Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-25T08:46:43.644Z Has data issue: false hasContentIssue false

Modelling for robust feedback control of fluid flows

Published online by Cambridge University Press:  25 March 2015

Bryn Ll. Jones*
Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S1 3JD, UK
P. H. Heins
Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S1 3JD, UK
E. C. Kerrigan
Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
J. F. Morrison
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
A. S. Sharma
Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK
Email address for correspondence:


This paper addresses the problem of designing low-order and linear robust feedback controllers that provide a priori guarantees with respect to stability and performance when applied to a fluid flow. This is challenging, since whilst many flows are governed by a set of nonlinear, partial differential–algebraic equations (the Navier–Stokes equations), the majority of established control system design assumes models of much greater simplicity, in that they are: firstly, linear; secondly, described by ordinary differential equations (ODEs); and thirdly, finite-dimensional. With this in mind, we present a set of techniques that enables the disparity between such models and the underlying flow system to be quantified in a fashion that informs the subsequent design of feedback flow controllers, specifically those based on the $\mathscr{H}_{\infty }$ loop-shaping approach. Highlights include the application of a model refinement technique as a means of obtaining low-order models with an associated bound that quantifies the closed-loop degradation incurred by using such finite-dimensional approximations of the underlying flow. In addition, we demonstrate how the influence of the nonlinearity of the flow can be attenuated by a linear feedback controller that employs high loop gain over a select frequency range, and offer an explanation for this in terms of Landahl’s theory of sheared turbulence. To illustrate the application of these techniques, an $\mathscr{H}_{\infty }$ loop-shaping controller is designed and applied to the problem of reducing perturbation wall shear stress in plane channel flow. Direct numerical simulation (DNS) results demonstrate robust attenuation of the perturbation shear stresses across a wide range of Reynolds numbers with a single linear controller.

© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Aamo, O. M. & Krstic, M. 2003 Flow Control by Feedback: Stabilization and Mixing. Springer.Google Scholar
Antoulas, A. C. 2005 An overview of approximation methods for large scale dynamical systems. Annu. Rev. Control 29, 181190.CrossRefGoogle Scholar
Arthur, G. G., McKeon, B. J., Dearing, S. S., Morrison, J. F. & Cui, Z. 2006 Manufacture of micro-sensors and actuators for flow control. Microelectron. Engng 83, 12051208.Google Scholar
Åström, K. J. & Murray, R. M. 2008 Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press.Google Scholar
Balas, M. J. 1978 Feedback control of flexible systems. IEEE Trans. Autom. Control 23, 673679.CrossRefGoogle Scholar
Baramov, L., Tutty, O. R. & Rogers, E. 2004 $H_{\infty }$ control of nonperiodic two-dimensional channel flow. IEEE Trans. Control Syst. Technol. 12, 111122.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1956 Turbulent diffusion. In Surveys in Mechanics (ed. Batchelor, G. K. & Davies, R. M.), pp. 352399. Cambridge University Press.Google Scholar
Bewley, T. R. 2001 Flow control: new challenges for a new renaissance. Prog. Aerosp. Sci. 37, 2158.Google Scholar
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305349.Google Scholar
Bobba, K. M.2004 Robust flow stability: theory, computations and experiments in near wall turbulence. PhD thesis, California Institute of Technology, Pasadena, CA, USA.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Bradshaw, P. 1967 ‘Inactive’ motion and pressure fluctuations in turbulent boundary layers. J. Fluid Mech. 30, 241258.Google Scholar
Bushnell, D. M. 2003 Aircraft drag reduction: a review. J. Aerosp. Engng 217, 118.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 16371650.CrossRefGoogle Scholar
Chernyshenko, S. I. & Baig, M. F. 2005 The mechanism of streak formation in near-wall turbulence. J. Fluid Mech. 544, 99131.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2010 Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow. Phys. Rev. E 82 (6), 066302.CrossRefGoogle Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011 The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689, 221253.CrossRefGoogle Scholar
Choi, H., Moin, P. & Kim, J. 1993 Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503539.Google Scholar
Chu, Y., Glover, K. & Dowling, A. P. 2003 Control of combustion oscillations via $\mathscr{H}_{\infty }$ loop-shaping, ${\it\mu}$ -analysis and Integral Quadratic Constraints. Automatica 39, 219231.Google Scholar
Collis, S. S., Joslin, R. D., Seifert, A. & Theofilis, V. 2004 Issues in active flow control: theory, control, simulation and experiment. Prog. Aerosp. Sci. 40, 237289.Google Scholar
Corbett, J. J. & Koehler, H. W. 2003 Updated emissions from ocean shipping. J. Geophys. Res. 108 (D20), 46504664.Google Scholar
Couchman, I. J. & Kerrigan, E. C. 2010 Control of mixing in a Stokes’ fluid flow. J. Process Control 20, 11031115.Google Scholar
Curtain, R. & Morris, K. 2009 Transfer functions of distributed parameter systems: a tutorial. Automatica 45, 11011116.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, 360387.Google Scholar
Dai, L. 1989 Singular Control Systems. Springer.Google Scholar
Dullerud, G. & Paganini, F. 2000 A Course in Robust Control Theory: A Convex Approach. Springer.CrossRefGoogle Scholar
Dunn, D. C. & Morrison, J. F. 2003 Anisotropy and energy flux in wall turbulence. J. Fluid Mech. 491, 353378.Google Scholar
Farrell, B. & Ioannou, J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5 (11), 26002609.Google Scholar
Farrell, B. & Ioannou, J. 1996 Turbulence suppression by active control. Phys. Fluids 8 (5), 12571268.Google Scholar
Ferziger, J. H. & Perić, M. 1997 Computational Methods for Fluid Dynamics. Springer.Google Scholar
Fish, F. E. & Lauder, G. V. 2006 Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38, 193224.CrossRefGoogle Scholar
Frederick, M., Kerrigan, E. C. & Graham, J. M. R. 2010 Gust alleviation using rapidly deployed trailing-edge flaps. J. Wind Engng Ind. Aerodyn. 98 (12), 712723.Google Scholar
Gad-el-Hak, M. 2000 Flow Control: Passive, Active and Reactive Flow Management. Cambridge University Press.Google Scholar
Gallas, Q., Carroll, B., Cattafesta, L., Holman, R., Nishida, T. & Sheplak, M. 2003 Lumped element modeling of piezoelectric-driven synthetic jet actuators. AIAA J. 41, 240247.Google Scholar
Gerdin, M.2006 Identification and estimation for models described by differential-algebraic equations. PhD thesis, Linköping University, Sweden.Google Scholar
Gibson, J. F.2012 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep., University of New Hampshire.Google Scholar
Glad, T. & Ljung, L. 2000 Control Theory: Multivariable and Nonlinear Methods. Taylor & Francis.Google Scholar
Golub, G. H. & Van Loan, C. F. 1996 Matrix Computations, 3rd edn. The Johns Hopkins University Press.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 2006 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.Google Scholar
Hanson, R., Lavoie, P., Naguib, A. M. & Morrison, J. F. 2010 Transient growth instability cancelation by a plasma actuator array. Exp. Fluids 49 (6), 13391348.Google Scholar
Hœpffner, J., Chevalier, M., Bewley, T. R. & Henningson, D. S. 2005 State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows. J. Fluid Mech. 534, 263294.Google Scholar
Hogberg, M., Bewley, T. R. & Henningson, D. S. 2003 Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149175.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.Google Scholar
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and some of the ‘problems’ of turbulence. J. Fluid Mech. 212, 497532.Google Scholar
Hyde, R. A., Glover, K. & Shanks, G. T. 1995 VSTOL first flight of an $H$ -infinity control law. Comput. Control Engng J. 6 (1), 1116.Google Scholar
Iwamoto, K.2002 Database of fully developed channel flow. Tech. Rep. ILR-0201, Department of Mechanical Engineering, The University of Tokyo.Google Scholar
Jones, B. L. & Kerrigan, E. C. 2010 When is the discretization of a spatially distributed system good enough for control? Automatica 46 (9), 14621468.Google Scholar
Jones, B. L., Kerrigan, E. C., Morrison, J. F. & Zaki, T. A. 2011 Flow estimation of boundary layers using DNS based wall shear information. Intl J. Control 84, 13101325.Google Scholar
Jovanovic, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145183.Google Scholar
Kim, J. 1989 On the structure of pressure fluctuations in simulated turbulent channel flow. J. Fluid Mech. 205, 421451.CrossRefGoogle Scholar
Kim, J. 2003 Control of turbulent boundary layers. Phys. Fluids 15 (5), 10931105.Google Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383417.Google Scholar
Kim, J. & Lim, J. 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 18851888.Google Scholar
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3D numerical spectral simulations of plane channel flows. In Proceedings of the Third GAMM-Conference on Numerical Methods in Fluid Mechanics, pp. 165173. Vieweg.Google Scholar
Landahl, M. T. 1967 A wave guide model for turbulent shear flow. J. Fluid Mech. 29, 441459.Google Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28 (4), 735756.Google Scholar
Landahl, M. T. 1977 Dynamics of boundary layer turbulence and the mechanism of drag reduction. Phys. Fluids 20 (10), S55S63.Google Scholar
Lee, K. H., Cortelezzi, L., Kim, J. & Speyer, J. 2001 Application of reduced-order controller to turbulent flows for drag reduction. Phys. Fluids 13 (5), 13211330.CrossRefGoogle Scholar
Lee, M. J., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561583.Google Scholar
Lim, J.2003 Control of wall-bounded turbulent shear flows using modern control theory. PhD thesis, University of California, Los Angeles.Google Scholar
Lu, Q., Bowyer, R. & Jones, B. L. 2014 Analysis and design of Coleman transform-based individual pitch controllers for wind-turbine load reduction. Wind Energy doi:10.1002/we.1769.Google Scholar
Luaga, E. & Bewley, T. R. 2004 Performance of a linear robust control strategy on a nonlinear model of spatially developing flows. J. Fluid Mech. 512, 343374.Google Scholar
McFarlane, D. & Glover, K. 1992 A loop shaping design procedure using $\mathscr{H}_{\infty }$ synthesis. IEEE Trans. Autom. Control 37 (6), 759769.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.Google Scholar
McKeon, B. J., Sharma, A. S. & Jacobi, I. 2013 Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25 (3), 031301.CrossRefGoogle Scholar
McKernan, J.2006 Control of plane Poiseuille flow: a theoretical and computational investigation. PhD thesis, Cranfield University.Google Scholar
McKernan, J., Papadakis, G. & Whidborne, J. F. 2006 Linear state-space representation of plane Poiseuille flow for control design: a tutorial. Intl J. Model. Identif. Control 1 (4), 272280.CrossRefGoogle Scholar
Morrison, J. F. 2007 The interaction between inner and outer regions of turbulent wall-bounded flow. Phil. Trans. R. Soc. A 365 (1852), 683698.CrossRefGoogle ScholarPubMed
Moser, R., Kim, J. & Mansour, N. N. 1999 Direct numerical simulation of turbulent channel flow up to $Re_{{\it\tau}}=590$ . Phys. Fluids 11 (4), 943945.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105 (15), 154502.Google Scholar
Pringle, C. C. T., Willis, A. P. & Kerswell, R. R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702, 415443.Google Scholar
Reinschke, J. & Smith, M. C. 2003 Designing robustly stabilising controllers for LTI spatially distributed systems using coprime factor synthesis. Automatica 39, 193203.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601639.Google Scholar
Rowley, C. W. 2005 Model reduction for fluids, using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 9971013.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2000 Stability and Transition in Shear Flows. Springer.Google Scholar
Schön, T., Gerdin, M., Glad, T. & Gustafsson, F.2003 A modeling and filtering framework for linear implicit systems. In Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 892–897.Google 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, 63102.Google Scholar
Shahzad, A., Jones, B. L., Kerrigan, E. C. & Constantinides, G. A. 2011 An efficient algorithm for the solution of a coupled Sylvester equation appearing in descriptor systems. Automatica 47, 244248.Google Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196238.Google Scholar
Sharma, A. S., Morrison, J. F., McKeon, B. J., Limebeer, D. J. N., Koberg, W. H. & Sherwin, S. J. 2011 Relaminarisation of $Re_{{\it\tau}}=100$ channel flow with globally stabilising linear feedback control. Phys. Fluids 23 (12), 125105.CrossRefGoogle Scholar
Skogestad, S. & Postlethwaite, I. 2005 Multivariable Feedback Control. Wiley.Google Scholar
Sturzebecher, D. & Nitsche, W. 2003 Active cancellation of Tollmien–Schlichting instabilities on a wing using multi-channel sensor actuator systems. Intl J. Heat Fluid Flow 24, 572583.Google Scholar
Townsend, A. A. 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97120.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behaviour of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578584.Google Scholar
Vinnicombe, G. 2001 Uncertainty and Feedback. Imperial College Press.Google Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465519.Google Scholar
Willcox, K. & Peraire, J. 2002 Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (11), 23232330.Google Scholar
Zhou, K. & Doyle, J. C. 1998 Essentials of Robust Control. Prentice Hall.Google Scholar
Zhou, K., Doyle, J. C. & Glover, K. 1996 Robust and Optimal Control. Prentice Hall.Google Scholar