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Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators

Published online by Cambridge University Press:  22 February 2010

S. AHUJA  and
C. W. ROWLEY
S. AHUJA*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
C. W. ROWLEY
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present an estimator-based control design procedure for flow control, using reduced-order models of the governing equations linearized about a possibly unstable steady state. The reduced-order models are obtained using an approximate balanced truncation method that retains the most controllable and observable modes of the system. The original method is valid only for stable linear systems, and in this paper, we present an extension to unstable linear systems. The dynamics on the unstable subspace are represented by projecting the original equations onto the global unstable eigenmodes, assumed to be small in number. A snapshot-based algorithm is developed, using approximate balanced truncation, for obtaining a reduced-order model of the dynamics on the stable subspace.

The proposed algorithm is used to study feedback control of two-dimensional flow over a flat plate at a low Reynolds number and at large angles of attack, where the natural flow is vortex shedding, though there also exists an unstable steady state. For control design, we derive reduced-order models valid in the neighbourhood of this unstable steady state. The actuation is modelled as a localized body force near the trailing edge of the flat plate, and the sensors are two velocity measurements in the near wake of the plate. A reduced-order Kalman filter is developed based on these models and is shown to accurately reconstruct the flow field from the sensor measurements, and the resulting estimator-based control is shown to stabilize the unstable steady state. For small perturbations of the steady state, the model accurately predicts the response of the full simulation. Furthermore, the resulting controller is even able to suppress the stable periodic vortex shedding, where the nonlinear effects are strong, thus implying a large domain of attraction of the stabilized steady state.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 645 , 25 February 2010 , pp. 447 - 478
Copyright
Copyright © Cambridge University Press 2010

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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
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., Schlatter, P., Schmid, P. J. & Henningson, D. S. 2009 b Global stability of a jet in crossflow. J. Fluid Mech. 624, 33–44.CrossRefGoogle Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.CrossRefGoogle Scholar
Birch, J. M. & Dickinson, M. H. 2001 Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412, 729–733.CrossRefGoogle Scholar
Choi, H., Jeon, W.-P. & Kim, J. 2008 Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113–139.CrossRefGoogle Scholar
Colonius, T. & Taira, K. 2008 A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Meth. Appl. Mech. Engng 197 (25–28), 2131–2146.CrossRefGoogle Scholar
Cortelezzi, L. 1996 Nonlinear feedback control of the wake past a plate with a suction point on the downstream wall. J. Fluid Mech. 327, 303–324.CrossRefGoogle Scholar
Cortelezzi, L., Chen, Y.-C. & Chang, H.-L. 1997 Nonlinear feedback control of the wake past a plate: from a low-order model to a higher-order model. Phys. Fluids 9 (7), 2009–2022.CrossRefGoogle Scholar
Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3 (10), 2337–2354.CrossRefGoogle Scholar
Ellington, C. P., van der Berg, C., Willmott, A. P. & Thomas, A. L. R. 1996 Leading-edge vortices in insect flight. Nature 384, 626–630.CrossRefGoogle Scholar
Gillies, E. A. 1998 Low-dimensional control of the circular cylinder wake. J. Fluid Mech. 371, 157–178.CrossRefGoogle Scholar
Gloerfelt, X. 2008 Compressible proper orthogonal decomposition/Galerkin reduced-order model of self-sustained oscillations in a cavity. Phys. Fluids 20, 115105.CrossRefGoogle Scholar
Graham, W. R., Peraire, J. & Tang, K. Y. 1999 Optimal control of vortex shedding using low-order models. Part 1. Open-loop model development. Intl J. Numer. Meth. Engng 44 (7), 945–972.3.0.CO;2-F>CrossRefGoogle Scholar
Greenblatt, D. & Wygnanski, I. J. 2000 The control of flow separation by periodic excitation. Prog. Aerosp. Sci. 36, 487–545.CrossRefGoogle Scholar
Henningson, D. S. & Åkervik, E. 2008 The use of global modes to understand transition and perform flow control. Phys. Fluids 20, 031302.CrossRefGoogle Scholar
Ilak, M. & Rowley, C. W. 2008 modelling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20, 034103.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
Kelley, C. T. 1995 Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics 16. SIAM.CrossRefGoogle Scholar
Kelley, C. T., Kevrekidis, I. G. & Qiao, L. 2004 Newton–Krylov solvers for timesteppers. http://arxiv.org/math/0404374.CrossRefGoogle Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users' Guide. SIAM.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Lumley, J. L. & Blossey, P. 1998 Control of turbulence. Annu. Rev. Fluid Mech. 30, 311–327.CrossRefGoogle Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. 2009 Reduced order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. Manuscript submitted for publication.Google Scholar
Moore, B. C. 1981 Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26 (1), 17–32.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, 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., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flow. J. Fluid Mech. 523, 339–365.CrossRefGoogle Scholar
Noack, B. R., Pelivan, I., Tadmor, G., Morzyński, M. & Comte, P. 2004 Robust low-dimensional Galerkin models of natural and actuated flows. In Fourth Aeroacoustics Workshop. RWTH Aachen, Aachen, Germany.Google Scholar
Pastoor, M., Henning, L., Noack, B. R., King, R. & Tadmor, G. 2008 Feedback shear layer control for bluff body drag reduction. J. Fluid Mech. 608, 161–196.CrossRefGoogle Scholar
Peskin, C. S. 1972 Flow patterns around heart valves: a numerical method. J. Comput. Phys. 11, 479–517.Google Scholar
Prabhu, R. D., Collis, S. S. & Chang, Y. 2001 The influence of control on proper orthogonal decomposition of wall-bounded turbulent flows. Phys. Fluids 12 (2), 520–537.CrossRefGoogle Scholar
Protas, B. 2008 Vortex dynamics models in flow control problems. Nonlinearity, 21 (9), R203–R250.CrossRefGoogle Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.CrossRefGoogle Scholar
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 997–1013.CrossRefGoogle Scholar
Rowley, C. W. & Juttijudata, V. 2005 Model-based control and estimation of cavity flow oscillations. In Proceedings of the 44th IEEE Conf. on Decision and Control. Seville, Spain.Google Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7 (3), 856–869.CrossRefGoogle Scholar
Scherpen, J. M. A. 1993 Balancing for nonlinear systems. Syst. Control Lett. 21 (2), 143–153.CrossRefGoogle Scholar
Siegel, S. G., Seidel, J., Fagley, C., Luchtenburg, D. M., Cohen, K. & McLaughlin, T. 2008 Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition. J. Fluid Mech. 610, 1–42.CrossRefGoogle Scholar
Skogestad, S. & Postlethwaite, I. 2005 Multivariable Feedback Control: Analysis and Design, 2nd edn. John Wiley and Sons.Google Scholar
Tadmor, G., Centuori, M., Noack, B., Luchtenburg, M., Lehmann, O. & Morzyński, M. 2007 Low order Galerkin models for the actuated flow around 2-D airfoils. AIAA Paper 2007-1313. Forty-fifth AIAA Aerospace Sciences Meeting and Exhibit.CrossRefGoogle Scholar
Taira, K. & Colonius, T. 2009 a Effect of tip vortices in low-Reynolds-number poststall flow control. AIAA J. 47 (3), 749–756.CrossRefGoogle Scholar
Taira, K. & Colonius, T. 2009 b Three-dimensional flows around low-aspect-ratio flat-plate wings at low Reynolds numbers. J. Fluid Mech. 623, 187–207.CrossRefGoogle Scholar
Trefethen, L. N. & Bau, D. III 1997 Numerical Linear Algebra. SIAM.CrossRefGoogle Scholar
Tuckerman, L. & Barkley, D. 2000 Bifurcation analysis for timesteppers. In Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Volumes in Applied Mathematics and Applications (ed. Doedel, E. & Tuckerman, L. S.) vol. 19, pp. 453–466. Springer.Google Scholar
Wang, Y., Haller, G., Banaszuk, A. & Tadmor, G. 2003 Closed-loop Lagrangian separation control in a bluff body shear flow model. Phys. Fluids 15 (8), 2251–2266.CrossRefGoogle Scholar
Williams, D., Collins, J., Jankhot, C., Colonius, T. & Tadmor, G. 2008 Control of flow structure on a semi-circular planform wing. AIAA Paper 2008–0597. Forty-sixth AIAA Aerospace Sciences Meeting and Exhibit.CrossRefGoogle Scholar
Zannetti, L. & Iollo, A. 2003 Passive control of the vortex wake past a flat plate at incidence. Theor. Comput. Fluid Dyn. 16, 211–230.CrossRefGoogle Scholar
Zhou, K., Salomon, G. & Wu, E. 1999 Balanced realization and model reduction for unstable systems. Intl J. Robust Nonlinear Control 9 (3), 183–198.3.0.CO;2-E>CrossRefGoogle Scholar