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Proper orthogonal decomposition for optimality systems

Published online by Cambridge University Press:  12 January 2008

Karl Kunisch
Affiliation:
Karl-Franzens-Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstrasse 36, 8010 Graz, Austria. [email protected]; [email protected]
Stefan Volkwein
Affiliation:
Karl-Franzens-Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstrasse 36, 8010 Graz, Austria. [email protected]; [email protected]
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Abstract

Proper orthogonal decomposition (POD) is apowerful technique for model reduction of non-linear systems. Itis based on a Galerkin type discretization with basis elementscreated from the dynamical system itself. In the context ofoptimal control this approach may suffer from the fact that thebasis elements are computed from a reference trajectory containingfeatures which are quite different from those of the optimallycontrolled trajectory. A method is proposed which avoids thisproblem of unmodelled dynamics in the proper orthogonaldecomposition approach to optimal control. It is referred to asoptimality system proper orthogonal decomposition (OS-POD).

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2008

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References

K. Afanasiev and M. Hinze, Adaptive control of a wake flow using proper orthogonal decomposition, in Lecture Notes in Pure and Applied Mathematics 216, Marcel Dekker (2001) 317–332.
E. Arian, M. Fahl and E. Sachs, Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE (2000).
P. Astrid, S. Weiland, K. Willcox and T. Backx, Missing point estimation in models described by proper orthogonal decomposition, in 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas (2004).
Banks, H.T., Joyner, M.L., Winchesky, B. and Winfree, W.P., Nondestructive evaluation using a reduced-order computational methodology. Inverse Problems 16 (2000) 117. CrossRef
G. Berkooz, P. Holmes and J.L. Lumley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, Cambridge University Press (1996).
P. Constantin and C.Foias, Navier-Stokes Equations. Chicago Lectures in Mathematics, University of Chicago Press, Chicago (1989).
Grepl, M.A. and Patera, A.T., A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157181. CrossRef
Gugercin, S. and Antoulas, A.C., A survey of model reduction by balanced truncation and some new results. Int. J. Control 77 (2004) 748766. CrossRef
T. Henri, Réduction de modéles par des méthodes de décomposition orthogonal propre. Ph.D. thesis, Université de Rennes, France (2004).
Homescu, C., Petzold, L.R. and Serban, R., Error estimation for reduced order models of dynamical systems. SIAM J. Numer. Anal. 43 (2005) 16931714. CrossRef
Ito, K. and Ravindran, S.S., Reduced basis method for unsteady viscous flows. Int. J. Comp. Fluid Dynam. 15 (2001) 97113. CrossRef
Kunisch, K. and Volkwein, S., Control of Burgers' equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theor. Appl. 102 (1999) 345371. CrossRef
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 92515. CrossRef
Kunisch, K., Volkwein, S. and Xie, L., HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dynam. Syst. 4 (2004) 701722. CrossRef
Lall, S., Marsden, J.E. and Glavaški, S., A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12 (2002) 519535. CrossRef
Ly, H.V. and Tran, H.T., Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quarterly Appl. Math. 60 (2002) 631656. CrossRef
Maurer, H. and Zowe, J., First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Programming 16 (1979) 98110. CrossRef
B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automatic Control AC-26 (1981) 17–31.
Ravindran, S.S., Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput. 23 (2002) 19241942. CrossRef
Rovas, D.V., Machiels, L. and Maday, Y., Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423445. CrossRef
Rowley, C.W., Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifurcation Chaos 15 (2005) 9971013. CrossRef
L. Sirovich, Turbulence and the dynamics of coherent structures, parts I-III. Quarterly Appl. Math. XLV (1987) 561–590.
K.Y. Tan, W.R. Graham and J. Peraire, Active flow control using a reduced order model and optimum control. AIAA (1996).
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer Verlag, Berlin (1988).
Volkwein, S., Second-order conditions for boundary control problems of the Burgers equation. Control Cybern. 30 (2001) 249278.
S. Volkwein, Boundary control of the Burgers equation: optimality conditions and reduced-order approach, in Optimal Control of Complex Structures, K.-H. Hoffmann, I. Lasiecka, G. Leugering, J. Sprekels and F. Tröltzsch Eds., International Series of Numerical Mathematics 139 (2001) 267–278.
Volkwein, S., Lagrange-SQP techniques for the control constrained optimal boundary control problems for the Burgers equation. Comput. Optim. Appl. 26 (2003) 253284. CrossRef
K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, in 15th AIAA Computational Fluid Dynamics Conference, Anaheim, USA (June 2001).