Abstract
The mitigation of oscillatory vortex shedding behind a cylinder is chosen as a well-investigated benchmark problem to compare model-based feedback flow control approaches. The flow is sensed by a single velocity signal in the wake and is manipulated via a single volume force actuator. A low-dimensional proper orthogonal decomposition Galerkin model is adopted as a control-oriented fluid flow representation. An extended Kalman filter is used as an effective means for online dynamic state estimation. Investigated strategies of linear and nonlinear controller design include pole placement, linear parameter-varying, input–output linearization, Lyapunov-based backstepping, and nonlinear model predictive control. These strategies are applicable to a large class of flows with oscillatory dynamics and to experimental conditions, where variants have already been used. Controllers are evaluated and compared based on their application to the full plant, that is, to the direct numerical simulation of the wake, emulating an experiment with a single hot-wire sensor. Overall, nonlinear closed-loop control is shown to be distinctly superior to linear approaches. As is often the case, physics dictates a similarity of successful control commands, irrespective of the design approach, and differentiates these controllers, as a group, from less successful approaches.
Similar content being viewed by others
References
Afanasiev, K., Hinze, M.: Adaptive control of a wake flow using proper orthogonal decomposition. In: Cagnol, J., Polis, M.P., Zolesio, J.P. (eds.) Shape Optimization & Optimal Design, Lecture Notes in Pure and Applied Mathematics, vol. 216, pp. 317–332. Marcel Dekker, New York (2001)
Allan, B., Juang, J.N., Raney, D.L., Seifert, A., Pack, L.G., Brown, D.E.: Closed loop separation control using oscillatory flow excitations. Tech. Rep. 37, ICASE (2000)
Apkarian, P., Gahinet, P., Beckeri, G.: Self-scheduled H ∞ control of linear parameter-varying systems. In: Proceedings of the American Control Conference, pp. 856–860 (1994)
Becker R., Garwon M., Gutknecht C., Bärwolff G., King R.: Robust control of separated shear flows in simulation and experiment. J. Process Control 15(6), 691–700 (2005)
Becker, R., Garwon, M., King, R.: Development of model-based sensors and their use for closed-loop control of separated shear flows. In: Proceedings of the European Control Conference. University of Cambridge, UK, September 1–4 (2003)
Bergmann, M., Cordier, L., Brancher, J.P.: Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced order model. Phys. Fluids 17, 097,101–097,121 (2005)
Bewley T.R., Liu S.: Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349 (1998)
Bloor S.: The transition of turbulence in the wake of a circular cylinder. J. Fluid Mech. 19, 290–304 (1964)
Brogan W.L.: Modern control theory. Prentice-Hall, Englewood (1985)
Cordier, L., Bergmann, M.: Post-Processing of Experimental and Numerical Data, chap. Proper Orthogonal Decomposition: An Overview. VKI Lecture Series 2003-03. Von Kármán Institut for Fluid Dynamics (2003)
Cottet G.H., Koumoutsakos P.: Vortex Methods—Theory and Practice. Cambridge University Press, Cambridge (2000)
Crowley, J.L., Demazeau, Y.: Principles and techniques for sensor data fusion. Signal Processing 32(1–2), 5–27 (1993). doi:10.1016/0165-1684(93)90034-8. Intelligent Systems for Signal and Image Understanding
Deane A.E., Kevrekidis I.G., Karniadakis G.E., Orszag S.A.: Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Phys. Fluids A 3(10), 2337–2354 (1991)
Detemple-Laake E., Eckelmann H.: Phenomenology of Kármán vortex streets in oscillatory flow. Exps. Fluids 7, 217–227 (1989)
Dorf R., Bishop R.: Modern Control Systems, 10th edn. Pearson Prentice Hall, Englewood Cliffs (2005)
Fiedler H., Fernholz H.: On the management and control of turbulent shear flows. Progr. Aeronaut. Sci. 27, 305–387 (1990)
Fletcher C.A.J.: Computational Galerkin Methods. Springer, New York (1984)
Gad-el-Hak M.: Flow Control: Passive, Active and Reactive Flow Management, 1st edn. Cambridge University Press, Cambridge (2000)
Gad-el-Hak M., Pollard A., Bonnet J.P.: Flow Control—Fundamentals and practices, 1st edn. Springer, Berlin (1998)
Garwon, M., Urzynicok, F., Darmadi, L.H.., King, R., Bärwolff, G.: Adaptive control of separated flows. In: Proceedings of the European Control Conference. University of Cambridge, UK, September 1–4 (2003)
Gelb A.: Applied Optimal Estimation, 2nd edn. MIT Press, Cambridge (1986)
Gerhard, J., Pastoor, M., King, R., Noack, B.R., Dillmann, A., Morzyński, M., Tadmor, G.: Model-based control of vortex shedding using low-dimensional Galerkin models. In: 33rd AIAA Fluids Conference and Exhibit. Orlando, Florida, USA, June 23–26, 2003 (2003). Paper 2003-4262
Glezer A., Amitay M., Honohan A.M.: Aspects of low- and high-frequency actuation for aerodynamic flow control. AIAA J. 43(7), 1501–1511 (2005)
Henning, L., King, R.: Multivariable closed-loop control of the reattachment length downstream of a backward-facing step. In: Proceedings of the 16th IFAC World Congress. 3–8 July, 2005, Prague, Czechia (2005). IFAC Paper 02575
Hinze M., Kunisch K.: Three control methods for time—dependent fluid flow. Flow Turbul. Combust. 65, 273–298 (2000)
Holmes P., Lumley J.L., Berkooz G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 1st edn. Cambridge University Press, Cambridge (1998)
Isidori, A.: Nonlinear Control Systems: An Introduction, 2nd edn. Communications and Control Engineering Series. Springer (1989)
Jackson C.P.: A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45 (1987)
Khalil H.K.: Nonlinear Systems. MacMillan, New York (1992)
Kim, B.H., Williams, D.R.: Nonlinear coupling of fluctuating drag and lift on cylinders undergoing forced oscillations. J. Fluid Mech. 559, 335–353 (2006)
King, R., Aleksic, K., Gelbert G., Losse, N., Muminovic, R., Brunn, A., Nitsche, W., Bothien, M.R., Moeck, J.P., Paschereit, C.O., Noack, B.R., Rist, U., Zeng, M.: Model predictive flow control—invited paper. In: 4th AIAA Flow Control Conference, pp. 1–19. June 23–26, 2008, Seattle (2008). AIAA-Paper 2008-3975
King, R., Becker, R., Feuerbach, G., Henning, L., Petz, R., Nitsche, W., Lemke, O., Neise, W.: Adaptive flow control using slope seeking. Mediterr. Conf. Control Autom. 1–6 (2006). doi:10.1109/MED.2006.328753
King, R., Becker, R., Garwon, M., Henning, L.: Robust and adaptive closed-loop control of separated shear flows. In: 2nd AIAA Flow Control Conference. Portland, Oregon, USA, 28.6.–1.7.2004 (2004). Paper 2004-2519
King, R., Seibold, M., Lehmann, O., Noack, B.R., Morzyński, M., Tadmor, G.: Nonlinear flow control based on a low dimensional model of fluid flow. In: Meurer, T., Graichen, K., Gilles, E. (eds.) Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, LNCIS 322, pp. 369–386. Springer (2005)
King, R.E.: Active Flow Control. Notes on Numerical Fluid Mechanics and Interdisciplinary Design, vol. (95). Springer, Berlin (2007)
Krstić M., Wang H.H.: Stability of extremum seeking feedback for general nonlinear dynamic systems. Automatica 36, 595–601 (2000)
Landau L.D., Lifshitz E.M.: Fluid Mechanics. Pergamon Press, Londres (1959)
Lehmann, O., Luchtenburg, M., Noack, B.R., King, R., Morzyński, M., Tadmor, G.: Wake stabilization using pod galerkin models with interpolated modes. In: 44th IEEE Conference on Decision and Control and European Control Conference ECC. Seville, Spain, 12–15 Dec. 2005 (2005). Invited Paper MoA15.2
Leith D.J., Leithead W.E.: Survey of gain-scheduling analysis and design. Int. J. Control 73(11), 1001–1025 (2000)
Luchtenburg, D.M., Günter, B., Noack, B.R., King, R., Tadmor, G.: A generalized mean-field model of the natural and actuated flows around a high-lift configuration. J. Fluid Mech. submitted, 1–50 (2008)
Morzyński, M.: Methods of the flow control analysed with the global stability method. Tech. rep., Inst. of Combustion Engines and Basics of Machine Design, Poznan University of Technology, Poland (1997)
Morzyński M., Afanasiev K., Thiele F.: Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Meth. Appl. Mech. Eng. 169, 161–176 (1999)
Noack, B., Tadmor, G., Morzyński, M.: Actuation models and dissipative control in empirical Galerkin models of fluid flows. In: The 2004 American Control Conference, pp. 0001–0006. Boston, MA, USA, June 30–July 2, 2004 (2004). Paper FrP15.6
Noack, B.R.: Niederdimensionale Galerkin-Modelle für laminare und transitionelle freie Scherströmungen (transl.: low-dimensional Galerkin models of laminar and transitional free shear flows) (2006). Habilitation thesis, Berlin Institute of Technology, Germany
Noack B.R., Afanasiev K., Morzyński M., Tadmor G., Thiele F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)
Noack B.R., Papas P., Monkewitz P.A.: The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365 (2005)
Noack B.R., Schlegel M., Ahlborn B., Mutschke G., Morzyński M., Comte P., Tadmor G.: A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilibr. Thermodyn. 33(2), 103–148 (2008)
Ogata K.: Modern Control Engineering. Prentice-Hall, Englewood (1970)
Pastoor M., Henning L., Noack B.R., King R., Tadmor G.: Feedback shear layer control for bluff body drag reduction. J. Fluid Mech. 608, 161–196 (2008)
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, New York (1992)
Protas B.: Linear feedback stabilization of laminar vortex shedding based on a point vortex model. Phys. Fluids 16(12), 4473–4488 (2004)
Protas B., Styczek A.: Optimal rotary control of the cylinder wake in the laminar regime. Phys. Fluids 14(7), 2073–2087 (2002)
Rebbeck H., Choi K.S.: A wind-tunnel experiment on real-time opposition control of turbulence. Phys. Fluids 18(035103), 1–13 (2006)
Rempfer D., Fasel H.F.: Dynamics of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 275, 257–283 (1994). doi:10.1017/S0022112094002351
Rempfer D., Fasel H.F.: Evolution of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 260, 351–375 (1994). doi:10.1017/S0022112094003551
Roussopoulos K.: Feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 248, 267–296 (1993)
Rowley C.W.: Model reduction for fluids using balanced proper orthogonal decomposition. J. Int. Bifurcation Chaos 15(3), 997–1013 (2005)
Rugh W.J., Shamma J.: Research on gain scheduling. Automatica 36, 1401–1425 (2000)
Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., Little, J.: Reduced-order model-based feedback control of subsonic cavity flows - an experimental approach. In: Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol. 25, pp. 211–230. Springer, Berlin (2007)
Siegel, S., Aradag, S., Seigel, J., Cohen, K., McLaughlin, T.: Low dimensional pod based estimation of a 3d turbulent separated flow. In: 45th AIAA Aerospace Sciences Meeting and Exhibit. Reno, NV, USA, Jan. 8–11, 2007 (2007). AIAA Paper 2007-0112
Siegel, S., Cohen, K., McLaughlin, T.: Feedback control of a circular cylinder wake in experiment and simulation. In: 33rd AIAA Fluids Conference and Exhibit. Orlando, Florida, USA, June 23–26, 2003 (2003). Paper No 2003-3571
Siegel S., Seidel J., Fagley C., Luchtenburg D.M., Cohen K., McLaughlin T.: Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition. J. Fluid Mech. 610(−1), 1–42 (2008). doi:10.1017/S0022112008002115
Slotine J.J.R., Li W.: Applied Nonlinear Control, First edn. Pearson Prentice Hall, Englewood Cliffs (1991)
Smith T.R., Moehlis J., Holmes P.: Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial. Nonlinear Dyn. 41, 275–307 (2005)
Strykowski, P.J., Sreenivasan, K.R.: On the formation and suppression of vortex shedding at low Reynolds numbers. J. Fluid Mech. 218, 71–107 (1990)
Tadmor, G., Lehmann, O., Noack, B.R., Morzyński, M.: Mean field representation of the natural and actuated cylinder wake. Phys. Fluids 22(3), 034102 (2010). doi:10.1063/1.3298960. http://link.aip.org/link/?PHF/22/034102/1
Tadmor, G., Noack, B.R.: Dynamic estimation for reduced Galerkin models of fluid flows. In: The 2004 American Control Conference, pp. 0001–0006. Boston, MA, U.S.A., June 30–July 2, 2004 (2004). Paper WeM18.1
Tadmor, G., Noack, B.R., Lehmann, O., Morzynski, M., Stankiewicz, W., Gonzalez, J.: Shift Modes and Transient Dynamics in Low Order, Design Oriented Galerkin Models. In: 45th AIAA Aerospace Sciences Meeting and Exhibit (8–11 January, 2007). Paper AIAA 2007-111
Thiria B., Goujon-Durand S., Wesfreid J.E.: The wake of a cylinder performing rotary oscillations. J. Fluid Mech. 560, 123–147 (2006)
Unal M.F., Rockwell D.: On the vortex formation from a cylinder; Part2. Control by a splitter-plate interference. J. Fluid Mech. 190, 513–529 (1987)
Wiederhold, O., King, R., Noack, B.R., Neuhaus, L., Neise W., Enghard, L., Swoboda, M.: Extensions of extremum-seeking control to improve the aerodynamic performance of axial turbomachines. In: 39th AIAA Fluid Dynamics Conference, pp. 1–19. Seattle, WA, USA (2008). AIAA-Paper 092407
Wiener N.: Cybernetics: or Control and Communication in the Animal and the Machine, 1st edn. MIT Press, Boston (1948)
Wu J., Wang L., Tadmor J.Z.: Suppression of the von karman vortex street behind a circular cylinder by a traveling wave generated by a flexible surface. J. Fluid. Mech. 574, 365–391 (2007)
Zebib A.: Stability of viscous flow past a circular cylinder. J. Eng. Math. 21, 155–165 (1987)
Zhang H.Q., Fey U., Noack B.R., König M., Eckelmann H.: On the transition of the cylinder wake. Phys. Fluids 7(4), 779–795 (1995)
Zhang T., Ge S.S., Hang C.C.: Adaptive neural network control for strict-feedback nonlinear systems using backstepping control. Automatica 36, 1835–1846 (2000)
Zheng A.: Some practical issues and possible solutions for nonlinear model predictive control. Prog. Syst. Control Theory 26, 129–145 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hussaini.
Rights and permissions
About this article
Cite this article
Aleksić-Roeßner, K., King, R., Lehmann, O. et al. On the need of nonlinear control for efficient model-based wake stabilization. Theor. Comput. Fluid Dyn. 28, 23–49 (2014). https://doi.org/10.1007/s00162-013-0299-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-013-0299-9