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Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form

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Dimension Reduction of Large-Scale Systems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 45))

Summary

In this paper we give a survey on balanced truncation model order reduction for linear time-invariant continuous-time systems in descriptor form. We first give a brief overview of the basic concepts from linear system theory and then present balanced truncation model reduction methods for descriptor systems and discuss their algorithmic aspects. The efficiency of these methods is demonstrated by numerical experiments.

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References

  1. Anderson, B.D.O., Antoulas, A.C.: Rational interpolation and state-variable realizations. Linear Algebra Appl., 137–138, 479–509 (1990)

    Article  MathSciNet  Google Scholar 

  2. Antoulas, A.C.: Lectures on the Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2004)

    Google Scholar 

  3. Antoulas, A.C., Sorensen, D.C., Gugercin, S.: A survey of model reduction methods for large-scale systems. In: Olshevsky, V. (ed) Structured Matrices in Mathematics, Computer Science and Engineering, Vol. I. Contemporary Mathematics Series, 280, pages 193–219. American Mathematical Society (2001)

    Google Scholar 

  4. Antoulas, A.C., Sorensen, D.C., Gugercin, S.: A modified low-rank Smith method for large-scale Lyapunov equations. Numerical Algorithms, 32, 27–55 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Antoulas, A.C., Sorensen, D.C., Zhou, Y.: On the decay rate of the Hankel singular values and related issues. Systems Control Lett., 46, 323–342 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bai, Z.: Krylov subspace techniques for reduced-order modeling of largescale dynamical systems. Appl. Numer. Math., 43, 9–44 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bunse-Gerstner, A., Byers, R., Mehrmann, V., Nichols, N.K.: Feedback design for regularizing descriptor systems. Linear Algebra Appl., 299, 119–151 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beattie, Ch.: Potential theory in the analysis of iterative methods. Lecture Notes, Technische Universität Berlin, Berlin (2004)

    Google Scholar 

  9. Brenan, K.E., Campbell, S.L., Petzold, L.R.: The Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Elsevier, North-Holland, New York (1989)

    Google Scholar 

  10. Bojanczyk, A.W., Ewerbring, L.M., Luk, F.T., Van Dooren, P.: An accurate product SVD algorithm. Signal Process., 25, 189–201 (1991)

    Article  MATH  Google Scholar 

  11. Bender, D.J.: Lyapunov-like equations and reachability/observability Gramians for descriptor systems. IEEE Trans. Automat. Control, 32, 343–348 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bernert, K.: Differenzenverfahren zur Lösung der Navier-Stokes-Gleichungen über orthogonalen Netzen. Wissenschaftliche Schriftenreihe 10/1990, Technische Universität Chemnitz (1990) [German]

    Google Scholar 

  13. Bai, Z., Freund, R.W.: A partial Padé-via-Lanczos method for reduced-order modeling. Linear Algebra Appl., 332–334, 141–166 (2001)

    MathSciNet  Google Scholar 

  14. Baker Jr., G.A., Graves-Morris P.R.: Padé Approximants. Second edition. Encyclopedia of Mathematics and its Applications, 59. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  15. Benner, P., Quintana-Ortí, E.S., Quintana-Ortí, G.: Parallel model reduction of large-scale linear descriptor systems via balanced truncation. In: High Performance Computing for Computational Science. Proceedings of the 6th International Meeting VECPAR'04 (Valencia, Spain, June 28–30, 2004), pages 65–78 (2004)

    Google Scholar 

  16. Bai, Z., Slone, R.D., Smith, W.T., Ye, Q.: Error bound for reduced system model by Padé approximation via the Lanczos process. IEEE Trans. Comput. Aided Design, 18, 133–141 (1999)

    Article  Google Scholar 

  17. Beelen, T., Van Dooren, P.: An improved algorithm for the computation of Kronecker's canonical form of a singular pencil. Linear Algebra Appl., 105, 9–65 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Campbell, S.L.: Singular Systems of Differential Equation, I. Pitman, San Francisco (1980)

    Google Scholar 

  19. Cobb, D.: Controllability, observability, and duality in singular systems. IEEE Trans. Automat. Control, 29, 1076–1082 (1984)

    Article  MathSciNet  Google Scholar 

  20. Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, 118. Springer, Berlin Heidelberg New York (1989)

    Google Scholar 

  21. Demmel, J.W., Kågström, B.: The generalized Schur decomposition of an arbitrary pencil A — λB: robust software with error bounds and applications. Part I: Theory and algorithms. ACM Trans. Math. Software, 19, 160–174 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Demmel, J.W., Kågström, B.: The generalized Schur decomposition of an arbitrary pencil A — λB: robust software with error bounds and applications. Part II: Software and applications. ACM Trans. Math. Software, 19, 175–201 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Doetsch, G.: Guide to the Applications of the Laplace and Z-Transforms. Van Nostrand Reinhold Company, London (1971)

    MATH  Google Scholar 

  24. Drmač, Z.: New accurate algorithms for singular value decomposition of matrix triplets. SIAM J. Matrix Anal. Appl., 21, 1026–1050 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Enns, D.: Model reduction with balanced realization: an error bound and a frequency weighted generalization. In: Proceedings of the 23rd IEEE Conference on Decision and Control (Las Vegas, 1984), pages 127–132. IEEE, New York (1984)

    Google Scholar 

  26. Estévez Schwarz, D., Tischendorf, C.: Structural analysis for electric circuits and consequences for MNA. Int. J. Circ. Theor. Appl., 28, 131–162 (2000)

    Article  MATH  Google Scholar 

  27. Feldmann, P., Freund, R.W.: Efficient linear circuit analysis by Padé approximation via the Lanczos process. IEEE Trans. Computer-Aided Design, 14, 639–649 (1995)

    Article  Google Scholar 

  28. Freund, R.W.: Krylov-subspace methods for reduced-order modeling in circuit simulation. J. Comput. Appl. Math., 123, 395–421 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Fortuna, L., Nunnari, G., Gallo, A.: Model Order Reduction Techniques with Applications in Electrical Engineering. Springer, London (1992)

    Google Scholar 

  30. Günther, M., Feldmann, U.: CAD-based electric-circuit modeling in industry. I. Mathematical structure and index of network equations. Surveys Math. Indust., 8, 97–129 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Gallivan, K., Grimme, E., Van Dooren, P.: Asymptotic waveform evaluation via a Lanczos method. Appl. Math. Lett., 7, 75–80 (1994)

    Article  MATH  Google Scholar 

  32. Gallivan, K., Grimme, E., Van Dooren, P.: A rational Lanczos algorithm for model reduction. Numerical Algorithms, 12, 33–63 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gragg, W.B., Lindquist, A.: On the partial realization problem. Linear Algebra Appl., 50, 277–319 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L ∞-errors bounds. Internat. J. Control, 39, 1115–1193 (1984)

    MATH  MathSciNet  Google Scholar 

  35. Grasedyck, L.: Existence of a low rank of H-matrix approximation to the solution of the Sylvester equation. Numer. Linear Algebra Appl., 11, 371–389 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Grimme, E.: Krylov projection methods for model reduction. Ph.D. Thesis, Iniversity of Illinois, Urbana-Champaign (1997)

    Google Scholar 

  37. Golub, G.H., Sølna, K., Van Dooren, P.: Computing the SVD of a general matrix product/quotient. SIAM J. Matrix Anal. Appl., 22, 1–19 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gugercin, S.: Projection methods for model reduction of large-scale dynamical systems. Ph.D. Thesis, Rice University, Houston (2003)

    Google Scholar 

  39. Golub, G.H., Van Loan, C.F.: Matrix Computations. 3rd ed. The Johns Hopkins University Press, Baltimore, London (1996)

    MATH  Google Scholar 

  40. Hackbusch, W.: A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices. Computing, 62, 89–108 (2000)

    Article  MathSciNet  Google Scholar 

  41. Hammarling, S.J.: Numerical solution of the stable non-negative definite Lyapunov equation. IMA J. Numer. Anal., 2, 303–323 (1982)

    MATH  MathSciNet  Google Scholar 

  42. Hackbusch, W., Grasedyck, L., Börm, S.: An introduction to hierarchical matrices. Math. Bohem., 127, 229–241 (2002)

    MathSciNet  MATH  Google Scholar 

  43. Jonsson, I., Kågström, B.: Recursive blocked algorithms for solving triangular systems — Part I: One-sided and coupled Sylvester-type matrix equations. ACM Trans. Math. Software, 28, 392–415 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kailath, T.: Linear Systems. Prentice-Hall Information and System Sciences Series. Prentice Hall, Englewood Cliffs (1980)

    Google Scholar 

  45. Kågström, B., Van Dooren, P.: A generalized state-space approach for the additive decomposition of a transfer function. J. Numer. Linear Algebra Appl., 1, 165–181 (1992)

    MathSciNet  Google Scholar 

  46. Kågström, B., Westin, L.: Generalized Schur methods with condition estimators for solving the generalized Sylvester equation. IEEE Trans. Automat. Control, 34, 745–751 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  47. Liu, Y., Anderson, B.D.O.: Singular perturbation approximation of balanced systems. Internat. J. Control, 50, 1379–1405 (1989)

    MathSciNet  MATH  Google Scholar 

  48. Laub, A.J., Heath, M.T., Paige, C.C., Ward, R.C.: Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Automat. Control, 32, 115–122 (1987)

    Article  MATH  Google Scholar 

  49. Li, J.-R.: Model reduction of large linear systems via low rank system Gramians. Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge (2000)

    Google Scholar 

  50. Liu, W.Q., Sreeram, V.: Model reduction of singular systems. In: Proceedings of the 39th IEEE Conference on Decision and Control (Sydney, Australia, 2000), pages 2373–2378. IEEE (2000)

    Google Scholar 

  51. Li, J.-R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl., 24, 260–280 (2002)

    Article  MathSciNet  Google Scholar 

  52. Li, J.-R., Wang, F., White, J.: An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect. In: Proceedings of the 36th Design Automation Conference (New Orleans, USA, 1999), pages 1–6. IEEE (1999)

    Google Scholar 

  53. März, R.: Canonical projectors for linear differential algebraic equations. Comput. Math. Appl., 31, 121–135 (1996)

    Article  MATH  Google Scholar 

  54. Moore, B.C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Control, 26, 17–32 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  55. Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math., 8, 33–48 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  56. Penzl, T.: A cyclic low-rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput., 21, 1401–1418 (1999/2000)

    Article  MathSciNet  Google Scholar 

  57. Penzl, T.: Algorithms for model reduction of large dynamical systems. Preprint SFB393/99-40, Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany, (1999). Available from http://www.tu-chemnitz.de/sfb393/sfb99pr.html

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  59. Penzl, T.: LYAPACK Users Guide. Preprint SFB393/00-33, Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany (2000). Available from http://www.tu-chemnitz.de/sfb393/sfb00pr.html

    Google Scholar 

  60. Perev, K., Shafai, B.: Balanced realization and model reduction of singular systems. Internat. J. Systems Sci., 25, 1039–1052 (1994)

    MathSciNet  MATH  Google Scholar 

  61. Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  62. Ruhe, A.: Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl., 58, 391–405 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  63. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996)

    MATH  Google Scholar 

  64. Safonov, M.G., Chiang R.Y.: A Schur method for balanced-truncation model reduction. IEEE Trans. Automat. Control, 34, 729–733 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  65. Schüpphaus, R.: Regelungstechnische Analyse und Synthese von Mehrkörpersystemen in Deskriptorform. Ph.D. Thesis, Fachbereich Sicherheitstechnik, Bergische Universität-GesamthochschuleWuppertal. Fortschritt-Berichte VDI, Reihe 8, Nr. 478. VDI Verlag, Düsseldorf (1995) [German]

    Google Scholar 

  66. Sokolov, V.I.: On realization of rational matrices. Technical Report 31-2003, Institut für Mathematik, Technische Universität Berlin, D-10263 Berlin, Germany (2003)

    Google Scholar 

  67. Stewart, G.W., Sun, J.-G.: Matrix Perturbation Theory. Academic Press, New York (1990)

    MATH  Google Scholar 

  68. Stykel, T.: Analysis and numerical solution of generalized Lyapunov Equations. Ph.D. Thesis, Institut für Mathematik, Technische Universität Berlin, Berlin (2002)

    Google Scholar 

  69. Stykel, T.: Numerical solution and perturbation theory for generalized Lyapunov equations. Linear Algebra Appl., 349, 155–185 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  70. Stykel, T.: Input-output invariants for descriptor systems. Preprint PIMS-03-1, The Pacific Institute for the Mathematical Sciences, Canada (2003)

    Google Scholar 

  71. Stykel, T.: Balanced truncation model reduction for semidiscretized Stokes equation. To appear in Linear Algebra Appl. (2004)

    Google Scholar 

  72. Stykel, T.: Gramian-based model reduction for descriptor systems. Math. Control Signals Systems, 16, 297–319 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  73. Stykel, T.: Low rank iterative methods for projected generalized Lyapunov equations. Preprint 198, DFG Research Center Matheon, Technische Universität Berlin (2005).

    Google Scholar 

  74. Sorensen, D.C., Zhou, Y.: Bounds on eigenvalue decay rates and sensitivity of solutions of Lyapunov equations. Technical Report TR02-07, Department of Computational and Applied Mathematics, Rice University, Houston (2002). Available from http://www.caam.rice.edu/caam/trs/2002/TR02-07.pdf

    Google Scholar 

  75. Tombs, M.S., Postlethweite, I.: Truncated balanced realization of a stable non-minimal state-space system. Internat. J. Control, 46, 1319–1330 (1987)

    MathSciNet  MATH  Google Scholar 

  76. Varga, A.: Efficient minimal realization procedure based on balancing. In: EL Moudni, A., Borne, P., Tzafestas, S.G. (eds) Proceedings of the IMACS/IFAC Symposium on Modelling and Control of Technological Systems (Lille, France, May 7–10, 1991), volume 2, pages 42–47 (1991)

    Google Scholar 

  77. Varga, A.: Computation of coprime factorizations of rational matrices. Linear Algebra Appl., 271, 83–115 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  78. Varga, A.: A Descriptor Systems Toolbox for MATLAB. In: Proceedings of the 2000 IEEE International Symposium on Computer Aided Control System Design (Anchorage, Alaska, September 25–27, 2000), pages 150–155 (2000). Available from http://www.robotic.dlr.de/control/publications/2000/varga cacsd2000p2.pdf

    Google Scholar 

  79. Verghese, G.C., Lévy, B.C., Kailath, T.: A generalized state-space for singular systems. IEEE Trans. Automat. Control, 26, 811–831 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  80. Watkins, D.S.: Performance of the QZ algorithm in the presence of infinite eigenvalues. SIAM J. Matrix Anal. Appl., 22, 364–375 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  81. Yip, E.L., Sincovec, R.F.: Solvability, controllability and observability of continuous descriptor systems. IEEE Trans. Automat. Control, 26, 702–707 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  82. Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice Hall, Upper Saddle River (1996)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Mathematik, MA 4–5, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Volker Mehrmann

  2. Institut für Mathematik, MA 3–3, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Tatjana Stykel

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  1. Volker Mehrmann
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  2. Tatjana Stykel
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Editors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Peter Benner

  2. Department of Computational and Applied Mathematics, Rice University, Main Street 6100, 77005-1892, Houston, TX, USA

    Danny C. Sorensen

  3. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Volker Mehrmann

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Mehrmann, V., Stykel, T. (2005). Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form. In: Benner, P., Sorensen, D.C., Mehrmann, V. (eds) Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27909-1_3

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