×

Nonlinear control of incompressible fluid flow: Application to Burgers’ equation and 2D channel flow. (English) Zbl 1011.76018

Summary: This paper proposes a methodology for synthesis of nonlinear finite-dimensional feedback controllers for incompressible Newtonian fluid flows described by two-dimensional Navier-Stokes equations. Combination of Galerkin’s method with approximate inertial manifolds is employed for the derivation of low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the flow. These ODE systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow the reference input asymptotically. The method is used to synthesize nonlinear finite-dimensional output feedback controllers for Burgers’ equation and for a two-dimensional channel flow that enhance the convergence rate to the spatially uniform steady-state and the parabolic velocity profile, respectively. The performance of the proposed controllers is successfully tested through simulations, and is shown to be superior to the one of linear controllers.

MSC:

76D55 Flow control and optimization for incompressible viscous fluids
93C20 Control/observation systems governed by partial differential equations
93B52 Feedback control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Armaou, A.; Christofides, P. D., Wave suppression by nonlinear finite-dimensional control, Chem. Engrg. Sci., 55, 2627-2640 (2000)
[2] Baker, J.; Christofides, P. D., Finite dimensional approximation and control of nonlinear parabolic PDE systems, Internat. J. Control, 73, 439-456 (2000) · Zbl 1001.93034
[3] Bakewell, P.; Lumley, J. L., Viscous sublayer and adjacent wall region in turbulent pipe flow, Phys. Fluids, 10, 1880-1889 (1967)
[4] Balas, M. J., Feedback control of linear diffusion processes, Internat. J. Control, 29, 523-533 (1979) · Zbl 0398.93027
[5] Balas, M. J., The Galerkin method and feedback control of linear distributed parameter systems, J. Math. Anal. Appl., 91, 527-546 (1983) · Zbl 0552.93034
[6] Balas, M. J., Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite dimensional controllers, J. Math. Anal. Appl., 114, 17-36 (1986) · Zbl 0595.93032
[7] Bangia, A. K.; Batcho, P. F.; Kevrekidis, I. G.; Karniadakis, G. E., Unsteady 2-D flows in complex geometries: Comparative bifurcation studies with global eigenfunction expansion, SIAM J. Sci. Comput., 18, 775-805 (1997) · Zbl 0873.58058
[8] Beringen, S., Active control of transition by periodic suction and blowing, Phys. Fluids, 27, 1345-1348 (1984)
[9] J. A. Burns and B. B. King, Optimal sensor location for robust control of distributed parameter systems, in; J. A. Burns and B. B. King, Optimal sensor location for robust control of distributed parameter systems, in
[10] J. A. Burns and Y.-R. Ou, Feedback control of the driven cavity problem using LQR designs, in; J. A. Burns and Y.-R. Ou, Feedback control of the driven cavity problem using LQR designs, in
[11] Bushnell, D. M.; McGinley, C. B., Turbulence control in wall flows, Ann. Rev. Fluid Mech., 21, 1-20 (1989) · Zbl 0663.76051
[12] Carlson, H. A.; Lumley, J. L., Active control in the turbulent wall layer of a minimal flow unit, J. Fluid Mech., 329, 341-371 (1996) · Zbl 0893.76033
[13] Choi, H.; Moin, P.; Kim, J., Active turbulence control for drag reduction in wall-bounded flows, J. Fluid Mech., 262, 75-110 (1994) · Zbl 0800.76191
[14] Choi, H.; Temam, R.; Moin, P.; Kim, J., Feedback control for unsteady flow and its application to the stochastic burger’s equation, J. Fluid Mech., 253, 509-543 (1993) · Zbl 0810.76012
[15] P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes, Birkhäuser, Boston, in press.; P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes, Birkhäuser, Boston, in press. · Zbl 1018.93001
[16] Christofides, P. D.; Daoutidis, P., Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds, J. Math. Anal. Appl., 216, 398-420 (1997) · Zbl 0890.93051
[17] Cortelezzi, L.; Lee, K. H.; Kim, J.; Speyer, J. L., Skin-friction drag reduction via reduced-order linear feedback control, Internat. J. Comput. Fluid Dynam., 11, 79-92 (1998) · Zbl 0941.76024
[18] Cortelezzi, L.; Speyer, J. L., Robust reduced-order controller of laminar boundary layer transition, Phys. Rev. E, 58, 1906-1910 (1998)
[19] Crawford, C. W.; Karniadakis, G. E., Reynolds stress analysis of EMHD-controlled wall turbulence. Part I. Streamwise forcing, Phys. Fluids, 9, 788-806 (1997)
[20] 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, 2337-2354 (1991) · Zbl 0746.76021
[21] Desai, M.; Ito, K., Optimal controls of Navier-Stokes equations, SIAM J. Control Optim., 32, 1428-1446 (1994) · Zbl 0813.35078
[22] Foias, C.; Jolly, M. S.; Kevrekidis, I. G.; Sell, G. R.; Titi, E. S., On the computation of inertial manifolds, Phys. Lett. A, 131, 433-437 (1989)
[23] Foias, C.; Sell, G. R.; Titi, E. S., Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dynam. Differential Equations, 1, 199-244 (1989) · Zbl 0692.35053
[24] Foias, C.; Témam, R., Algebraic approximation of attractors: The finite dimensional case, Physica D, 32, 163-182 (1988) · Zbl 0671.58024
[25] Fukunaga, K., Introduction to Statistical Pattern Recognition (1990), Academic Press: Academic Press New York · Zbl 0711.62052
[26] Gad-el-Hak, M., Flow control, Appl. Mech. Rev., 42, 261-293 (1989)
[27] Gad-el-Hak, M., Interactive control of turbulent boundary layers: A futuristic overview, AIAA J., 32, 1753-1765 (1994)
[28] Gad-el-Hak, M.; Bushnell, D. M., Separation control, J. Fluids Engrg., 113, 5-30 (1991)
[29] Guckenheimer, J., Strange attractors in fluids: Another view, Ann. Rev. Fluid Mech., 18, 15-32 (1986) · Zbl 0634.76058
[30] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer-Verlag: Springer-Verlag Berlin/Heidelberg · Zbl 0456.35001
[31] Holmes, P.; Lumley, J. L.; Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry (1996), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0890.76001
[32] Hou, L. S.; Yan, Y., Dynamics for controlled Navier-Stokes systems with distributed controls, SIAM J. Control Optim., 35, 654-677 (1997) · Zbl 0871.49008
[33] Isodori, A., Nonlinear Control Systems: An Introduction (1989), Springer-Verlag: Springer-Verlag Berlin/Heidelberg
[34] K. Ito and S. S. Ravindran, Optimal control of compressible Navier-Stokes equations, in; K. Ito and S. S. Ravindran, Optimal control of compressible Navier-Stokes equations, in
[35] K. Ito and S. S. Ravindran, A reduced order method for control of fluid flows, in; K. Ito and S. S. Ravindran, A reduced order method for control of fluid flows, in
[36] K. Ito and S. S. Ravindran, Reduced order methods for nonlinear infinite dimensional control systems, in; K. Ito and S. S. Ravindran, Reduced order methods for nonlinear infinite dimensional control systems, in
[37] Jones, D. A.; Titi, E. S., A remark on quasi-stationary approximate inertial manifolds for the Navier-Stokes equations, SIAM J. Math. Anal., 25, 894-914 (1994) · Zbl 0808.35102
[38] S. S. Joshi, J. L. Speyer, and J. Kim, Modeling and control of two dimensional Poiseuille flow, in; S. S. Joshi, J. L. Speyer, and J. Kim, Modeling and control of two dimensional Poiseuille flow, in
[39] Joshi, S. S.; Speyer, J. L.; Kim, J., A system theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flows, J. Fluid Mech., 332, 157-184 (1997) · Zbl 0904.76022
[40] S. Kang and K. Ito, A feedback control law for systems arising in fluid dynamics, in; S. Kang and K. Ito, A feedback control law for systems arising in fluid dynamics, in
[41] B. B. King and Y. Qu, Nonlinear dynamic compensator design for flow control in a driven cavity, in; B. B. King and Y. Qu, Nonlinear dynamic compensator design for flow control in a driven cavity, in
[42] Kokotovic, P. V.; Khalil, H. K.; O’Reilly, J., Singular Perturbations in Control: Analysis and Design (1986), Academic Press: Academic Press London
[43] Lumley, J. L., Drag reduction in turbulent flow by polymer additives, J. Polymer Sci. D Macromol. Rev., 7, 263-290 (1973)
[44] Rajaee, M.; Karlsson, S. K.F; Sirovich, L., Low-dimensional description of free shear flow coherent structures and their dynamical behavior, J. Fluid Mech., 258, 1-20 (1994) · Zbl 0800.76190
[45] Shvartsman, S. Y.; Kevrekidis, I. G., Nonlinear model reduction for control of distributed parameter systems: A computer assisted study, AIChE J., 44, 1579-1595 (1998)
[46] Singh, S. N.; Bandyopadhyay, P. R., Linear feedback control of boundary layer using electromagnetic microtiles, Trans. ASME, 119, 852-858 (1997)
[47] Sirovich, L., Turbulence and the dynamics of coherent structures. Part I. Coherent structures, Quart. Appl. Math., 45, 561-571 (1987) · Zbl 0676.76047
[48] Sirovich, L., Turbulence and the dynamics of coherent structures. Part II. Symmetries and transformations, Quart. Appl. Math., 45, 573-582 (1987)
[49] Sirovich, L., Turbulence and the dynamics of coherent structures. Part III. Dynamics and scaling, Quart. Appl. Math., 45, 583-590 (1987) · Zbl 0676.76047
[50] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0662.35001
[51] Titi, E. S., On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. Appl., 149, 540-557 (1990) · Zbl 0723.35063
[52] Virk, P. S., Drag reduction fundamentals, AIChE J., 21, 625-656 (1975)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.