Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled dynamics in the proper orthogonal decomposition approach to optimal control. It is referred to as optimality system proper orthogonal decomposition (OS-POD).
@article{M2AN_2008__42_1_1_0,
author = {Kunisch, Karl and Volkwein, Stefan},
title = {Proper orthogonal decomposition for optimality systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {42},
year = {2008},
pages = {1-23},
doi = {10.1051/m2an:2007054},
mrnumber = {2387420},
zbl = {1141.65050},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2008__42_1_1_0}
}
Kunisch, Karl; Volkwein, Stefan. Proper orthogonal decomposition for optimality systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 1-23. doi : 10.1051/m2an:2007054. http://gdmltest.u-ga.fr/item/M2AN_2008__42_1_1_0/
[1] and , Adaptive control of a wake flow using proper orthogonal decomposition, in Lecture Notes in Pure and Applied Mathematics 216, Marcel Dekker (2001) 317-332. | MR 1816851 | Zbl 1013.76028
[2] , and , Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE (2000).
[3] , , and , Missing point estimation in models described by proper orthogonal decomposition, in 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas (2004).
[4] , , and , Nondestructive evaluation using a reduced-order computational methodology. Inverse Problems 16 (2000) 1-17. | MR 1776475 | Zbl 0960.35107
[5] , and , Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, Cambridge University Press (1996). | MR 1422658 | Zbl 0890.76001
[6] and , Navier-Stokes Equations. Chicago Lectures in Mathematics, University of Chicago Press, Chicago (1989). | MR 972259 | Zbl 0687.35071
[7] and , A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157-181. | Numdam | MR 2136204 | Zbl 1079.65096
[8] and , A survey of model reduction by balanced truncation and some new results. Int. J. Control 77 (2004) 748-766. | MR 2072207 | Zbl 1061.93022
[9] , Réduction de modéles par des méthodes de décomposition orthogonal propre. Ph.D. thesis, Université de Rennes, France (2004).
[10] , and , Error estimation for reduced order models of dynamical systems. SIAM J. Numer. Anal. 43 (2005) 1693-1714. | MR 2182145 | Zbl 1096.65076
[11] and , Reduced basis method for unsteady viscous flows. Int. J. Comp. Fluid Dynam. 15 (2001) 97-113. | MR 1895508 | Zbl 1036.76011
[12] and , Control of Burgers' equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theor. Appl. 102 (1999) 345-371. | MR 1706822 | Zbl 0949.93039
[13] and , Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 92-515. | MR 1921667 | Zbl 1075.65118
[14] , and , HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dynam. Syst. 4 (2004) 701-722. | MR 2111244 | Zbl 1058.35061
[15] , and , A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12 (2002) 519-535. | MR 1895968 | Zbl 1006.93010
[16] and , Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quarterly Appl. Math. 60 (2002) 631-656. | MR 1939004 | Zbl 1146.76631
[17] and , First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Programming 16 (1979) 98-110. | MR 517762 | Zbl 0398.90109
[18] , Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automatic Control AC-26 (1981) 17-31. | MR 609248 | Zbl 0464.93022
[19] , Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput. 23 (2002) 1924-1942. | MR 1923719 | Zbl 1026.76015
[20] , and , Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423-445. | MR 2241309 | Zbl 1101.65099
[21] , Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifurcation Chaos 15 (2005) 997-1013. | MR 2136757 | Zbl 1140.76443
[22] , Turbulence and the dynamics of coherent structures, parts I-III. Quarterly Appl. Math. XLV (1987) 561-590. | MR 910462 | Zbl 0676.76047
[23] , and , Active flow control using a reduced order model and optimum control. AIAA (1996).
[24] , Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer Verlag, Berlin (1988). | MR 953967 | Zbl 0662.35001
[25] , Second-order conditions for boundary control problems of the Burgers equation. Control Cybern. 30 (2001) 249-278. | Zbl 1001.93033
[26] , 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. | MR 1901647 | Zbl 1024.49024
[27] , Lagrange-SQP techniques for the control constrained optimal boundary control problems for the Burgers equation. Comput. Optim. Appl. 26 (2003) 253-284. | MR 2013365 | Zbl 1077.49024
[28] and , Balanced model reduction via the proper orthogonal decomposition, in 15th AIAA Computational Fluid Dynamics Conference, Anaheim, USA (June 2001).