On the numerical solution of large-scale sparse discrete-time Riccati equations

Article

Abstract

We discuss the numerical solution of large-scale discrete-time algebraic Riccati equations (DAREs) as they arise, e.g., in fully discretized linear-quadratic optimal control problems for parabolic partial differential equations (PDEs). We employ variants of Newton’s method that allow to compute an approximate low-rank factor of the solution of the DARE. The principal computation in the Newton iteration is the numerical solution of a Stein (aka discrete Lyapunov) equation in each step. For this purpose, we present a low-rank Smith method as well as a low-rank alternating-direction-implicit (ADI) iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.

Keywords

Riccati equation  Discrete-time control Large Sparse Algebraic Riccati equation Newton’s method Smith iteration ADI iteration Low-rank factor 

Mathematics Subject Classifications (2010)

65F30 15A24 49M15 93C20 93C05 

References

  1. 1.
    Abels, J., Benner, P.: CAREX—a collection of benchmark examples for continuous-time algebraic Riccati equations (version 2.0). SLICOT Working Note 1999-14. Available from www.slicot.org (1999)
  2. 2.
    Armstrong, E., Rublein, G.T.: A stabilization algorithm for linear discrete constant systems. IEEE Trans. Automat. Contr. AC–21, 629–631 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arnold, W. III, Laub, A.: Generalized eigenproblem algorithms and software for algebraic Riccati equations. Proc. IEEE 72, 1746–1754 (1984)CrossRefGoogle Scholar
  4. 4.
    Banks, H., Kunisch, K.: The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22, 684–698 (1984)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Barraud, A.Y.: A numerical algorithm to solve A T X A − X = Q. IEEE Trans. Automat. Contr. AC–22, 883–885 (1977)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bartels, R., Stewart, G.: Solution of the matrix equation AX + XB = C: Algorithm 432. Commun. ACM 15, 820–826 (1972)CrossRefGoogle Scholar
  7. 7.
    Benner, P.: Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems. Logos–Verlag, Berlin, Germany (1997) (also: Dissertation, Fakultät für Mathematik, TU Chemnitz–Zwickau (1997))Google Scholar
  8. 8.
    Benner, P.: Solving large-scale control problems. IEEE Control Syst. Mag. 14(1), 44–59 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Benner, P.: Editorial: large-scale matrix equations of special type (special issue). Numer. Linear Algebra Appl. 15(9), 747–754 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Benner, P., Faßbender, H.: Initializing Newton’s method for discrete-time algebraic Riccati equations using the butterfly SZ algorithm. In: Gonzalez, O. (ed.) Proc. 1999 IEEE Intl. Symp. CACSD, Kohala Coast-Island of Hawai’i, Hawai’i, USA, 22–27 August 1999 (CD-Rom), pp. 70–74 (1999)Google Scholar
  11. 11.
    Benner, P., Faßbender, H.: A hybrid method for the numerical solution of discrete-time algebraic Riccati equations. Contemp. Math. 280, 255–269 (2001)Google Scholar
  12. 12.
    Benner, P., Faßbender, H., Watkins, D.: SR and SZ algorithms for the symplectic (butterfly) eigenproblem. Linear Algebra Appl. 287, 41–76 (1999)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Benner, P., Li, J.R., Penzl, T.: Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems. Numer. Linear Algebra Appl. 15(9), 755–777 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Benner, P., Quintana-Ortí, E., Quintana-Ortí, G.: Numerical solution of discrete stable linear matrix equations on multicomputers. Parallel Algorithms Appl. 17(1), 127–146 (2002)MathSciNetMATHGoogle Scholar
  15. 15.
    Benner, P., Quintana-Ortí, E., Quintana-Ortí, G.: Solving linear-quadratic optimal control problems on parallel computers. Optim. Methods Softw. 23(6), 879–909 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Byers, R., Mackey, D., Mehrmann, V., Xu, H.: Symplectic, BVD, and palindromic approaches to discrete-time control problems. In: Collection of Papers Dedicated to the 60-th Anniversary of Mihail Konstantinov, pp. 81–102. Publ. House RODINA, Sofia (2009)Google Scholar
  17. 17.
    Calvetti, D., Levenberg, N., Reichel, L.: Iterative methods for X − AXB = C. J. Comput. Appl. Math. 86(1), 73–101 (1997)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Calvetti, D., Reichel, L.: Application of ADI itertaive methods to the restoration of noisy images. SIAM J. Matrix Anal. Appl. 17, 165–186 (1996)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Chahlaoui, Y., Van Dooren, P.: A collection of benchmark examples for model reduction of linear time invariant dynamical systems. SLICOT Working Note 2002–2. Available from www.slicot.org (2002)
  20. 20.
    Chan, T.: Rank revealing QR factorizations. Linear Algebra Appl. 88/89, 67–82 (1987)CrossRefGoogle Scholar
  21. 21.
    Datta, B.: Numerical Methods for Linear Control Systems. Elsevier Academic Press (2004)Google Scholar
  22. 22.
    Dennis, J., Schnabel, R.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, Englewood Cliffs, New Jersey (1983)MATHGoogle Scholar
  23. 23.
    Faßbender, H.: Symplectic Methods for the Symplectic Eigenproblem. Kluwer Academic/Plenum Publisher (2000)Google Scholar
  24. 24.
    Feitzinger, F., Hylla, T., Sachs, E.: Inexact Kleinman-Newton method for Riccati equations. SIAM J. Matrix Anal. Appl. 31, 272–288 (2009)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Gallivan, K., Rao, X., van Dooren, P.: Singular Riccati equations stabilizing large-scale systems. Linear Algebra Appl. 415, 359–372 (2006)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gardiner, J., Laub, A.: Parallel algorithms for algebraic Riccati equations. Int. J. Control 54(6), 1317–1333 (1991)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Gardiner, J., Laub, A., Amato, J., Moler, C.: Solution of the Sylvester matrix equation AXB + CXD = E. ACM Trans. Math. Softw. 18, 223–231 (1992)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Gardiner, J., Wette, M., Laub, A., Amato, J., Moler, C.: Algorithm 705: a Fortran-77 software package for solving the Sylvester matrix equation AXB T + CXD T = E. ACM Trans. Math. Softw. 18, 232–238 (1992)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  30. 30.
    Gugercin, S., Sorensen, D., Antoulas, A.: A modified low-rank Smith method for large-scale Lyapunov equations. Numer. Algorithms 32(1), 27–55 (2003)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Hammarling, S.: Numerical solution of the discrete-time, convergent, non-negative definite Lyapunov equation. Syst. Control Lett. 17, 137–139 (1991)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Hewer, G.: An iterative technique for the computation of steady state gains for the discrete optimal regulator. IEEE Trans. Automat. Contr. AC–16, 382–384 (1971)CrossRefGoogle Scholar
  33. 33.
    Heyouni, M., Jbilou, K.: An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation. Electron. Trans. Numer. Anal. 33, 53–62 (2008)MathSciNetMATHGoogle Scholar
  34. 34.
    Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press (1985)Google Scholar
  35. 35.
    Ionescu, V., Oarǎ, C., Weiss, M.: General matrix pencil techniques for the solution of algebraic Riccati equations: a unified approach. IEEE Trans. Automat. Contr. 42(8), 1085–1097 (1997)MATHCrossRefGoogle Scholar
  36. 36.
    Ito, K.: Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type approximation. SIAM J. Control Optim. 28, 1251–1269 (1990)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Kleinman, D.: On an iterative technique for Riccati equation computations. IEEE Trans. Automat. Contr. AC–13, 114–115 (1968)CrossRefGoogle Scholar
  38. 38.
    Kleinman, D.: Stabilizing a discrete, constant, linear system with application to iterative methods for solving the Riccati equation. IEEE Trans. Automat. Contr. AC–19, 252–254 (1974)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Lancaster, P., Rodman, L.: The Algebraic Riccati Equation. Oxford University Press, Oxford (1995)Google Scholar
  40. 40.
    Lasiecka, I., Triggiani, R.: Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lecture Notes in Control and Information Sciences, vol. 164. Springer, Berlin (1991)MATHCrossRefGoogle Scholar
  41. 41.
    Li, J.R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24(1), 260–280 (2002)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Lin, W.W., Xu, S.F.: Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM J. Matrix Anal. Appl. 28(1), 26–39 (2006)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Mehrmann, V.: Existence, uniqueness and stability of solutions to singular, linear-quadratic control problems. Linear Algebra Appl. 121, 291–331 (1989)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Mehrmann, V.: The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Lecture Notes in Control and Information Sciences, No. 163. Springer, Heidelberg (1991)MATHCrossRefGoogle Scholar
  45. 45.
    Mehrmann, V., Tan, E.: Defect correction methods for the solution of algebraic Riccati equations. IEEE Trans. Automat. Contr. 33, 695–698 (1988)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Pappas, T., Laub, A., Sandell, N.: On the numerical solution of the discrete-time algebraic Riccati equation. IEEE Trans. Automat. Contr. AC–25, 631–641 (1980)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Peaceman, D., Rachford, H.: The numerical solution sof parabolic and elliptic differential equations. J. SIAM 3, 28–41 (1955)MathSciNetMATHGoogle Scholar
  48. 48.
    Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math. 8, 33–48 (1997)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Penzl, T.: A cycli low-rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21(4), 1401–1418 (2000)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Penzl, T.: Lyapack Users Guide. Tech. Rep. SFB393/00-33, Sonderforschungsbereich 393, Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, 09107 Chemnitz, FRG. Available from http://www.tu-chemnitz.de/sfb393/sfb00pr.html (2000)
  51. 51.
    Roberts, J.: Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Int. J. Control 32, 677–687 (1980) (Reprint of Technical Report No. TR-13, CUED/B-Control, Cambridge University, Engineering Department (1971))MATHCrossRefGoogle Scholar
  52. 52.
    Sima, V.: Algorithms for Linear-Quadratic Optimization. Pure and Applied Mathematics, vol. 200. Marcel Dekker, Inc., New York, NY (1996)MATHGoogle Scholar
  53. 53.
    Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29, 1268–1288 (2007)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Smith, R.: Matrix equation XA + BX = C. SIAM J. Appl. Math. 16(1), 198–201 (1968)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Sun, X., Quintana-Ortí, E.: Spectral division methods for block generalized Schur decompositions. Math. Comput. 73, 1827–1847 (2004)MATHCrossRefGoogle Scholar
  56. 56.
    Varga, A.: A note on Hammarling’s algorithm for the discrete Lyapunov equation. Syst. Control Lett. 15(3), 273–275 (1990)MATHCrossRefGoogle Scholar
  57. 57.
    Wachspress, E.: Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 107, 87–90 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Research Group Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  2. 2.Carl-Friedrich-Gauß-Fakultät, Institut Computational Mathematics AG NumerikTU BraunschweigBraunschweigGermany

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