Parallel Model Reduction of Large Linear Descriptor Systems via Balanced Truncation

  • Peter Benner
  • Enrique S. Quintana-Ortí
  • Gregorio Quintana-Ortí
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3402)


In this paper we investigate the use of parallel computing to deal with the high computational cost of numerical algorithms for model reduction of large linear descriptor systems. The state-space truncation methods considered here are composed of iterative schemes which can be efficiently implemented on parallel architectures using existing parallel linear algebra libraries. Our experimental results on a cluster of Intel Pentium processors show the performance of the parallel algorithms.


Model reduction balanced truncation linear descriptor systems parallel algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  2. 2.
    Antoulas, A.C.: Lectures on the Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia (to appear)Google Scholar
  3. 3.
    Bai, Z., Demmel, J., Dongarra, J., Petitet, A., Robinson, H., Stanley, K.: The spectral decomposition of nonsymmetric matrices on distributed memory parallel computers. SIAM J. Sci. Comput. 18, 1446–1461 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bai, Z., Demmel, J., Gu, M.: An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems. Numer. Math. 76(3), 279–308 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bartels, R.H., Stewart, G.W.: Solution of the matrix equation AX + XB = C: Algorithm 432. Comm. ACM 15, 820–826 (1972)CrossRefGoogle Scholar
  6. 6.
    Benner, P.: Spectral projection methods for model reduction of descriptor systems (In Preparation)Google Scholar
  7. 7.
    Benner, P., Claver, J.M., Quintana-Ortí, E.S.: Efficient solution of coupled Lyapunov equations via matrix sign function iteration. In: Dourado, A., et al. (eds.) Proc. 3rd Portuguese Conf. on Automatic Control CONTROLO 1998, Coimbra, pp. 205–210 (1998)Google Scholar
  8. 8.
    Benner, P., Quintana-Ortí, E.S.: Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms 20(1), 75–100 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Benner, P., Quintana-Ortí, E.S., Quintana-Ortí, G.: State-space truncation methods for parallel model reduction of large-scale systems. Parallel Comput. 29, 1701–1722 (2003)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Benner, P., Quintana-Ortí, E.S., Quintana-Ortí, G.: Parallel model reduction of large-scale linear descriptor systems via Balanced Truncation. In: DFG Research Center “Mathematics for Key Technologies”, Berlin, Germany (2004) (Preprint #53),
  11. 11.
    Blackford, L.S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.C.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)zbMATHCrossRefGoogle Scholar
  12. 12.
    Cheng, J., Ianculescu, G., Kenney, C.S., Laub, A.J., Papadopoulos, P.M.: Control-structure interaction for space station solar dynamic power module. IEEE Control Systems, 4–13 (1992)Google Scholar
  13. 13.
    Chu, P.Y., Wie, B., Gretz, B., Plescia, C.: Approach to large space structure control system design using traditional tools. AIAA J. Guidance, Control, and Dynamics 13, 874–880 (1990)CrossRefGoogle Scholar
  14. 14.
    Fortuna, L., Nummari, G., Gallo, A.: Model Order Reduction Techniques with Applications in Electrical Engineering. Springer, Heidelberg (1992)Google Scholar
  15. 15.
    Freund, R.: Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation. In: Datta, B.N. (ed.) Applied and Computational Control, Signals, and Circuits, vol. 1, ch. 9, pp. 435–498. Birkhäuser, Boston (1999)Google Scholar
  16. 16.
    Gardiner, J.D., Laub, A.J., Amato, J.J., Moler, C.B.: Solution of the Sylvester matrix equation AXB + CXD = E. ACM Trans. Math. Software 18, 223–231 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  18. 18.
    Malyshev, A.N.: Parallel algorithm for solving some spectral problems of linear algebra. Linear Algebra Appl. 188/189, 489–520 (1993)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Moore, B.C.: Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Automat. Control, AC 26, 17–32 (1981)zbMATHCrossRefGoogle Scholar
  20. 20.
    Obinata, G., Anderson, B.D.O.: Model Reduction for Control System Design. Communications and Control Engineering Series. Springer, London (2001)zbMATHGoogle Scholar
  21. 21.
    Papadopoulos, P., Laub, A.J., Ianculescu, G., Ly, J., Kenney, C.S., Pandey, P.: Optimal control study for the space station solar dynamic power module. In: Proc. Conf. on Decision and Control CDC 1991, Brighton, December 1991, pp. 2224–2229 (1991)Google Scholar
  22. 22.
    Paul, C.R.: Analysis of Multiconductor Transmission Lines. Wiley–Interscience, Singapur (1994)Google Scholar
  23. 23.
    Quintana-Ortí, E.S.: Algoritmos Paralelos Para Resolver Ecuaciones Matriciales de Riccati en Problemas de Control. PhD thesis, Universidad Politécnica de Valencia (1996)Google Scholar
  24. 24.
    Quintana-Ortí, G., Sun, X., Bischof, C.H.: A BLAS-3 version of the QR factorization with column pivoting. SIAM J. Sci. Comput. 19, 1486–1494 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Roberts, J.D.: Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Internat. J. Control, 32, 677–687 (1980) Reprint of Technical Report No. TR-13, CUED/B-Control, Cambridge University, Engineering Department (1971)Google Scholar
  26. 26.
    Safonov, M.G., Chiang, R.Y.: A Schur method for balanced-truncation model reduction. IEEE Trans. Automat. Control, AC 34, 729–733 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Stykel, T.: Model reduction of descriptor systems. Technical Report 720-2001, Institut für Mathematik, TU Berlin, D-10263 Berlin, Germany (2001)Google Scholar
  28. 28.
    Stykel, T.: Analysis and Numerical Solution of Generalized Lyapunov Equations. Dissertation, TU Berlin (2002)Google Scholar
  29. 29.
    Stykel, T.: Balanced truncation model reduction for semidiscretized Stokes equation. Technical Report 04-2003, Institut für Mathematik, TU Berlin, D-10263 Berlin, Germany (2003)Google Scholar
  30. 30.
    Sun, X., Quintana-Ortí, E.S.: Spectral division methods for block generalized Schur decompositions. Mathematics of Computation 73, 1827–1847 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Tombs, M.S., Postlethwaite, I.: Truncated balanced realization of a stable non-minimal state-space system. Internat. J. Control 46(4), 1319–1330 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    van de Geijn, R.A.: Using PLAPACK: Parallel Linear Algebra Package. MIT Press, Cambridge (1997)Google Scholar
  33. 33.
    Varga, A.: Efficient minimal realization procedure based on balancing. In: Prepr. of the IMACS Symp. on Modelling and Control of Technological Systems, vol. 2, pp. 42–47 (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Peter Benner
    • 1
  • Enrique S. Quintana-Ortí
    • 2
  • Gregorio Quintana-Ortí
    • 2
  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany
  2. 2.Depto. de Ingeniería y Ciencia de ComputadoresUniversidad Jaume ICastellónSpain

Personalised recommendations