# 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:
 [1] K. Afanasiev and M. Hinze, Adaptive control of a wake flow using proper orthogonal decomposition,, Lect. Notes Pure Appl. Math., 216 (2001), 317. [2] A. C. Antoulas, "Approximation of Large-Scale Dynamical Systems,", SIAM, (2005). doi: 10.1137/1.9780898718713. [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. doi: 10.1023/A:1020576801966. [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. doi: 10.1137/09075041X. [5] P. Benner and J. Saak, A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations,, 2010. Available from: , (). [6] P. Benner and E. S. Quintana-Ortí, Model reduction based on spectral projection methods,, In, 45 (2005), 5. [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. doi: 10.1016/j.cam.2008.12.029. [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. doi: 10.1007/BF01215994. [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. doi: 10.1080/00207178408933239. [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, 349 (2011), 873. [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. doi: 10.1137/060666123. [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. doi: 10.1007/s10589-007-9058-4. [13] P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry,", Cambridge Univ. Press, (1996). doi: 10.1017/CBO9780511622700. [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. [15] C. Joerres, G. Vossen and M. Herty, On an inexact gradient method using POD for a parabolic optimal control problem,, submitted, (2011). [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. doi: 10.1109/TAC.1983.1103159. [17] E. Kammann, F. Tröltzsch and S. Volkwein, A method of a-posteriori error estimation with application to proper orthogonal decomposition,, submitted, (2011). [18] D. Kubalinska, "Optimal Interpolation-Based Model Reduction,", PhD thesis, (2008). [19] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems,, Numerische Mathematik, 90 (2001), 117. doi: 10.1007/s002110100282. [20] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems,, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), 1. doi: 10.1051/m2an:2007054. [21] E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction,, Statistical Forecasting Scientific Rep. 1, (1956). [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. [23] K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems,, in, (1997), 253. [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. [25] L. Meier and D. Luenberger, Approximation of linear constant systems,, IEEE Transactions on Automatic Control, 12 (1967), 585. doi: 10.1109/TAC.1967.1098680. [26] B. C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction,, IEEE Trans. Automatic Control, 26 (1981), 17. doi: 10.1109/TAC.1981.1102568. [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). [28] S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD,, SIAM J. Sci. Comput., 15 (2000), 457. [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. doi: 10.1080/10556788.2010.526756. [30] J. Saak, "Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction,", PhD thesis, (2009). [31] E. W. Sachs and M. Schu, A priori error estimates for reduced order models in finance,, submitted, (2011). [32] T. Stykel, Gramian-based model reduction for descriptor systems,, Math. Control Signals Systems, 16 (2004), 297. doi: 10.1007/s00498-004-0141-4. [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, 17 (2011), 355. [34] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems,, Computational Optimization and Applications, 44 (2009), 83. doi: 10.1007/s10589-008-9224-3. [35] F. Tröltzsch., "Optimal Control of Partial Differential Equations. Theory, Methods and Applications,", American Math. Society, 112 (2010). [36] R. Usmani, Inversion of a tridiagonal Jacobi matrix,, Linear Algebra Appl., 212/213 (1994), 413. doi: 10.1016/0024-3795(94)90414-6. [37] S. Volkwein, Model reduction using proper orthogonal decomposition,, Lecture Notes, (2011). [38] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems,, to appear in Control and Cybernetics, (2012). [39] G. Vossen, $\mathcal H_{2,\alpha}$-norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations,, submitted, (2011).

show all references

##### 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. [2] A. C. Antoulas, "Approximation of Large-Scale Dynamical Systems,", SIAM, (2005). doi: 10.1137/1.9780898718713. [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. doi: 10.1023/A:1020576801966. [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. doi: 10.1137/09075041X. [5] P. Benner and J. Saak, A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations,, 2010. Available from: , (). [6] P. Benner and E. S. Quintana-Ortí, Model reduction based on spectral projection methods,, In, 45 (2005), 5. [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. doi: 10.1016/j.cam.2008.12.029. [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. doi: 10.1007/BF01215994. [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. doi: 10.1080/00207178408933239. [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, 349 (2011), 873. [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. doi: 10.1137/060666123. [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. doi: 10.1007/s10589-007-9058-4. [13] P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry,", Cambridge Univ. Press, (1996). doi: 10.1017/CBO9780511622700. [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. [15] C. Joerres, G. Vossen and M. Herty, On an inexact gradient method using POD for a parabolic optimal control problem,, submitted, (2011). [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. doi: 10.1109/TAC.1983.1103159. [17] E. Kammann, F. Tröltzsch and S. Volkwein, A method of a-posteriori error estimation with application to proper orthogonal decomposition,, submitted, (2011). [18] D. Kubalinska, "Optimal Interpolation-Based Model Reduction,", PhD thesis, (2008). [19] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems,, Numerische Mathematik, 90 (2001), 117. doi: 10.1007/s002110100282. [20] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems,, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), 1. doi: 10.1051/m2an:2007054. [21] E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction,, Statistical Forecasting Scientific Rep. 1, (1956). [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. [23] K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems,, in, (1997), 253. [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. [25] L. Meier and D. Luenberger, Approximation of linear constant systems,, IEEE Transactions on Automatic Control, 12 (1967), 585. doi: 10.1109/TAC.1967.1098680. [26] B. C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction,, IEEE Trans. Automatic Control, 26 (1981), 17. doi: 10.1109/TAC.1981.1102568. [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). [28] S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD,, SIAM J. Sci. Comput., 15 (2000), 457. [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. doi: 10.1080/10556788.2010.526756. [30] J. Saak, "Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction,", PhD thesis, (2009). [31] E. W. Sachs and M. Schu, A priori error estimates for reduced order models in finance,, submitted, (2011). [32] T. Stykel, Gramian-based model reduction for descriptor systems,, Math. Control Signals Systems, 16 (2004), 297. doi: 10.1007/s00498-004-0141-4. [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, 17 (2011), 355. [34] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems,, Computational Optimization and Applications, 44 (2009), 83. doi: 10.1007/s10589-008-9224-3. [35] F. Tröltzsch., "Optimal Control of Partial Differential Equations. Theory, Methods and Applications,", American Math. Society, 112 (2010). [36] R. Usmani, Inversion of a tridiagonal Jacobi matrix,, Linear Algebra Appl., 212/213 (1994), 413. doi: 10.1016/0024-3795(94)90414-6. [37] S. Volkwein, Model reduction using proper orthogonal decomposition,, Lecture Notes, (2011). [38] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems,, to appear in Control and Cybernetics, (2012). [39] G. Vossen, $\mathcal H_{2,\alpha}$-norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations,, submitted, (2011).

Impact Factor: