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Cluster-based control of a separating flow over a smoothly contoured ramp

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Abstract

The ability to manipulate and control fluid flows is of great importance in many scientific and engineering applications. The proposed closed-loop control framework addresses a key issue of model-based control: The actuation effect often results from slow dynamics of strongly nonlinear interactions which the flow reveals at timescales much longer than the prediction horizon of any model. Hence, we employ a probabilistic approach based on a cluster-based discretization of the Liouville equation for the evolution of the probability distribution. The proposed methodology frames high-dimensional, nonlinear dynamics into low-dimensional, probabilistic, linear dynamics which considerably simplifies the optimal control problem while preserving nonlinear actuation mechanisms. The data-driven approach builds upon a state space discretization using a clustering algorithm which groups kinematically similar flow states into a low number of clusters. The temporal evolution of the probability distribution on this set of clusters is then described by a control-dependent Markov model. This Markov model can be used as predictor for the ergodic probability distribution for a particular control law. This probability distribution approximates the long-term behavior of the original system on which basis the optimal control law is determined. We examine how the approach can be used to improve the open-loop actuation in a separating flow dominated by Kelvin–Helmholtz shedding. For this purpose, the feature space, in which the model is learned, and the admissible control inputs are tailored to strongly oscillatory flows.

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References

  1. Brunton, S.L., Noack, B.R: Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67(5), 050801:01 (2015)

  2. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 (1963)

  3. Lasota, A ., Mackey, M.C.: Chaos, Fractals, and Noise, 2nd edn. Springer, New York (1994)

    Book  MATH  Google Scholar 

  4. Kaiser, E., Noack, B.R., Cordier, L., Spohn, A., Segond, M., Abel, M., Daviller, G., Östh, J., Krajnović, S., Niven, R.K.: Cluster-based reduced-order modeling of a mixing layer. J. Fluid Mech. 754, 365 (2014)

  5. Froyland, G.: Extracting dynamical behavior via Markov models. In: Mees, A.I. (ed.) Nonlinear Dynamics and Statistics, pp. 281–321. Birkhäuser, Boston (2001)

    Chapter  Google Scholar 

  6. Hopf, E.: Statistical hydromechanics and functional analysis. J. Ration. Mech. Anal. 1, 87 (1952)

    MATH  Google Scholar 

  7. Noack, B.R., Niven, R.K.: Maximum-entropy closure for a Galerkin system of incompressible shear flow. J. Fluid Mech. 700, 187 (2012)

    Article  MATH  Google Scholar 

  8. Munowitz, M., Pines, A., Mehring, M.: Multiple-quantum dynamics in NMR: a directed walk through Liouville space. J. Chem. Phys. 86, 3172 (1987)

    Article  Google Scholar 

  9. Brockett, R.W.: On the control of a flock by a leader. In: Proceedings of the Steklov Institute of Mathematics, vol. 268, pp. 49–57 (2010)

  10. Brockett, R.W.: Minimizing attention in a motion control context. In: Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 3349–3352 (2003)

  11. Majumdar, A., Vasudevan, R., Tobenkin, M.M., Tedrake, R.: Convex optimization of nonlinear feedback controllers via occupation measures. Int. J. Robot. Res. 33, 1209 (2014)

    Article  Google Scholar 

  12. Brockett, R.: Notes on the control of the Liouville equation. In: Cannarsa, P., Coron, J .M. (eds.) Control of Partial Differential Equations. Cetraro, Italy 2010, pp. 101–129. Springer, Berlin Heidelberg (2012)

  13. Bollt, E.M., Santitissadeekorn, N.: Applied and Computational Measurable Dynamics. Society for Industrial and Applied Mathematics, Philadelphia (2013)

  14. Iversion, K.E.: A Programming Language, 2nd edn. Wiley, New York, NY, USA (1962)

  15. Ulam, S.: A collection of Mathematical Problems. Interscience Publishers, New York (1960)

  16. Li, T.Y.: Finite approximation for the Frobenius-Perron operator: a solution to Ulam’s conjecture. J. Approx. Theory 17(2), 177 (1976)

  17. Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2007)

    MATH  Google Scholar 

  18. Lloyd, S.: Least squares quantization in PCM. IEEE Trans. Inf. Theory 28, 129 (1956). (Originally as an unpublished Bell laboratories Technical Note (1957))

  19. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (2000)

    Book  Google Scholar 

  20. Bang-Jensen, J., Gutin, G., Yeo, A.: When the greedy algorithm fails. Discrete Optim. 1, 121 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Powell, Warren P.: Approximate Dynamic Programming: Solving the Curses of Dimensionality. Wiley, Hoboken, New Jersey (2007)

  22. Sutton, R.S., Barto, A.G.: Reinforcement Learning: an Introduction. MIT Press, Cambridge (1998)

    Google Scholar 

  23. Sommer, F.: Mehrfachlösungen bei laminaren Strömungen mit druckinduzierter Ablösung: eine Kuspen-Katastrophe (trans.: Multiple solutions of laminar flows with pressure induced separation: a cusp catastrophe). Technical Report 7:206, Fortschrittberichte VDI, VDI Verlag, Düsseldorf (1992)

  24. Hood, P., Taylor, C.: Finite element methods in flow problems. In: Oden, J.T., Gallagher, R.H., Zienkiewicz, O.C., Taylor, C. (eds) Navier-Stokes equations using mixed interpolation, pp. 121–132. Huntsville Press, University of Alabama (1974) (1974)

  25. Morzyński, M.: Numerical solution of Navier-Stokes equations by the finite element method. In: Proceedings of SYMKOM 87, Compressor and Turbine Stage Flow Path—Theory and Experiment, pp. 119–128. (1987)

  26. Afanasiev, K.: Stabilitätsanalyse, niedrigdimensionale modellierung und optimale kontrolle der kreiszylinderumströmung (trans.: Stability analysis, low-dimensional modeling, and optimal control of the flow around a circular cylinder). Ph.D. thesis, Fakultät Maschinenwesen, Technische Universität Dresden (2003)

  27. Bao, F., Dallmann, U.C.: Some physical aspects of separation bubble on a rounded backward-facing step. Aerosp. Sci. Technol. 8, 83 (2004)

    Article  Google Scholar 

  28. Ho, C.M., Huerre, P.: Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365 (1984)

    Article  Google Scholar 

  29. Holmes, P., Lumley, J.L., Berkooz, G., Rowley, C.W.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  30. Du, Qiang, Gunzburger, Max D.: Centroidal Voronoi Tessellation Based Proper Orthogonal Decomposition Analysis. Birkhäuser Basel, Basel (2003)

    Book  MATH  Google Scholar 

  31. Schneider, T.M., Eckhardt, B., Vollmer, J.: Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75, 066313 (2007)

    Article  MathSciNet  Google Scholar 

  32. Giannakis, D., Majda, A.J.: Quantifying the predictive skill in long-range forecasting. Part I: Coarse-grained predictions in a simple ocean model. J. Climate 25 (6), 1793–1813 1793 (2012)

  33. Brunton, S.L., Brunton, B.W., Proctor, J.L., Kaiser, E., Kutz, J.N.: Chaos as an Intermittently Forced Linear System. arXiv:1608.05306v1 (2016)

  34. Froyland, G.: Statistically optimal almost-invariant sets. Physica D 200(3), 205 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Bellman, R.E.: Adaptive Control Processes. Princeton University Press, New York (1961)

    Book  MATH  Google Scholar 

  36. Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. II, 4th edn. Athena Scientific, Boston, MA (2012)

  37. Guéniat, F., Mathelin, L., Hussaini, M.Y.: A statistical learning strategy for closed-loop control of fluid flows. Theor. Comput. Fluid Dyn. 30, 497 (2016)

    Article  Google Scholar 

  38. Wahde, M.: Biologically Inspired Optimization Methods: An Introduction. WIT Press, Southampton, UK (2008)

  39. Amsallem, D., Cortial, J., Farhat, C.: In: 7th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition AIAA 2009-800, Orlando, Florida, 5–8 Jan 2009 (2009)

  40. Amsallem, David, Zahr, Matthew J., Farhat, Charbel: Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Methods Eng. 92(10), 891 (2012). doi:10.1002/nme.4371

    Article  MATH  MathSciNet  Google Scholar 

  41. Brunton, B.W., Brunton, J.L., Proctor, S.L., Kutz, J.N.: Sparse sensor placement optimization for classification. SIAM. J. Appl. Math. 76, 2099–2122 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  42. Brunton, S.L., Proctor, J.L., Tu, J.H., Kutz, J.N.: Compressive sampling and dynamic mode decomposition. J. Comp. Dyn. 2(2), 165–191 (2016)

  43. Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data: sparse identification of nonlinear dynamical systems. arXiv:1509.03580 (2015)

  44. Tedrake, R., Jackowski, Z., Cory, R., Roberts, J. W., Hoburg, W.: Learning to fly like a bird. In: 14th International Symposium of Robotics Research (ISRR 2009), Lucerne, Aug 31–Sept 3, 2009 (2016)

  45. Duriez, T., Parezanovic, V., Laurentie, J.C., Fourment, C., Delville, J., Bonnet, J.P., Cordier, L., Noack, B.R., Segond, M., Abel, M.W., Gautier, N., Aider, J.L., Raibaudo, C., Cuvier, C., Stanislas, M., Brunton, S.L.: Closed-loop control of experimental shear layers using machine learning (Invited). In: 7th AIAA Flow Control Conference AIAA Paper, Atlanta, Georgi (2014)

  46. Parezanović, V., Cordier, L., Spohn, A., Duriez, T., Noack, B.R., Bonnet, J.P., Segond, M., Abel, M., Brunton, S.L.: Frequency selection by feedback control in a turbulent shear flow. J. Fluid Mech. 797, 247 (2016)

    Article  Google Scholar 

  47. Mardia, K.V., Kent, J.T., Bibby, J.M.: Multivariate Analysis. Academic Press, Cambridge (1979)

    MATH  Google Scholar 

  48. Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, Monographs on Statistics and Applied Probability, vol. 88, 2nd edn. Chapman and Hall, London (2000)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mechanical Engineering Department, University of Washington, Seattle, WA, 98195, USA

    Eurika Kaiser

  2. Institut PPRIME, UPR 3346 CNRS – Université de Poitiers – ENSMA, 86961, Futuroscope Chasseneuil, France

    Eurika Kaiser, Bernd R. Noack & Andreas Spohn

  3. LIMSI-CNRS, UPR 3251, 91405, Orsay Cedex, France

    Bernd R. Noack

  4. Technische Universität Braunschweig, 38108, Braunschweig, Germany

    Bernd R. Noack

  5. Florida Center for Advanced Aero-Propulsion, Florida State University, Tallahassee, FL, 32310, USA

    Louis N. Cattafesta

  6. Chair of Virtual Engineering, Poznan University of Technology, 60-965, Poznań, Poland

    Marek Morzyński

Authors
  1. Eurika Kaiser
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  2. Bernd R. Noack
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  3. Andreas Spohn
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  4. Louis N. Cattafesta
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  5. Marek Morzyński
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Corresponding author

Correspondence to Eurika Kaiser.

Additional information

Communicated by Ati Sharma.

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Kaiser, E., Noack, B.R., Spohn, A. et al. Cluster-based control of a separating flow over a smoothly contoured ramp. Theor. Comput. Fluid Dyn. 31, 579–593 (2017). https://doi.org/10.1007/s00162-016-0419-4

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  • DOI: https://doi.org/10.1007/s00162-016-0419-4

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