### “Stabilizing a CFD model of an unstable system through model reduction”

**Authors:**Svein Hovland and Jan T. Gravdahl,

**Affiliation:**NTNU, Department of Engineering Cybernetics

**Reference:**2006, Vol 27, No 3, pp. 171-180.

**Keywords:**Computational fluid dynamics, model-order reduction, reduced-order control

**Abstract:**We demonstate stabilization of a computational fluid dynamics model of an unstable system. The unstable heating of a two-dimensional plate is used as a case study. Active control is introduced by cooling parts of the boundaries of the plate. The high order of the original model is reduced by proper orthogonal decomposition, giving an unstable reduced order model with a state space structure convenient for controller design. A stabilizing controller based on pole placement is designed for the reduced order model and integral action is included to enhance performance. The controller is then applied to the full model, where it is shown through simulations to stabilize the system. The demonstrated procedure makes it possible to analyze stability properties and design control systems for a class of systems that would otherwise be very computationally demanding.

PDF (519 Kb) DOI: 10.4173/mic.2006.3.3

**References:**

[1] AAMO, OLE MORTEN KRSTIC, MIROSLAV (2002). Flow Control by Feedback: Stabilization and Mixing, Springer.

[2] ANDERSON, J. D. (1995). Computational Fluid Dynamics, McGraw-Hill International Editions.

[3] ANTOULAS, A. C., SORENSEN, D.C. GUGERCIN, S. (2001). A survey of model reduction methods for large-scale systems, V. Olshevsky.Ed., Contemporary Mathematics, 280, pp. 193 - 219.

[4] ASTRID, P., HUISMAN, L., WEILAND, S. BACKX, A. C. P. M. (2002). Reduction and predictive control design for a computational fluid dynamics model, In Proc. 41st IEEE Conf. on Decision and Control, volume 3, pp. 3378 - 3383, Las Vegas, NV.

[5] ASTRID, PATRICIA (2004). Reduction of process simulation models, PhD thesis, Eindhoven University of Technology.

[6] BERKOOZ, G., Holmes, P. LUMLEY, J. (1993). The proper orthogonal decomposition in the analysis of turbulent flows, Annual Review of Fluid Mechanics, 25, pp. 539 - 575 doi:10.1146/annurev.fl.25.010193.002543

[7] BEWLEY, THOMAS R. (2001). Flow control: new challenges for a new Renaissance, Progress in Aerospace Sciences, 37(1), pp. 21 - 58 doi:10.1016/S0376-0421(00)00016-6

[8] CANUTO, C., HUSSAINI, M. Y., QUERTERONI, A. ZANG, T. A. (1988). Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics.

[9] CONSTANTIN, P., FOIAS, C., NICOLAENKO, B. TEMAM, R. (1989). Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer Verlag.

[10] FERZIGER, J. H. PERIC, MILOVAN (2002). Computational Methods for Fluid Dynamics, Springer Verlag, 3rd edition.

[11] KARHUNEN, K. (1946). Zur spektral theorie stochasticher prozesse, Ann Acad Sci Fennicae, Ser A1 Math Phys, 34, pp. 1 - 7.

[12] KIRBY, MICHAEL (2001). Geometric Data Analysis, Wiley.

[13] KUNISCH, K. VOLKWEIN, S. (1999). Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition, Journal of Optimization Theory and Applications, 10.2, pp. 345 - 371, August doi:10.1023/A:1021732508059

[14] LOÈVE, M. (1946). Functiona aleatoire de second ordre, Revue Science, 84, pp. 195 - 206.

[15] LUMLEY, J. L. (1967). The structure of inhomogeneous turbulent flow, In Atmospheric Turbulence and Wave Propagation.

[16] PRAJNA, STEPHEN (2003). POD Model Reduction with Stability Guarantee, In Proc. 42nd IEEE Conf. on Decision and Control, Hawaii.

[17] RAVINDRAN, S. S. (2000). A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, International Journal for Numerical Methods in Fluids, 34, pp. 425 - 448.

[18] RAVINDRAN, S. S. (2002). Control of flow separation over a forward-facing step by Model reduction, Computer Methods in Applied Mechanics and Engineering, 191, pp. 4599 - 4617 doi:10.1016/S0045-7825(02)00395-X

[19] SIROVICH, L. (1987). Turbulence and the dynamics of coherent structures, Part 1: Coherent structures. Quarterly of Applied Mathematics, 4.3, pp. 561 - 571.

[20] VERSTEEG, H. K. MALALASEKERA, W. (1995). An Introduction to Computational Fluid Dynamics, Longman.

[21] VOLKWEIN, S. (1999). Proper orthogonal decomposition and singular value decomposition, Technical Report 103, SFB-Preprint.

**BibTeX:**

@article{MIC-2006-3-3,

title={{Stabilizing a CFD model of an unstable system through model reduction}},

author={Hovland, Svein and Gravdahl, Jan T.},

journal={Modeling, Identification and Control},

volume={27},

number={3},

pages={171--180},

year={2006},

doi={10.4173/mic.2006.3.3},

publisher={Norwegian Society of Automatic Control}

};