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Machine-learned control-oriented flow estimation for multi-actuator multi-sensor systems exemplified for the fluidic pinball

Published online by Cambridge University Press:  01 December 2022

Songqi Li Open the ORCID record for Songqi Li [Opens in a new window] ,
Wenpeng Li  and
Bernd R. Noack Open the ORCID record for Bernd R. Noack [Opens in a new window]
Songqi Li
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, 518055 Shenzhen, PR China
Wenpeng Li
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, 518055 Shenzhen, PR China
Bernd R. Noack*
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, 518055 Shenzhen, PR China
*
†Email address for correspondence: bernd.noack@hit.edu.cn
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Abstract

We propose the first machine-learned control-oriented flow estimation for multiple-input, multiple-output plants. The starting point is constant actuation with open-loop actuation commands leading to a database with simultaneously recorded actuation commands, sensor signals and flow fields. A key enabler is an estimator input vector comprising sensor signals and actuation commands. The mapping from the sensor signals and actuation commands to the flow fields is realized in an analytically simple, data-centric and general nonlinear approach. The analytically simple estimator generalizes linear stochastic estimation for actuation commands. The data-centric approach yields flow fields from estimator inputs by interpolating from the database – similar to Loiseau, Noack & Brunton (J. Fluid Mech., vol. 844, 2018, pp. 459–490) for unforced flow. The interpolation is performed with $k$ nearest neighbours ($k$NN). The general global nonlinear mapping from inputs to flow fields is obtained from a deep neural network (DNN) via an iterative training approach. The estimator comparison is performed for the fluidic pinball plant, which is a multiple-input, multiple-output wake control benchmark (Deng et al., J. Fluid Mech., vol. 884, 2020, A37) featuring rich dynamics under steady controls. We conclude that the machine learning methods clearly outperform the linear model. The performance of $k$NN and DNN estimators are comparable for periodic dynamics. Yet the DNN performs consistently better when the flow is chaotic. Moreover, a thorough comparison regarding the complexity, computational cost and prediction accuracy is presented to demonstrate the relative merits of each estimator. The proposed method can be generalized for closed-loop flow control plants.

JFM classification

Mathematical Foundations: Machine learning Mathematical Foundations: Big data
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 952 , 10 December 2022 , A36
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© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Abadi, M., et al. 2015 TensorFlow: large-scale machine learning on heterogeneous systems. Software available from tensorflow.org.Google Scholar
Adrian, R.J. 1994 Stochastic estimation of conditional structure: a review. Appl. Sci. Res. 53 (3–4), 291–303.CrossRefGoogle Scholar
Agarap, A.F. 2018 Deep learning using rectified linear units (ReLU). arXiv:1803.08375.Google Scholar
Amitay, M., Smith, D.R., Kibens, V., Parekh, D.E. & Glezer, A. 2001 Aerodynamic flow control over an unconventional airfoil using synthetic jet actuators. AIAA J. 39 (3), 361–370.CrossRefGoogle Scholar
Bansal, M.S. & Yarusevych, S. 2017 Experimental study of flow through a cluster of three equally spaced cylinders. Expl Therm. Fluid Sci. 80, 203–217.CrossRefGoogle Scholar
Barbagallo, A., Sipp, D. & Schmid, P.J. 2009 Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641, 1–50.CrossRefGoogle Scholar
Bearman, P.W. 1967 The effect of base bleed on the flow behind a two-dimensional model with a blunt trailing edge. Aeronaut. Q. 18 (3), 207–224.CrossRefGoogle Scholar
Bieker, K., Peitz, S., Brunton, S.L., Kutz, J.N. & Dellnitz, M. 2020 Deep model predictive flow control with limited sensor data and online learning. Theor. Comput. Fluid Dyn. 34 (4), 577–591.CrossRefGoogle Scholar
Blanchard, A.B., Maceda, G.Y.C., Fan, D., Li, Y., Zhou, Y., Noack, B.R. & Sapsis, T.P. 2021 Bayesian optimization for active flow control. Acta Mech. Sin. 37 (12), 1786–1798.CrossRefGoogle Scholar
Bonnet, J.P., Cole, D.R., Delville, J., Glauser, M.N. & Ukeiley, L.S. 1994 Stochastic estimation and proper orthogonal decomposition: complementary techniques for identifying structure. Exp. Fluids 17 (5), 307–314.CrossRefGoogle Scholar
Bourgeois, J.A., Martinuzzi, R.J. & Noack, B.R. 2013 Generalised phase average with applications to sensor-based flow estimation of the wall-mounted square cylinder wake. J. Fluid Mech. 736, 316–350.CrossRefGoogle Scholar
Bruce, L.M., Koger, C.H. & Li, J. 2002 Dimensionality reduction of hyperspectral data using discrete wavelet transform feature extraction. IEEE Trans. Geosci. Remote Sens. 40 (10), 2331–2338.CrossRefGoogle Scholar
Brunton, S.L. & Noack, B.R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67 (5), 050801.CrossRefGoogle Scholar
Brunton, S.L., Noack, B.R. & Koumoutsakos, P. 2020 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52 (1), 477–508.CrossRefGoogle Scholar
Burkardt, J., Gunzburger, M. & Lee, H.-C. 2006 a Centroidal Voronoi tessellation-based reduced-order modeling of complex systems. SIAM J. Sci. Comput. 28 (2), 459–484.CrossRefGoogle Scholar
Burkardt, J., Gunzburger, M. & Lee, H.-C. 2006 b POD and CVT-based reduced-order modeling of Navier–Stokes flows. Comput. Meth. Appl. Mech. Engng 196 (1–3), 337–355.CrossRefGoogle Scholar
Burkov, A. 2019 The Hundred-Page Machine Learning Book. Andriy Burkov.Google Scholar
Castellanos, R., Maceda, G.Y.C., de la Fuente, I., Noack, B.R., Ianiro, A. & Discetti, S. 2022 Machine-learning flow control with few sensor feedback and measurement noise. Phys. Fluids 34 (4), 047118.CrossRefGoogle Scholar
Cattafesta, L.N. & Sheplak, M. 2011 Actuators for active flow control. Annu. Rev. Fluid Mech. 43 (1), 247–272.CrossRefGoogle Scholar
Chamorro, L.P., Arndt, R.E.A. & Sotiropoulos, F. 2013 Drag reduction of large wind turbine blades through riblets: evaluation of riblet geometry and application strategies. Renew. Energy 50, 1095–1105.CrossRefGoogle Scholar
Chen, W., Ji, C., Alam, M.M., Williams, J. & Xu, D. 2020 Numerical simulations of flow past three circular cylinders in equilateral-triangular arrangements. J. Fluid Mech. 891, A14.CrossRefGoogle Scholar
Chen, J., Raiola, M. & Discetti, S. 2022 Pressure from data-driven estimation of velocity fields using snapshot PIV and fast probes. Exp. Therm. Fluid Sci. 136, 110647.CrossRefGoogle Scholar
Chen, K.K., Tu, J.H. & Rowley, C.W. 2012 Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887–915.CrossRefGoogle Scholar
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech. 262, 75–110.CrossRefGoogle Scholar
Clark, H., Naghib-Lahouti, A. & Lavoie, P. 2014 General perspectives on model construction and evaluation for stochastic estimation, with application to a blunt trailing edge wake. Exp. Fluids 55 (7), 1756.CrossRefGoogle Scholar
Cornejo Maceda, G.Y., Li, Y., Lusseyran, F., Morzyński, M. & Noack, B.R. 2021 Stabilization of the fluidic pinball with gradient-enriched machine learning control. J. Fluid Mech. 917, A42.CrossRefGoogle Scholar
Cortina-Fernández, J., Vila, C.S., Ianiro, A. & Discetti, S. 2021 From sparse data to high-resolution fields: ensemble particle modes as a basis for high-resolution flow characterization. Exp. Therm. Fluid Sci. 120, 110178.CrossRefGoogle Scholar
Cranmer, M., Sanchez Gonzalez, A., Battaglia, P., Xu, R., Cranmer, K., Spergel, D. & Ho, S. 2020 Discovering symbolic models from deep learning with inductive biases. In Adv. Neural Inf. Process. Syst. (ed. H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan & H. Lin), vol. 33, pp. 17429–17442. Curran Associates.Google Scholar
Deng, Z., Chen, Y., Liu, Y. & Kim, K. 2019 Time-resolved turbulent velocity field reconstruction using a long short-term memory (LSTM)-based artificial intelligence framework. Phys. Fluids 31 (7), 075108.Google Scholar
Deng, N., Noack, B.R., Morzyński, M. & Pastur, L.R. 2020 Low-order model for successive bifurcations of the fluidic pinball. J. Fluid Mech. 884, A37.CrossRefGoogle Scholar
Deng, N., Noack, B.R., Morzyński, M. & Pastur, L.R. 2021 a Galerkin force model for transient and post-transient dynamics of the fluidic pinball. J. Fluid Mech. 918, A4.CrossRefGoogle Scholar
Deng, N., Noack, B.R., Morzyński, M. & Pastur, L.R. 2022 Cluster-based hierarchical network model of the fluidic pinball – cartographing transient and post-transient, multi-frequency, multi-attractor behaviour. J. Fluid Mech. 934, A24.CrossRefGoogle Scholar
Deng, N., Pastur, L.R., Tuckerman, L.S. & Noack, B.R. 2021 b Coinciding local bifurcations in the Navier–Stokes equations. Europhys. Lett. 135 (2), 24002.CrossRefGoogle Scholar
Deng, Z., Zhu, X., Cheng, D., Zong, M. & Zhang, S. 2016 Efficient $k$NN classification algorithm for big data. Neurocomputing 195, 143–148.CrossRefGoogle Scholar
Evrard, A., Cadot, O., Herbert, V., Ricot, D., Vigneron, R. & Délery, J. 2016 Fluid force and symmetry breaking modes of a 3D bluff body with a base cavity. J. Fluids Struct. 61, 99–114.CrossRefGoogle Scholar
Farzamnik, E., Ianiro, A., Discetti, S., Deng, N., Oberleithner, K., Noack, B.R. & Guerrero, V. 2022 From snapshots to manifolds – a tale of shear flows. arXiv:2203.14781.Google Scholar
Fukami, K., Nakamura, T. & Fukagata, K. 2020 Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data. Phys. Fluids 32 (9), 095110.CrossRefGoogle Scholar
Gerhard, J., Pastoor, M., King, R., Noack, B.R., Dillmann, A., Morzyński, M. & Tadmor, G. 2003 Model-based control of vortex shedding using low-dimensional Galerkin models. In 33rd AIAA Fluid Dynamics Conference and Exhibit. AIAA Paper 2003-4262.Google Scholar
Geropp, D. & Odenthal, H.-J. 2000 Drag reduction of motor vehicles by active flow control using the Coanda effect. Exp. Fluids 28 (1), 74–85.CrossRefGoogle Scholar
Ghraieb, H., Viquerat, J., Larcher, A., Meliga, P. & Hachem, E. 2021 Single-step deep reinforcement learning for open-loop control of laminar and turbulent flows. Phys. Rev. Fluids 6 (5), 053902.Google Scholar
Glorot, X. & Bengio, Y. 2010 Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (ed. Y.W. Teh & M. Titterington), Proc. Mach. Learn. Res. 9, 249–256.Google Scholar
Guastoni, L., Güemes, A., Ianiro, A., Discetti, S., Schlatter, P., Azizpour, H. & Vinuesa, R. 2021 Convolutional-network models to predict wall-bounded turbulence from wall quantities. J. Fluid Mech. 928, A27.CrossRefGoogle Scholar
Güemes, A., Discetti, S. & Ianiro, A. 2019 Sensing the turbulent large-scale motions with their wall signature. Phys. Fluids 31 (12), 125112.CrossRefGoogle Scholar
Güemes, A., Discetti, S., Ianiro, A., Sirmacek, B., Azizpour, H. & Vinuesa, R. 2021 From coarse wall measurements to turbulent velocity fields through deep learning. Phys. Fluids 33 (7), 075121.CrossRefGoogle Scholar
Hasegawa, K., Fukami, K., Murata, T. & Fukagata, K. 2020 CNN-LSTM based reduced order modeling of two-dimensional unsteady flows around a circular cylinder at different Reynolds numbers. Fluid Dyn. Res. 52 (6), 065501.CrossRefGoogle Scholar
Hess, M., Alla, A., Quaini, A., Rozza, G. & Gunzburger, M. 2019 A localized reduced-order modeling approach for PDEs with bifurcating solutions. Comput. Meth. Appl. Mech. Engng 351, 379–403.Google Scholar
Hornik, K., Stinchcombe, M. & White, H. 1989 Multilayer feedforward networks are universal approximators. Neural Netw. 2 (5), 359–366.CrossRefGoogle Scholar
Howell, J., Sheppard, A. & Blakemore, A. 2003 Aerodynamic drag reduction for a simple bluff body using base bleed. SAE Trans., 1085–1091.Google Scholar
Ishar, R., Kaiser, E., Morzyński, M., Fernex, D., Semaan, R., Albers, M., Meysonnat, P.S., Schröder, W. & Noack, B.R. 2019 Metric for attractor overlap. J. Fluid Mech. 874, 720–755.Google Scholar
Joslin, R.D. & Miller, D.N. 2009 Fundamentals and Applications of Modern Flow Control. AIAA.CrossRefGoogle Scholar
Kingma, D.P. & Ba, J. 2014 Adam: a method for stochastic optimization. arXiv:1412.6980.Google Scholar
Kumar, Y., Bahl, P. & Chakraborty, S. 2022 State estimation with limited sensors – a deep learning based approach. J. Comput. Phys. 457, 111081.CrossRefGoogle Scholar
Lasagna, D., Orazi, M. & Iuso, G. 2013 Multi-time delay, multi-point linear stochastic estimation of a cavity shear layer velocity from wall-pressure measurements. Phys. Fluids 25 (1), 017101.CrossRefGoogle Scholar
LeCun, Y., Bengio, Y. & Hinton, G. 2015 Deep learning. Nature 521 (7553), 436–444.Google ScholarPubMed
Lee, C., Kim, J., Babcock, D. & Goodman, R. 1997 Application of neural networks to turbulence control for drag reduction. Phys. Fluids 9 (6), 1740–1747.CrossRefGoogle Scholar
Li, Y., Cui, W., Jia, Q., Li, Q., Yang, Z., Morzyński, M. & Noack, B.R. 2022 Explorative gradient method for active drag reduction of the fluidic pinball and slanted Ahmed body. J. Fluid Mech. 932, A7.CrossRefGoogle Scholar
Lin, Q. 2021 Fully automated control-oriented reduced-order modeling exemplified for the fluidic pinball. Master thesis, Harbin Institute of Technology.Google Scholar
Loiseau, J.-C., Noack, B.R. & Brunton, S.L. 2018 Sparse reduced-order modelling: sensor-based dynamics to full-state estimation. J. Fluid Mech. 844, 459–490.CrossRefGoogle Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A.M. Yaglom & V.I. Tartarsky), pp. 166–178.Google Scholar
McKay, M.D., Beckman, R.J. & Conover, W.J. 1979 A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21 (2), 239–245.Google Scholar
Menier, E., Bucci, M.A., Yagoubi, M., Mathelin, L. & Schoenauer, M. 2022 CD-ROM: complementary deep-reduced order model. arXiv:2202.10746v2.Google Scholar
Modi, V.J. 1997 Moving surface boundary-layer control: a review. J. Fluids Struct. 11 (6), 627–663.CrossRefGoogle Scholar
Mukut, A.N.M.M.I. & Abedin, M.Z. 2019 Review on aerodynamic drag reduction of vehicles. Intl J. Engng Mater. Manuf. 4 (1), 1–14.Google Scholar
Murata, T., Fukami, K. & Fukagata, K. 2020 Nonlinear mode decomposition with convolutional neural networks for fluid dynamics. J. Fluid Mech. 882, A13.CrossRefGoogle Scholar
Murray, N. & Ukeiley, L. 2002 Estimating the shear layer velocity field above an open cavity from surface pressure measurements. In 32nd AIAA Fluid Dynamics Conference and Exhibit. AIAA Paper 2002-2866.CrossRefGoogle Scholar
Nair, N.J. & Goza, A. 2020 Leveraging reduced-order models for state estimation using deep learning. J. Fluid Mech. 897, R1.CrossRefGoogle Scholar
Noack, B.R., Afabasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Noack, B.R. & Morzyński, M. 2017 The fluidic pinball – a toolkit for multiple-input multiple-output flow control (version 1.0). Tech. Rep. Institute of Combustion Engines and Transport, Poznan University of Technology.Google Scholar
Noack, B.R., Stankiewicz, W., Morzyński, M. & Schmid, P.J. 2016 Recursive dynamic mode decomposition of transient and post-transient wake flows. J. Fluid Mech. 809, 843–872.CrossRefGoogle Scholar
Pedregosa, F., et al. 2011 Scikit-learn: machine learning in Python. J. Machine Learning Res. 12, 2825–2830.Google Scholar
Peitz, S., Otto, S.E. & Rowley, C.W. 2020 Data-driven model predictive control using interpolated Koopman generators. SIAM J. Appl. Dyn. Syst. 19 (3), 2162–2193.CrossRefGoogle Scholar
Pfeiffer, J. & King, R. 2012 Multivariable closed-loop flow control of drag and yaw moment for a 3D bluff body. In 6th AIAA Flow Control Conf. (ed. B. Ganapathisubramani). AIAA Paper 2012-2802.CrossRefGoogle Scholar
Raibaudo, C. & Martinuzzi, R.J. 2021 Unsteady actuation and feedback control of the experimental fluidic pinball using genetic programming. Exp. Fluids 62 (10), 219.Google Scholar
Raibaudo, C., Zhong, P., Noack, B.R. & Martinuzzi, R.J. 2020 Machine learning strategies applied to the control of a fluidic pinball. Phys. Fluids 32 (1), 015108.CrossRefGoogle Scholar
Raissi, M., Perdikaris, P. & Karniadakis, G.E. 2019 Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707.CrossRefGoogle Scholar
Ramchoun, H., Amine, M., Idrissi, J., Ghanou, Y. & Ettaouil, M. 2016 Multilayer perceptron: architecture optimization and training. Intl J. Interact. Multimed. Artif. Intell. 4 (1), 26–30.Google Scholar
Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., Little, J. & Myatt, J.H. 2007 Feedback control of subsonic cavity flows using reduced-order models. J. Fluid Mech. 579, 315–346.CrossRefGoogle Scholar
Seifert, J. 2012 A review of the Magnus effect in aeronautics. Prog. Aerosp. Sci. 55, 17–45.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Maths 45 (3), 561–571.CrossRefGoogle Scholar
Stalnov, O., Palei, V., Fono, I., Cohen, K. & Seifert, A. 2007 Experimental estimation of a D-shaped cylinder wake using body-mounted sensors. Exp. Fluids 42 (4), 531–542.Google Scholar
Stodolsky, F. & Gaines, L. 2003 Railroad and locomotive technology roadmap. Tech. Rep. Argonne National Lab. (ANL).Google Scholar
Stone, C.J. 1977 Consistent nonparametric regression. Ann. Stat. 5 (4), 595–620.CrossRefGoogle Scholar
Sudin, M.N., Abdullah, M.A., Shamsuddin, S.A., Ramli, F.R. & Tahir, M.M. 2014 Review of research on vehicles aerodynamic drag reduction methods. Intl J. Mech. Mechatron. Engng 14 (2), 37–47.Google Scholar
Tadmor, G., Noack, B.R., Morzynski, M. & Siegel, S. 2004 Low-dimensional models for feedback flow control. Part II: Control design and dynamic estimation. In 2nd AIAA Flow Control Conference. American Institute of Aeronautics and Astronautics.Google Scholar
Taylor, J.A. & Glauser, M.N. 2004 Towards practical flow sensing and control via POD and LSE based low-dimensional tools. Trans. ASME J. Fluids Engng 126 (3), 337–345.Google Scholar
Tinney, C.E., Coiffet, F., Delville, J., Hall, A.M., Jordan, P. & Glauser, M.N. 2006 On spectral linear stochastic estimation. Exp. Fluids 41 (5), 763–775.CrossRefGoogle Scholar
Tu, J.H., Griffin, J., Hart, A., Rowley, C.W., Cattafesta, L.N. & Ukeiley, L.S. 2013 Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields. Exp. Fluids 54 (2), 1429.CrossRefGoogle Scholar
Tung, T.C. & Adrian, R.J. 1980 Higher-order estimates of conditional eddies in isotropic turbulence. Phys. Fluids 23 (7), 1469.CrossRefGoogle Scholar
Wang, J., Choi, K., Feng, L., Jukes, T.N. & Whalley, R.D. 2013 Recent developments in DBD plasma flow control. Prog. Aerosp. Sci. 62, 52–78.CrossRefGoogle Scholar
Weller, J., Lombardi, E. & Iollo, A. 2009 Robust model identification of actuated vortex wakes. Physica D 238, 416–427.CrossRefGoogle Scholar
Wood, C.J. 1964 The effect of base bleed on a periodic wake. J. R. Aeronaut. Soc. 68 (643), 477–482.CrossRefGoogle Scholar
Zaman, K.B.M.Q., Bridges, J.E. & Huff, D.L. 2011 Evolution from ‘tabs’ to ‘chevron technology’ – a review. Intl J. Aeroacoust. 10 (5–6), 685–709.CrossRefGoogle Scholar
Zhang, S., Li, X., Zong, M., Zhu, X. & Wang, R. 2018 Efficient $k$NN classification with different numbers of nearest neighbors. IEEE Trans. Neural Networks Learn. Syst. 29 (5), 1774–1785.CrossRefGoogle Scholar