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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold


IMPACT FACTOR 2018: 3.107

CiteScore 2018: 2.43

SCImago Journal Rank (SJR) 2018: 1.252
Source Normalized Impact per Paper (SNIP) 2018: 1.618

Mathematical Citation Quotient (MCQ) 2017: 1.68

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1569-3953
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Volume 26, Issue 1

Issues

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Peter Benner
  • Corresponding author
  • Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106, Magdeburg, Germany
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/ Hermann Mena
Published Online: 2016-05-04 | DOI: https://doi.org/10.1515/jnma-2016-1039

Abstract

The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.

Keywords: Riccati differential equation; Rosenbrock method; BDF method; LQR problem; parabolic control problem

MSC 2010: 65L06; 65N12; 49N05; 93C20; 93D15

References

  • [1]

    H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser, Basel, Switzerland, 2003.Google Scholar

  • [2]

    A. C. Antoulas, D. C. Sorensen, and Y. Zhou, On the decay rate of Hankel singular values and related issues, Syst. Contr. Lett. 46 (2000), No. 5, 323–342.Google Scholar

  • [3]

    E. Arias, V. Hernández, J. Ibanes, and J. Peinado, A family of BDF algorithms for solving Differential Matrix Riccati Equations using adaptive techniques, Procedia Computer Science 1 (2010), No. 1, 2569–2577.CrossrefGoogle Scholar

  • [4]

    U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, PA, SIAM, Philadelphia, 1998.Google Scholar

  • [5]

    H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Cont. Optim. 22 (1984), 684–698.CrossrefGoogle Scholar

  • [6]

    P. Benner, Solving large-scale control problems, IEEE Control Systems Magazine 14 (2004), No. 1, 44–59.Google Scholar

  • [7]

    P. Benner and S. Hein, Model predictive control for nonlinear parabolic differential equations based on a linear quadratic Gaussian design, In: Proc. in Applied Mathematics and Mechanics, 9, pp. 613–614, 2009.Google Scholar

  • [8]

    P. Benner and S. Hein, Model Predictive Control Based on an LQG Design for Time-Varying Linearizations, Chemnitz Scientific Computing Preprints, Report No. 09-07, TU Chemnitz (Germany), 2010.Google Scholar

  • [9]

    P. Benner and S. Hein, MPC/LQG for infinite-dimensional systems using time-invariant linearizations, In: System Modeling and Optimization. 25th IFIP TC 7 Conference, Berlin, Germany, September 12-16, 391, pp. 217–224, IFIP AICT, 2011.Google Scholar

  • [10]

    P. Benner, P. Kürschner, and J. Saak, Eflcient handling of complex shift parameters in the low-rank cholesky factor ADI method, Numerical Algorithms 62 (2012), No. 2, 225–251.Google Scholar

  • [11]

    P. Benner, J. R. Li, and T. Penzl, Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems, Numer. Lin. Alg. Appl. 15 (2008), No. 9, 755–777.CrossrefGoogle Scholar

  • [12]

    P. Benner, V. Mehrmann, and D. Sorensen (eds.), Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering 45, Springer-Verlag, Berlin/Heidelberg, Germany, 2005.Google Scholar

  • [13]

    P. Benner and H. Mena, BDF methods for large-scale differential Riccati equations, In: Proc. of Mathematical Theory of Network and Systems, MTNS 2004 (Eds. B. De Moor, B. Motmans, J.Willems, P. Van Dooren, and V. Blondel), 2004.Google Scholar

  • [14]

    P. Benner and H. Mena, Numerical Solution of the Infinite-Dimensional LQR-Problem and the Associated Differential Riccati Equations, Max Planck Institute Magdeburg, Preprint No. MPIMD/12-13, August 2012, Available from http://www.mpimagdeburg.mpg.de/preprints/.

  • [15]

    P. Benner and H. Mena, Rosenbrock methods for solving differential Riccati equations, IEEE Trans. Automat. Control 58 (2013), No. 11, 2950–2957.CrossrefWeb of ScienceGoogle Scholar

  • [16]

    P. Benner and J. Saak, Suboptimality estimates for the semi-discretized LQR probelm for parabolic PDEs, In: Proc. in Applied Mathematics and Mechanics, 10, pp. 591–592, 2010.Google Scholar

  • [17]

    P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM Mitteilungen 36 (2013), No. 1, 32–52.CrossrefGoogle Scholar

  • [18]

    A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Volume I, Systems & Control: Foundations & Applications, Birkäuser, Boston–Basel–Berlin, 1992.Google Scholar

  • [19]

    A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Volume II, Systems & Control: Foundations & Applications, Birkäuser, Boston–Basel–Berlin, 1992.Google Scholar

  • [20]

    J. G. Blom, W. Hundsdorfer, E. J. Spee, and J. G. Verwer, A second order Rosenbrock method applied to photochemical dispersion problems, SIAM J. Sci. Comput. 20 (1999), No. 4, 1456–1480.CrossrefGoogle Scholar

  • [21]

    F. Bornemann and P. Deuflhard, Scientific Computing with Ordinary Differential Equations, Text in Applied Mathematics 42, Springer, New York, 2002.Google Scholar

  • [22]

    C. Choi and A. J. Laub, Eflcient matrix-valued algorithms for solving stiff Riccati differential equations, IEEE Trans. Automat. Control 35 (1990), 770–776.CrossrefGoogle Scholar

  • [23]

    R. F. Curtain and A. J. Pritchard, Infinite-dimensional Riccati equation for systems defined by evolution operators, SIAM J. Cont. Optim. 14 (1976), 951–983.CrossrefGoogle Scholar

  • [24]

    R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear System Theory, Texts in Applied Mathematics, Springer-Verlag, New York, 1995.Google Scholar

  • [25]

    L. Dieci, Numerical integration of the differential Riccati equation and some related issues, SIAM J. Numer. Anal. 29 (1992), No. 3, 781–815.CrossrefGoogle Scholar

  • [26]

    E. Eich, Projizierende Mehrschrittverfahren zur numerischen Lösung von Bewegungsgleichungen technischer Mehrkörpersysteme mit Zwangsbedingungen und Unstetigkeiten, Ph.D. thesis, University of Augsburg, 1991.Google Scholar

  • [27]

    J. S. Gibson, The Riccati integral equation for optimal control problems in Hilbert spaces, SIAM J. Cont. Optim. 17 (1979), No. 4, 537–565.CrossrefGoogle Scholar

  • [28]

    L. Grasedyck, Existence of a low rank or H-matrix approximant to the solution of a Sylvester equation, Numer. Lin. Alg. Appl. 11 (2004), 371–389.CrossrefGoogle Scholar

  • [29]

    E. Hairer and G.Wanner, Solving Ordinary Differential Equations II-Stiff and Differerntial Algebraic Problems, Springer Series in Computational Mathematics, Springer-Verlag, New York, 2000.Google Scholar

  • [30]

    S. Hein, MPC/LQG-Based Optimal Control of Nonlinear Parabolic PDEs, Ph.D. thesis, TU Chemnitz, February 2010, available from http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201000134.

  • [31]

    K. Ito and K. Kunisch, Receding horizon optimal control for infinite dimensional systems, ESAIM Control Optim. Calc. Var. 8 (2002), 741–760.CrossrefGoogle Scholar

  • [32]

    K. Ito and K. Kunisch, Receding horizon control with incomplete observations, SIAM J. Cont. Optim. 45 (2006), No. 1, 207–225.CrossrefGoogle Scholar

  • [33]

    M. Kroller and K. Kunisch, Convergence rates for the feedback operators arising in the linear quadratic regulator problem governed by parabolic equations, SIAM J. Numer. Anal. 28 (1991), No. 5, 1350–1385.CrossrefGoogle Scholar

  • [34]

    N. Lang, H. Mena, and J. Saak, On the benefits of the LDL factorization for large-scale differential matrix equation solvers, Lin. Alg. Appl. 480 (2015), 44–71.CrossrefGoogle Scholar

  • [35]

    I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences 164, Springer, Berlin, 1991.Google Scholar

  • [36]

    I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I. Abstract Parabolic Systems, Cambridge University Press, Cambridge, UK, 2000.Google Scholar

  • [37]

    J. R. Li and J. White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl. 24 (2002), No. 1, 260–280.CrossrefGoogle Scholar

  • [38]

    H. Mena, Numerical Solution of Differential Riccati Equations Arising in Optimal Control Problems for Parabolic Partial Differential Equations, Ph.D. thesis, Escuela Politecnica Nacional, 2007.Google Scholar

  • [39]

    T. Penzl, Numerische Lösung großer Lyapunov-Gleichungen, Ph.D. thesis, Technische Universität Chemnitz, 1998.Google Scholar

  • [40]

    T. Penzl, Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case, Sys. Control Lett. 40 (2000), 139–144.CrossrefGoogle Scholar

  • [41]

    T. Penzl, Lyapack Users Guide, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, Report No. SFB393/00-33, 09107 Chemnitz, Germany, 2000, Available from http://www.tu-chemnitz.de/sffi393/sffi00pr.html.

  • [42]

    I. R. Petersen, V. A. Ugrinovskii, and A. V. Savkin, Robust Control Design Using H Methods, Springer-Verlag, London, UK, 2000.Google Scholar

  • [43]

    A. J. Pritchard and D. Salamon, The linear quadratic control problem for infinite dimensional systems with unbounded input and output operators, SIAM J. Cont. Optim. 25 (1987), 121–144.CrossrefGoogle Scholar

  • [44]

    J. Saak, Eflziente numerische Lösung eines Optimalsteuerungsproblems für die Abkühlung von Stahlprofilen, Diplomarbeit, Fachbereich 3/Mathematik und Informatik, Universität Bremen, D-28334 Bremen, 2003.Google Scholar

  • [45]

    J. Sabino, Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method, Ph.D. thesis, Rice University, Houston, Texas, 2007.Google Scholar

  • [46]

    R. Schneider, FEINS: Finite element solver for shape optimization with adjoint equations, In: Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry, Springer, 2011, Software available at http://www.feins.org/.

  • [47]

    V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput. 29 (2007), No. 3, 1268–1288.CrossrefWeb of ScienceGoogle Scholar

  • [48]

    F. Tröltzsch and A. Unger, Fast solution of optimal control problems in the selective cooling of steel, Z. Angew. Math. Mech. 81 (2001), 447–456.CrossrefGoogle Scholar

  • [49]

    X.Wu, B. Jacob, and H. Elbern, Optimal Actuator and Observation Location for Time-Varying Systems on a Finite-Time Horizon, arXiv:1503.09031, March 2015.Google Scholar

  • [50]

    J. Zabczyk, Remarks on the algebraic Riccati equation, Appl. Math. Optim. 2 (1976), 251–258.Google Scholar

About the article

Received: 2016-04-26

Revised: 2016-04-28

Accepted: 2016-05-08

Published Online: 2016-05-04

Published in Print: 2018-03-26


Citation Information: Journal of Numerical Mathematics, Volume 26, Issue 1, Pages 1–20, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2016-1039.

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