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Resolvent-based tools for optimal estimation and control via the Wiener–Hopf formalism

Published online by Cambridge University Press:  28 February 2022

Eduardo Martini Open the ORCID record for Eduardo Martini [Opens in a new window] ,
Junoh Jung Open the ORCID record for Junoh Jung [Opens in a new window] ,
André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window] ,
Peter Jordan Open the ORCID record for Peter Jordan [Opens in a new window]  and
Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window]
Eduardo Martini*
Affiliation:
Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos/SP, Brazil Département Fluides, Thermique et Combustion, Institut Pprime, CNRS, Université de Poitiers, ENSMA, 86000 Poitiers, France
Junoh Jung
Affiliation:
University of Michigan, Ann Arbor, MI 48109, USA*
André V.G. Cavalieri
Affiliation:
Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos/SP, Brazil
Peter Jordan
Affiliation:
Département Fluides, Thermique et Combustion, Institut Pprime, CNRS, Université de Poitiers, ENSMA, 86000 Poitiers, France
Aaron Towne
Affiliation:
University of Michigan, Ann Arbor, MI 48109, USA*
*
†Email address for correspondence: eduardo.martini@univ-poitiers.fr
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Abstract

The application of control tools to complex flows frequently requires approximations, such as reduced-order models and/or simplified forcing assumptions, where these may be considered low rank or defined in terms of simplified statistics (e.g. white noise). In this work we propose a resolvent-based control methodology with causality imposed via a Wiener–Hopf formalism. Linear optimal causal estimation and control laws are obtained directly from full-rank, globally stable systems with arbitrary disturbance statistics, circumventing many drawbacks of alternative methods. We use efficient, matrix-free methods to construct the matrix Wiener–Hopf problem, and we implement a tailored method to solve the problem numerically. The approach naturally handles forcing terms with space–time colour; it allows inexpensive parametric investigation of sensor/actuator placement in scenarios where disturbances/targets are low rank; it is directly applicable to complex flows disturbed by high-rank forcing; it has lower cost in comparison to standard methods; it can be used in scenarios where an adjoint solver is not available; or it can be based exclusively on experimental data. The method is particularly well suited for the control of amplifier flows, for which optimal control approaches are typically robust. Validation of the approach is performed using the linearized Ginzburg–Landau equation. Flow over a backward-facing step perturbed by high-rank forcing is then considered. Sensor and actuator placement are investigated for this case, and we show that while the flow response downstream of the step is dominated by the Kelvin–Helmholtz mechanism, it has a complex, high-rank receptivity to incoming upstream perturbations, requiring multiple sensors for control.

JFM classification

Flow Control: Control theory Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A19
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

*

The online version of this article has been updated since original publication. A notice detailing the change has also been published.

References

REFERENCES

Åkervik, E., Hœpffner, J., Ehrenstein, U. & Henningson, D.S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314.CrossRefGoogle Scholar
Amaral, F.R., Cavalieri, A.V., Martini, E., Jordan, P. & Towne, A. 2021 Resolvent-based estimation of turbulent channel flow using wall measurements. J. Fluid Mech. 927, A17.CrossRefGoogle Scholar
Åström, K.J. & Murray, R.M. 2010 Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press.Google Scholar
Åström, K.J. & Wittenmark, B. 2013 Computer-Controlled Systems: Theory and Design. Courier Corporation.Google Scholar
Atkinson, K.E. & Shampine, L.F. 2008 Algorithm 876: solving Fredholm integral equations of the second kind in Matlab. ACM Trans. Math. Softw. 34 (4), 1–20.CrossRefGoogle Scholar
Bagheri, S., Brandt, L. & Henningson, D.S. 2009 a Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263–298.CrossRefGoogle Scholar
Bagheri, S. & Henningson, D.S. 2011 Transition delay using control theory. Phil. Trans. R. Soc. A: Math. Phys. Engng Sci. 369 (1940), 1365–1381.CrossRefGoogle ScholarPubMed
Bagheri, S., Henningson, D.S., Hœpffner, J. & Schmid, P.J. 2009 b Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2), 020803.CrossRefGoogle Scholar
Barbagallo, A., Dergham, G., Sipp, D., Schmid, P.J. & Robinet, J.-C. 2012 Closed-loop control of unsteadiness over a rounded backward-facing step. J. Fluid Mech. 703, 326–362.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
Barrett, J.F. & Moir, T.J. 1987 A unified approach to multivariable discrete-time filtering based on the Wiener-theory. Kybernetika 23 (3), 177–197.Google Scholar
Belson, B.A., Semeraro, O., Rowley, C.W. & Henningson, D.S. 2013 Feedback control of instabilities in the two-dimensional Blasius boundary layer: the role of sensors and actuators. Phys. Fluids 25 (5), 054106.CrossRefGoogle Scholar
Bewley, T.R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349.CrossRefGoogle Scholar
Brito, P.P.C., Morra, P., Cavalieri, A.V.G., Araújo, T.B., Henningson, D.S. & Hanifi, A. 2021 Experimental control of Tollmien–Schlichting waves using pressure sensors and plasma actuators. Exp. Fluids 62 (2), 32.CrossRefGoogle Scholar
Brunton, S.L. & Noack, B.R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67, 050801.CrossRefGoogle Scholar
Cavalieri, A.V.G., Jordan, P. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71 (2), 020802.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Bewley, T.R. & Henningson, D.S. 2006 State estimation in wall-bounded flow systems. Part 2. Turbulent flows. J. Fluid Mech. 552, 167–187.CrossRefGoogle Scholar
Claerbout, J.F. 1976 Fundamentals of Geophysical Data Processing, vol. 274. Citeseer.Google Scholar
Colburn, C.H., Cessna, J.B. & Bewley, T.R. 2011 State estimation in wall-bounded flow systems. Part 3. The ensemble Kalman filter. J. Fluid Mech. 682, 289–303.CrossRefGoogle Scholar
Crighton, D. & Leppington, F. 1970 Scattering of aerodynamic noise by a semi-infinite compliant plate. J. Fluid Mech. 43 (4), 721–736.CrossRefGoogle Scholar
Dahan, J.A., Morgans, A.S. & Lardeau, S. 2012 Feedback control for form-drag reduction on a bluff body with a blunt trailing edge. J. Fluid Mech. 704, 360–387.CrossRefGoogle Scholar
Daniele, V. 1978 On the factorization of Wiener-Hopf matrices in problems solvable with Hurd's method. IEEE Trans. Antennas Propag. 26 (4), 614–616.CrossRefGoogle Scholar
Daniele, V. & Lombardi, G. 2007 Fredholm factorization of Wiener-Hopf scalar and matrix kernels. Radio Sci. 42 (06), 1–19.CrossRefGoogle Scholar
Daniele, V.G. & Zich, R. 2014 The Wiener-Hopf Method in Electromagnetics. SciTech Publishing Incorporated.CrossRefGoogle Scholar
Fabbiane, N., Semeraro, O., Bagheri, S. & Henningson, D.S. 2014 Adaptive and model-based control theory applied to convectively unstable flows. Appl. Mech. Rev. 66 (6), 060801.CrossRefGoogle Scholar
Fabbiane, N., Simon, B., Fischer, F., Grundmann, S., Bagheri, S. & Henningson, D.S. 2015 On the role of adaptivity for robust laminar flow control. J. Fluid Mech. 767, R1.CrossRefGoogle Scholar
Faranosov, G., Belyaev, I., Bychkov, O., Kopiev, V., Kopiev, V.A., Moralev, I. & Kazansky, P. 2019 Plasma-based active closed-loop control of instability waves in unexcited turbulent jet. Part 1. Free jet. In 25th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2019-2557.CrossRefGoogle Scholar
Farghadan, A., Towne, A., Martini, E. & Cavalieri, A.V.G. 2021 A randomized time-domain algorithm for efficiently computing resolvent modes. AIAA Paper 2021–2896.CrossRefGoogle Scholar
Fischer, P.F. 1998 Projection techniques for iterative solution of $Ax= b$ with successive right-hand sides. Comput. Meth. Appl. Mech. Engng 163 (1-4), 193–204.CrossRefGoogle Scholar
Fischer, P.F. & Patera, A.T. 1989 Parallel spectral element methods for the incompressible Navier–Stokes equations. In Solution of Superlarge Problems in Computational Mechanics, pp. 49–65. Springer.CrossRefGoogle Scholar
Freire, G.A., Cavalieri, A.V.G., Silvestre, F.J., Hanifi, A. & Henningson, D.S. 2020 Actuator and sensor placement for closed-loop control of convective instabilities. Theor. Comput. Fluid Dyn. 34 (5), 619–641.CrossRefGoogle Scholar
Grimble, M. 1979 Solution of the discrete-time stochastic optimal control problem in the 2-domain. Intl J. Syst. Sci. 10 (12), 1369–1390.CrossRefGoogle Scholar
Hanson, R.E., Bade, K.M., Belson, B.A., Lavoie, P., Naguib, A.M. & Rowley, C.W. 2014 Feedback control of slowly-varying transient growth by an array of plasma actuators. Phys. Fluids 26 (2), 024102.CrossRefGoogle Scholar
Hervé, A., Sipp, D., Schmid, P.J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 26–58.CrossRefGoogle Scholar
Högberg, M., Bewley, T.R. & Henningson, D.S. 2003 Relaminarization of $Re \tau =100$ turbulence using gain scheduling and linear state-feedback control. Phys. Fluids 15 (11), 3572–3575.CrossRefGoogle Scholar
Ilak, M. & Rowley, C.W. 2008 Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20 (3), 034103.CrossRefGoogle Scholar
Illingworth, S.J., Morgans, A.S. & Rowley, C.W. 2011 Feedback control of flow resonances using balanced reduced-order models. J. Sound Vib. 330 (8), 1567–1581.CrossRefGoogle Scholar
Jin, B., Illingworth, S.J. & Sandberg, R.D. 2020 Feedback control of vortex shedding using a resolvent-based modelling approach. J. Fluid Mech. 897, A26.CrossRefGoogle Scholar
Jones, B.L., Heins, P.H., Kerrigan, E.C., Morrison, J.F. & Sharma, A.S. 2015 Modelling for robust feedback control of fluid flows. J. Fluid Mech. 769, 687–722.CrossRefGoogle Scholar
Juang, J.-N. & Pappa, R.S. 1985 An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control Dyn. 8 (5), 620–627.CrossRefGoogle Scholar
Juillet, F., McKeon, B.J. & Schmid, P.J. 2014 Experimental control of natural perturbations in channel flow. J. Fluid Mech. 752, 296–309.CrossRefGoogle Scholar
Juillet, F., Schmid, P.J. & Huerre, P. 2013 Control of amplifier flows using subspace identification techniques. J. Fluid Mech. 725, 522–565.CrossRefGoogle Scholar
Karban, U., Bugeat, B., Martini, E., Towne, A., Cavalieri, A.V.G., Lesshafft, L., Agarwal, A., Jordan, P. & Colonius, T. 2020 Ambiguity in mean-flow-based linear analysis. J. Fluid Mech. 900, R5.CrossRefGoogle Scholar
Kim, J. & Bewley, T.R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
Kisil, A. 2016 Approximate Wiener-Hopf factorisation with stability analysis. PhD thesis, University of Cambridge.Google Scholar
Kopiev, V., Faranosov, G., Kopiev, V.A., Bychkov, O.P., Moralev, I. & Kazansky, P. 2019 Plasma-based active closed-loop control of instability waves in unexcited turbulent jet. Part 2. Installed jet. In 25th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2019-2558.CrossRefGoogle Scholar
Leclercq, C., Demourant, F., Poussot-Vassal, C. & Sipp, D. 2019 Linear iterative method for closed-loop control of quasiperiodic flows. J. Fluid Mech. 868, 26–65.CrossRefGoogle Scholar
Lesshafft, L. 2018 Artificial eigenmodes in truncated flow domains. Theor. Comput. Fluid Dyn. 32 (3), 245–262.CrossRefGoogle Scholar
Li, Y. & Gaster, M. 2006 Active control of boundary-layer instabilities. J. Fluid Mech. 550, 185–205.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493–517.CrossRefGoogle Scholar
Luhar, M., Sharma, A.S. & McKeon, B.J. 2014 Opposition control within the resolvent analysis framework. J. Fluid Mech. 749, 597–626.CrossRefGoogle Scholar
Ma, Z., Ahuja, S. & Rowley, C.W. 2011 Reduced-order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. 25 (1–4), 233–247.CrossRefGoogle Scholar
Maia, I.A., Jordan, P., Cavalieri, A.V.G., Martini, E., Sasaki, K. & Silvestre, F.J. 2021 Real-time reactive control of stochastic disturbances in forced turbulent jets. Phys. Rev. Fluids 6 (12), 123901.CrossRefGoogle Scholar
Martinelli, F. 2009 Feedback control of turbulent wall flows. PhD thesis, Politecnico di Milano.Google Scholar
Martini, E., Cavalieri, A.V.G., Jordan, P., Towne, A. & Lesshafft, L. 2020 Resolvent-based optimal estimation of transitional and turbulent flows. J. Fluid Mech. 900, A2.CrossRefGoogle Scholar
Martini, E., Rodríguez, D., Towne, A. & Cavalieri, A.V. 2021 Efficient computation of global resolvent modes. J. Fluid Mech. 919, A3.CrossRefGoogle Scholar
McKeon, B.J. & Sharma, A.S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Moir, T. & Barrett, J. 1989 Wiener theory of digital linear-quadratic control. Intl J. Control 49 (6), 2123–2155.CrossRefGoogle Scholar
Morari, M. & Zafiriou, E. 1989 Robust Process Control. Morari.Google Scholar
Morra, P., Nogueira, P.A.S., Cavalieri, A.V.G. & Henningson, D.S. 2021 The colour of forcing statistics in resolvent analyses of turbulent channel flows. J. Fluid Mech. 907, A24.CrossRefGoogle Scholar
Morra, P., Sasaki, K., Hanifi, A., Cavalieri, A.V. & Henningson, D.S. 2020 A realizable data-driven approach to delay bypass transition with control theory. J. Fluid Mech. 883, A33.CrossRefGoogle Scholar
Noack, B. & Eckelmann, H. 1993 Theoretical investigation of the cylinder wake with a low-dimensional Galerkin method. In Bluff-Body Wakes, Dynamics and Instabilities (ed. H. Eckelmann, J.M.R. Graham, P. Huerre & P.A. Monkewitz), pp. 143–146. Springer.CrossRefGoogle Scholar
Noble, B. 1959 Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press.CrossRefGoogle Scholar
Nogueira, P.A.S., Morra, P., Martini, E., Cavalieri, A.V.G. & Henningson, D.S. 2021 Forcing statistics in resolvent analysis: application in minimal turbulent Couette flow. J. Fluid Mech. 908, A32.CrossRefGoogle Scholar
Peake, N. 2004 On the unsteady motion of a long fluid-loaded elastic plate with mean flow. J. Fluid Mech. 507, 335–366.CrossRefGoogle Scholar
Rawlins, A. & Williams, W. 1981 Matrix wiener-hopf factorisation. Q. J. Mech. Appl. Maths 34 (1), 1–8.CrossRefGoogle Scholar
Rowley, C.W., Williams, D.R., Colonius, T., Murray, R.M. & Macmynowski, D.G. 2006 Linear models for control of cavity flow oscillations. J. Fluid Mech. 547, 317–330.CrossRefGoogle Scholar
Sasaki, K., Morra, P., Cavalieri, A.V.G., Hanifi, A. & Henningson, D.S. 2020 On the role of actuation for the control of streaky structures in boundary layers. J. Fluid Mech. 883, A34.CrossRefGoogle Scholar
Sasaki, K., Morra, P., Fabbiane, N., Cavalieri, A.V.G., Hanifi, A. & Henningson, D.S. 2018 a On the wave-cancelling nature of boundary layer flow control. Theor. Comput. Fluid Dyn. 32 (5), 593–616.CrossRefGoogle Scholar
Sasaki, K., Tissot, G., Cavalieri, A.V., Silvestre, F.J., Jordan, P. & Biau, D. 2016 Closed-loop control of wavepackets in a free shear-flow. In 22nd AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2016-2758.CrossRefGoogle Scholar
Sasaki, K., Tissot, G., Cavalieri, A.V.G., Silvestre, F.J., Jordan, P. & Biau, D. 2018 b Closed-loop control of a free shear flow: a framework using the parabolized stability equations. Theor. Comput. Fluid Dyn. 32 (6), 765–788.CrossRefGoogle Scholar
Schmid, P.J. & Sipp, D. 2016 Linear control of oscillator and amplifier flows. Phys. Rev. Fluids 1 (4), 040501.CrossRefGoogle Scholar
Semeraro, O., Bagheri, S., Brandt, L. & Henningson, D.S. 2011 Feedback control of three-dimensional optimal disturbances using reduced-order models. J. Fluid Mech. 677, 63–102.CrossRefGoogle Scholar
Semeraro, O., Pralits, J.O., Rowley, C.W. & Henningson, D.S. 2013 Riccati-less approach for optimal control and estimation: an application to two-dimensional boundary layers. J. Fluid Mech. 731, 394–417.CrossRefGoogle Scholar
Simon, B., Fabbiane, N., Nemitz, T., Bagheri, S., Henningson, D.S. & Grundmann, S. 2016 In-flight active wave cancelation with delayed-x-LMS control algorithm in a laminar boundary layer. Exp. Fluids 57 (10), 160.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Todoran, G., Holonec, R. & IAKAB, C. 2008 Discrete Hilbert transform. Numeric algorithms. Acta Electrotehn. 49 (4), 485–490.Google Scholar
Towne, A., Lozano-Durán, A. & Yang, X. 2020 Resolvent-based estimation of space–time flow statistics. J. Fluid Mech. 883, A17.CrossRefGoogle Scholar
Towne, A., Schmidt, O.T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867.CrossRefGoogle Scholar
Tuel, W. 1968 Computer algorithm for spectral factorization of rational matrices. IBM J. Res. Dev. 12 (2), 163–170.CrossRefGoogle Scholar
Wiener, N. 1942 The Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications. DIC Contract 6037. A Research Pursued on Behalf of the National Defense Research Council (Section D2) at the Massachusetts Institute of Technology. Massachusettes Insitute of Technology.Google Scholar
Youla, D., Jabr, H. & Bongiorno, J. 1976 Modern wiener-hopf design of optimal controllers–Part II: the multivariable case. IEEE Trans. Autom. Control 21 (3), 319–338.CrossRefGoogle Scholar
Zhou, C., Yang, L., Liu, Y. & Yang, Z. 2009 A novel method for computing the Hilbert transform with Haar multiresolution approximation. J. Comput. Appl. Maths 223 (2), 585–597.CrossRefGoogle Scholar