/6.2010-1085	conference	June 2012
Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses Chen, Kevin K.; Tu, Jonathan H.; Rowley, Clarence W. Journal of Nonlinear Science, Vol. 22, Issue 6, p. 887-915 https://doi.org/10.1007/s00332-012-9130-9	journal	April 2012
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis Towne, Aaron; Schmidt, Oliver T.; Colonius, Tim Journal of Fluid Mechanics, Vol. 847 https://doi.org/10.1017/jfm.2018.283	journal	May 2018
Reduced-order models for control of fluids using the eigensystem realization algorithm Ma, Zhanhua; Ahuja, Sunil; Rowley, Clarence W. Theoretical and Computational Fluid Dynamics, Vol. 25, Issue 1-4 https://doi.org/10.1007/s00162-010-0184-8	journal	February 2010
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions Mezić, Igor Nonlinear Dynamics, Vol. 41, Issue 1-3 https://doi.org/10.1007/s11071-005-2824-x	journal	August 2005
Estimation of perturbations in robotic behavior using dynamic mode decomposition Berger, Erik; Sastuba, Mark; Vogt, David Advanced Robotics, Vol. 29, Issue 5 https://doi.org/10.1080/01691864.2014.981292	journal	March 2015
Discovering dynamic patterns from infectious disease data using dynamic mode decomposition Proctor, Joshua L.; Eckhoff, Philip A. International Health, Vol. 7, Issue 2 https://doi.org/10.1093/inthealth/ihv009	journal	February 2015
A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions Colonius, Tim; Taira, Kunihiko Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 25-28 https://doi.org/10.1016/j.cma.2007.08.014	journal	April 2008
Balanced Model Reduction via the Proper Orthogonal Decomposition Willcox, K.; Peraire, J. AIAA Journal, Vol. 40, Issue 11 https://doi.org/10.2514/2.1570	journal	November 2002
Subspace methods for system identification Joe Qin, S. Automatica, Vol. 43, Issue 4 https://doi.org/10.1016/j.automatica.2006.07.027	journal	April 2007
System Identification So¨derstro¨m, T.; Stoica, P.; Johansson, Rolf Journal of Dynamic Systems, Measurement, and Control, Vol. 115, Issue 4 https://doi.org/10.1115/1.2899207	journal	December 1993
Balanced model reduction via the proper orthogonal decomposition Willcox, K.; Peraire, J. AIAA Journal, Vol. 40 https://doi.org/10.2514/3.15326	journal	January 2002
Cluster-based reduced-order modelling of a mixing layer Kaiser, Eurika; Noack, Bernd R.; Cordier, Laurent Universität Potsdam https://doi.org/10.25932/publishup-41611	text	January 2019
Balanced model reduction via the proper orthogonal decomposition Willcox, K.; Peraire, J. 15th AIAA Computational Fluid Dynamics Conference https://doi.org/10.2514/6.2001-2611	conference	August 2001
Estimation of perturbations in robotic behavior using dynamic mode decomposition Berger, Erik; Sastuba, Mark; Vogt, David Taylor & Francis https://doi.org/10.6084/m9.figshare.1348844	audiovisual	January 2015
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples Needell, D.; Tropp, J. A. Applied and Computational Harmonic Analysis, Vol. 26, Issue 3 https://doi.org/10.1016/j.acha.2008.07.002	journal	May 2009
Online Sparse System Identification and Signal Reconstruction using Projections onto Weighted $\ell_1$ Balls Kopsinis, Yannis; Slavakis, Konstantinos; Theodoridis, Sergios arXiv https://doi.org/10.48550/arxiv.1004.3040	text	January 2010
l_0 Norm Constraint LMS Algorithm for Sparse System Identification Gu, Yuantao; Jin, Jian; Mei, Shunliang arXiv https://doi.org/10.48550/arxiv.1303.2261	text	January 2013
State-space model identification and feedback control of unsteady aerodynamic forces Brunton, Steven L.; Dawson, Scott T. M.; Rowley, Clarence W. arXiv https://doi.org/10.48550/arxiv.1401.1473	text	January 2014
Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition Brunton, Bingni W.; Johnson, Lise A.; Ojemann, Jeffrey G. arXiv https://doi.org/10.48550/arxiv.1409.5496	text	January 2014
Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition Dawson, Scott T. M.; Hemati, Maziar S.; Williams, Matthew O. arXiv https://doi.org/10.48550/arxiv.1507.02264	text	January 2015
On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations Sharma, Ati S.; Mezić, Igor; McKeon, Beverley J. arXiv https://doi.org/10.48550/arxiv.1603.08378	text	January 2016
Chaos as an Intermittently Forced Linear System Brunton, Steven L.; Brunton, Bingni W.; Proctor, Joshua L. arXiv https://doi.org/10.48550/arxiv.1608.05306	text	January 2016
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control Korda, Milan; Mezić, Igor arXiv https://doi.org/10.48550/arxiv.1611.03537	text	January 2016
Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator Arbabi, Hassan; Mezić, Igor arXiv https://doi.org/10.48550/arxiv.1611.06664	text	January 2016
Modal Analysis of Fluid Flows: An Overview Taira, Kunihiko; Brunton, Steven L.; Dawson, Scott T. M. arXiv https://doi.org/10.48550/arxiv.1702.01453	preprint	January 2017
Dynamic mode decomposition using a Kalman filter for parameter estimation Nonomura, Taku; Shibata, Hisaichi; Takaki, Ryoji arXiv https://doi.org/10.48550/arxiv.1804.00143	text	January 2018
Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information Candes, Emmanuel; Romberg, Justin; Tao, Terence arXiv https://doi.org/10.48550/arxiv.math/0409186	preprint	January 2004
Nonlinear system identification using compressed sensing Naik, Manjish; Cochran, Douglas 2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR) https://doi.org/10.1109/acssc.2012.6489039	conference	November 2012
Sparse LMS for system identification 2009 IEEE International Conference on Acoustics, Speech and Signal Processing https://doi.org/10.1109/icassp.2009.4960286	conference	April 2009
Bayesian Dynamic Mode Decomposition Takeishi, Naoya; Kawahara, Yoshinobu; Tabei, Yasuo Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence https://doi.org/10.24963/ijcai.2017/392	conference	August 2017
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples Needell, D.; Tropp, J. A. arXiv https://doi.org/10.48550/arxiv.0803.2392	text	January 2008
Dynamic Mode Decomposition for Real-Time Background/Foreground Separation in Video Grosek, Jacob; Kutz, J. Nathan arXiv https://doi.org/10.48550/arxiv.1404.7592	preprint	January 2014
Data-Driven Sparse Sensor Placement for Reconstruction Manohar, Krithika; Brunton, Bingni W.; Kutz, J. Nathan arXiv https://doi.org/10.48550/arxiv.1701.07569	text	January 2017
Variable projection methods for an optimized dynamic mode decomposition Askham, Travis; Kutz, J. Nathan arXiv https://doi.org/10.48550/arxiv.1704.02343	preprint	January 2017
Subspace dynamic mode decomposition for stochastic Koopman analysis Takeishi, Naoya; Kawahara, Yoshinobu; Yairi, Takehisa arXiv https://doi.org/10.48550/arxiv.1705.04908	text	January 2017
Estimation of perturbations in robotic behavior using dynamic mode decomposition Berger, Erik; Sastuba, Mark; Vogt, David Taylor & Francis https://doi.org/10.6084/m9.figshare.1348844.v1	audiovisual	January 2015

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