# American Institute of Mathematical Sciences

2012, 2(3): 465-485. doi: 10.3934/naco.2012.2.465

## Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems

 1 Chair for Nonlinear Dynamics, Steinbachstr. 15, 52074 Aachen, Germany 2 Institut für Mathematik und Statistik, Universität Konstanz, D-78457 Konstanz, Germany

Received  November 2011 Revised  January 2012 Published  August 2012

The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal model reduction (H2), balanced truncation (BT), and proper orthogonal decomposition (POD) are studied. The proposed approach is based on semi-discretization of the underlying dynamics for the state and the adjoint equations as a large scale linear time-invariant (LTI) system. This system is reduced to a lower-dimensional one using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The size of the reduced-order system is iteratively increased until the error in the optimal control, computed with the a-posteriori error estimator, satisfies a given accuracy. The method is illustrated with numerical tests.
Citation: Georg Vossen, Stefan Volkwein. Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 465-485. doi: 10.3934/naco.2012.2.465
##### References:
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Kärcher, Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 873-877.  Google Scholar [11] S. Gugercin, A. C. Antoulas and C. A. Beattie, $H_2$ model reduction for large-scale linear dynamical systems, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 609-638. doi: 10.1137/060666123.  Google Scholar [12] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition, Comput. Optim. and Appl., 39 (2008), 319-345. doi: 10.1007/s10589-007-9058-4.  Google Scholar [13] P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry," Cambridge Univ. Press, New York, 1996. doi: 10.1017/CBO9780511622700.  Google Scholar [14] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semi-smooth Newton method, SIAM J. Optimization, 13 (2003), 865-888.  Google Scholar [15] C. Joerres, G. Vossen and M. Herty, On an inexact gradient method using POD for a parabolic optimal control problem, submitted, 2011. Google Scholar [16] E. A. Jonckheere and L. M. Silverman, A new set of invariants for linear systems - Application to reduced order compensator design, IEEE Trans. Automat. Control, 28 (1983), 953-964. doi: 10.1109/TAC.1983.1103159.  Google Scholar [17] E. Kammann, F. Tröltzsch and S. Volkwein, A method of a-posteriori error estimation with application to proper orthogonal decomposition, submitted, 2011. Google Scholar [18] D. Kubalinska, "Optimal Interpolation-Based Model Reduction," PhD thesis, University of Bremen, 2008. Google Scholar [19] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148. doi: 10.1007/s002110100282.  Google Scholar [20] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), 1-23. doi: 10.1051/m2an:2007054.  Google Scholar [21] E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction, Statistical Forecasting Scientific Rep. 1, Department of Meteorology, Massachusetts Institute of Technology, Cambridge, MA, 1956. Google Scholar [22] L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera and D. V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, CR Acad Sci Paris Series I, 331 (2000), 1531-1548. Google Scholar [23] K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems, in "Mathematical Programming with Data Perturbation" (eds. A. V. Fiacco and Marcel Dekker), Inc., New York, (1997), 253-284. Google Scholar [24] H. Maurer and J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Mathematical Programming, 16 (1979), 98-110.  Google Scholar [25] L. Meier and D. Luenberger, Approximation of linear constant systems, IEEE Transactions on Automatic Control, 12 (1967), 585-588. doi: 10.1109/TAC.1967.1098680.  Google Scholar [26] B. C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Automatic Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.  Google Scholar [27] A. T. Patera and G. Rozza, "Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations," MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2006. Google Scholar [28] S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, SIAM J. Sci. Comput., 15 (2000), 457-478.  Google Scholar [29] J. C. De Los Reyes and T. Stykel, A balanced truncation based strategy for optimal control of evolution problems, Optim. Methods Software, 26 (2011), 673-694. doi: 10.1080/10556788.2010.526756.  Google Scholar [30] J. Saak, "Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction," PhD thesis, TU Chemnitz, 2009. Google Scholar [31] E. W. Sachs and M. Schu, A priori error estimates for reduced order models in finance, submitted, 2011. Google Scholar [32] T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems, 16 (2004), 297-319. doi: 10.1007/s00498-004-0141-4.  Google Scholar [33] T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and POD a-posteriori error estimators for an elliptic linear quadratic optimal control problem, Mathematical and Computer Modelling of Dynamical Systems, Special Issue: Model order reduction of parameterized problems, 17 (2011), 355-369. Google Scholar [34] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems, Computational Optimization and Applications, 44 (2009), 83-115. doi: 10.1007/s10589-008-9224-3.  Google Scholar [35] F. Tröltzsch., "Optimal Control of Partial Differential Equations. Theory, Methods and Applications," American Math. Society, Providence, 112, 2010. Google Scholar [36] R. Usmani, Inversion of a tridiagonal Jacobi matrix, Linear Algebra Appl. , 212/213 (1994), 413-414. doi: 10.1016/0024-3795(94)90414-6.  Google Scholar [37] S. Volkwein, Model reduction using proper orthogonal decomposition, Lecture Notes, Institute of Mathematics and Statistics, University of Constance, 2011. Google Scholar [38] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, to appear in Control and Cybernetics, 2012. Google Scholar [39] G. Vossen, $\mathcal H_{2,\alpha}$-norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations, submitted, 2011. Google Scholar

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##### References:
 [1] K. Afanasiev and M. Hinze, Adaptive control of a wake flow using proper orthogonal decomposition, Lect. Notes Pure Appl. Math., 216 (2001), 317-332.  Google Scholar [2] A. C. Antoulas, "Approximation of Large-Scale Dynamical Systems," SIAM, Philadelphia, (2005). doi: 10.1137/1.9780898718713.  Google Scholar [3] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, 23 (2002), 201-219. doi: 10.1023/A:1020576801966.  Google Scholar [4] P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems, SIAM Journal on Control and Optimization , 49 (2011), 686-711. doi: 10.1137/09075041X.  Google Scholar [5] P. Benner and J. Saak, A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations,, 2010. Available from: , ().   Google Scholar [6] P. Benner and E. S. Quintana-Ortí, Model reduction based on spectral projection methods, In "Reduction of Large-Scale Systems" (eds. P. Benner, V. Mehrmann and D. C. Sorensen), Lecture Notes in Computational Science and Engineering, 45 (2005), 5-48. Google Scholar [7] A. Bunse-Gerstner, D. Kubalinska, G. Vossen and D. Wilczek, $h_2$-norm optimal model reduction for large-scale discrete dynamical MIMO systems, Journal of Computational and Applied Mathematics, 233 (2010), 1202-1216. doi: 10.1016/j.cam.2008.12.029.  Google Scholar [8] A. L. Dontchev, W. W. Hager, A. B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control, Appl. Math. and Optim., 31 (1995), 297-326. doi: 10.1007/BF01215994.  Google Scholar [9] K. Glover, All optimal Hankel-norm approximations of linear multi-variable systems and their $L_\infty$ error bounds, International Journal of Control, 39 (1984), 1115-1193. doi: 10.1080/00207178408933239.  Google Scholar [10] M. A. Grepl and M. Kärcher, Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 873-877.  Google Scholar [11] S. Gugercin, A. C. Antoulas and C. A. Beattie, $H_2$ model reduction for large-scale linear dynamical systems, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 609-638. doi: 10.1137/060666123.  Google Scholar [12] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition, Comput. Optim. and Appl., 39 (2008), 319-345. doi: 10.1007/s10589-007-9058-4.  Google Scholar [13] P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry," Cambridge Univ. Press, New York, 1996. doi: 10.1017/CBO9780511622700.  Google Scholar [14] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semi-smooth Newton method, SIAM J. Optimization, 13 (2003), 865-888.  Google Scholar [15] C. Joerres, G. Vossen and M. Herty, On an inexact gradient method using POD for a parabolic optimal control problem, submitted, 2011. Google Scholar [16] E. A. Jonckheere and L. M. Silverman, A new set of invariants for linear systems - Application to reduced order compensator design, IEEE Trans. Automat. Control, 28 (1983), 953-964. doi: 10.1109/TAC.1983.1103159.  Google Scholar [17] E. Kammann, F. Tröltzsch and S. Volkwein, A method of a-posteriori error estimation with application to proper orthogonal decomposition, submitted, 2011. Google Scholar [18] D. Kubalinska, "Optimal Interpolation-Based Model Reduction," PhD thesis, University of Bremen, 2008. Google Scholar [19] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148. doi: 10.1007/s002110100282.  Google Scholar [20] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), 1-23. doi: 10.1051/m2an:2007054.  Google Scholar [21] E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction, Statistical Forecasting Scientific Rep. 1, Department of Meteorology, Massachusetts Institute of Technology, Cambridge, MA, 1956. Google Scholar [22] L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera and D. V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, CR Acad Sci Paris Series I, 331 (2000), 1531-1548. Google Scholar [23] K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems, in "Mathematical Programming with Data Perturbation" (eds. A. V. Fiacco and Marcel Dekker), Inc., New York, (1997), 253-284. Google Scholar [24] H. Maurer and J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Mathematical Programming, 16 (1979), 98-110.  Google Scholar [25] L. Meier and D. Luenberger, Approximation of linear constant systems, IEEE Transactions on Automatic Control, 12 (1967), 585-588. doi: 10.1109/TAC.1967.1098680.  Google Scholar [26] B. C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Automatic Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.  Google Scholar [27] A. T. Patera and G. Rozza, "Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations," MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2006. Google Scholar [28] S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, SIAM J. Sci. Comput., 15 (2000), 457-478.  Google Scholar [29] J. C. De Los Reyes and T. Stykel, A balanced truncation based strategy for optimal control of evolution problems, Optim. Methods Software, 26 (2011), 673-694. doi: 10.1080/10556788.2010.526756.  Google Scholar [30] J. Saak, "Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction," PhD thesis, TU Chemnitz, 2009. Google Scholar [31] E. W. Sachs and M. Schu, A priori error estimates for reduced order models in finance, submitted, 2011. Google Scholar [32] T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems, 16 (2004), 297-319. doi: 10.1007/s00498-004-0141-4.  Google Scholar [33] T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and POD a-posteriori error estimators for an elliptic linear quadratic optimal control problem, Mathematical and Computer Modelling of Dynamical Systems, Special Issue: Model order reduction of parameterized problems, 17 (2011), 355-369. Google Scholar [34] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems, Computational Optimization and Applications, 44 (2009), 83-115. doi: 10.1007/s10589-008-9224-3.  Google Scholar [35] F. Tröltzsch., "Optimal Control of Partial Differential Equations. Theory, Methods and Applications," American Math. Society, Providence, 112, 2010. Google Scholar [36] R. Usmani, Inversion of a tridiagonal Jacobi matrix, Linear Algebra Appl. , 212/213 (1994), 413-414. doi: 10.1016/0024-3795(94)90414-6.  Google Scholar [37] S. Volkwein, Model reduction using proper orthogonal decomposition, Lecture Notes, Institute of Mathematics and Statistics, University of Constance, 2011. Google Scholar [38] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, to appear in Control and Cybernetics, 2012. Google Scholar [39] G. Vossen, $\mathcal H_{2,\alpha}$-norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations, submitted, 2011. Google Scholar

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