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Model Reduction for Circuit Simulation pp 53–83Cite as

Balancing-Related Model Reduction of Circuit Equations Using Topological Structure

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Abstract

In recent years, model order reduction has been recognized to be a powerful tool in analysis and simulation of integrated circuits. We consider balancing-related model reduction methods for differential-algebraic equations arising in circuit simulation. We show how positive real and bounded real balanced truncation can be used for passivity-preserving model reduction of circuit equations. These methods are based on balancing the solutions of projected Lur’e or Riccati matrix equations and admit computable error bounds. We also discuss efficient algorithms for solving such matrix equations that exploit the topological structure of circuit equations. Numerical experiments demonstrate the performance of the presented model reduction methods.

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References

  1. Anderson, B., Vongpanitlerd, S.: Network Analysis and Synthesis. Prentice Hall, Englewood Cliffs (1973)

    Google Scholar 

  2. Antoulas, A.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  3. Antoulas, A.: A new result on passivity preserving model reduction. Syst. Control Lett. 54(4), 361–374 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Bechtold, T., Verhoeven, A., ter Maten, E., Voss, T.: Model order reduction: an advanced, efficient and automated computational tool for microsystems. In: Cutello, V., Fotia, G., Puccio, L. (eds) Applied and Industrial Mathematics in Italy II, Selected contributions from the 8th SIMAI Conference, Series on Advances in Mathematics for Applied Sciences, vol. 75, pp. 113–124. World Scientific Publishing Co. Pte. Ltd, Singapore (2007)

    Chapter  Google Scholar 

  6. Benner, P: Numerical solution of special algebraic Riccati equations via exact line search method. In: Proceedings of the European Control Conference (ECC97), Paper 786. BELWARE Information Technology, Waterloo, Belgium (1997)

    Google Scholar 

  7. Benner, P.: Advances in balancing-related model reduction for circuit simulation. In: Roos, J., Costa, L. (eds) Scientific Computing in Electrical Engineering SCEE 2008, Mathematics in Industry, vol. 14, pp. 469–482. Springer, Berlin (2010)

    Chapter  Google Scholar 

  8. Benner, P., Quintana-Ortí, E.: Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms 20(1), 75–100 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Benner, P., Faßbender, H.: Numerische Methoden zur passivitätserhaltenden Modellreduktion (Numerical methods for passivity preserving model reduction). at-Automatisierungstechnik 54(4), 153–160 (2006) (In German)

    Article  Google Scholar 

  10. Benner, P., Stykel, T.: Numerical algorithms for projected generalized Riccati equations (in preparation)

    Google Scholar 

  11. Benner, P., Quintana-Ortí, E., Quintana-Ortí, G.: Efficient numerical algorithms for balanced stochastic truncation. Int. J. Appl. Math. Comput. Sci. 11(5), 1123–1150 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Benner, P., Hernández, V., Pastor, A.: The Kleinman iteration for nonstabilizable systems. Math. Control Signals Syst. 16, 76–93 (2003)

    MATH  Google Scholar 

  13. Benner, P., Quintana-Ortí, E., Quintana-Ortí, G.: Computing passive reduced-order models for circuit simulation. In: Proceedings of the International Conference on Parallel Computing in Electrical Engineering PARELEC 2004, pp. 146–151. IEEE Computer Society, Los Alamitos (2004)

    Google Scholar 

  14. Benner, P., Quintana-Ortí, E., Quintana-Ortí, G.: Parallel model reduction of large-scale linear descriptor systems via balanced truncation. In: Daydé, M., Dongarra, J., Hernández, V., Palma, J. (eds) High Performance Computing for Computational Science—VECPAR 2004. Lecture Notes in Computer Science, vol. 3402, pp. 340–353. Springer, Berlin (2004)

    Chapter  Google Scholar 

  15. Benner, P., Mehrmann, V., Sorensen, D. (eds.): Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45. Springer, Berlin (2005)

    Google Scholar 

  16. 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, 755–777 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Benner, P., Mena, H., Saak, J.: On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations. Electron. Trans. Numer. Anal. 29, 136–149 (2008)

    MathSciNet  Google Scholar 

  18. Benner, P., Li, R.C., Truhar, N.: On the ADI method for Sylvester equations. J. Comput. Appl. Math. 233(4), 1035–1045 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Brenan, K., Campbell, S., Petzold, L.: The Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations Classics in Applied Mathematics. SIAM, Philadelphia (1996)

    Google Scholar 

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

    Google Scholar 

  21. Chan, T.: Rank revealing QR factorizations. Linear Algebra Appl. 88/89, 67–82 (1987)

    Article  Google Scholar 

  22. Chua, L., Desoer, C., Kuh, E.: Linear and Nonlinear Circuits. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  23. Clements, D., Anderson, B., Laub, A., Matson, L.: Spectral factorization with imaginary-axis zeros. Linear Algebra Appl. 250, 225–252 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, vol. 118. Springer, Berlin (1989)

    Google Scholar 

  25. Davison, E.: A method for simplifying linear dynamical systems. IEEE Trans. Automat. Contr. 11, 93–101 (1966)

    Article  MathSciNet  Google Scholar 

  26. 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), pp. 127–132. IEEE, New York (1984)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  28. Faßbender, H., Benner, P.: Passivity preserving model reduction via a structured Lanczos method. In: Proceedings of the IEEE International Symposium on Computer-Aided Control Systems Design (Munich, Germany, October 4–6, 2006), pp. 8–13. IEEE (2006)

    Google Scholar 

  29. Ferrante, A.: Positive real lemma: necessary and sufficient conditions for the existence of solutions under virtually no assumptions. IEEE Trans. Automat. Contr. 50(5), 720–724 (2005)

    Article  MathSciNet  Google Scholar 

  30. Freund, R.: Model reduction methods based on Krylov subspaces. Acta Numerica 12, 267–319 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Freund, R.: SPRIM: structure-preserving reduced-order interconnect macromodeling. In: Technical Digest of the 2004 IEEE/ACM International Conference on Computer-Aided Design, pp. 80–87. IEEE Computer Society Press, Los Alamos (2004)

    Google Scholar 

  32. Freund, R.: Structure-preserving model order reduction of RCL circuit equations. In: Schilders, W., van der Vorst, H., Rommes, (J. (eds) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13, pp. 49–73. Springer, Berlin (2008)

    Chapter  Google Scholar 

  33. Freund, R., Feldmann, P.: The SyMPVL algorithm and its applications in interconnect simulation. In: Proceedings of the 1997 International Conference on Simulation of Semiconductor Processes and Devices, pp. 113–116. IEEE, New York (1997)

    Google Scholar 

  34. Gantmacher, F.: Theory of Matrices. Chelsea Publishing Company, New York (1959)

    MATH  Google Scholar 

  35. Glover, K.: All optimal Hankel-norm aproximations of linear multivariable systems and their L ∞-error bounds. Int. J. Control 39(6), 1115–1193 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  36. Golub, G., Van Loan, C.: Matrix Computations. 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  37. Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Teubner-Texte zur Mathematik, vol. 88. B.G. Teubner, Leipzig (1986)

    Google Scholar 

  38. Grimme, E.: Krylov projection methods for model reduction. Ph.D. thesis, University of Illinois, Urbana-Champaign (1997)

    Google Scholar 

  39. Gugercin, S., Antoulas, A.: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748–766 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  40. Ionescu, V., Oară, C., Weiss, M.: Generalized Riccati Theory and Robust Control: A Popov Function Approach. Wiley, Chichester (1999)

    MATH  Google Scholar 

  41. Ionutiu, R., Rommes, J., Antoulas, A.: Passivity-preserving model reduction using dominant spectral-zero interpolation. IEEE Trans. Comput. Aided Design Integr. Circuits Syst. 27(12), 2250–2263 (2008)

    Article  Google Scholar 

  42. Ishihara, J., Terra, M.: On the Lyapunov theorem for singular systems. IEEE Trans. Automat. Contr. 47(11), 1926–1930 (2002)

    Article  MathSciNet  Google Scholar 

  43. Jbilou, K.: ADI preconditioned Krylov methods for large Lyapunov matrix equations. Linear Algebra Appl. 432(10), 2473–2485 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Jungnickel, D.: Graphs, Network and Algorithms. Springer, Berlin (2005)

    Google Scholar 

  45. Knockaert, L., De Zutter, D.: Laguerre-SVD reduced-order modeling. IEEE Trans. Microw. Theory Tech. 48(9), 1469–1475 (2000)

    Article  Google Scholar 

  46. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)

    Book  MATH  Google Scholar 

  47. Lancaster, P., Rodman, L.: The Algebraic Riccati Equation. Oxford University Press, Oxford (1995)

    Google Scholar 

  48. Lewis, F.: A survey of linear singular systems. Circuits Syst. Signal Process 5(1), 3–36 (1986)

    Article  MATH  Google Scholar 

  49. Li, J.R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24(1), 260–280 (2002)

    Article  MathSciNet  Google Scholar 

  50. Marschall, S.: An approximate method for reducing the order of a linear system. Contr. Eng. 10, 642–648 (1966)

    Google Scholar 

  51. März, R.: Canonical projectors for linear differential algebraic equations. Comput. Math. Appl. 31(4/5), 121–135 (1996)

    Article  MATH  Google Scholar 

  52. Mehrmann, V., Stykel, T.: Balanced truncation model reduction for large-scale systems in descripter form. Chapter 3 (pp. 83–115) of [12]

    Google Scholar 

  53. Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Contr. 26(1), 17–32 (1981)

    Article  MATH  Google Scholar 

  54. Mullis, T., Roberts, R.: Synthesis of minimum roundoff noise fixed point digital filters. IEEE Trans. Circuits Syst. CAS-23(9), 551–562 (1976)

    Article  MathSciNet  Google Scholar 

  55. Ober, R.: Balanced parametrization of classes of linear systems. SIAM J. Control Optim. 29(6), 1251–1287 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  56. Odabasioglu, A., Celik, M., Pileggi, L.: PRIMA: passive reduced-order interconnect macromodeling algorithm. IEEE Trans. Comput. Aided Design Integr. Circuits Syst. 17(8), 645–654 (1998)

    Article  Google Scholar 

  57. Penzl, T.: A cyclic low-rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21(4), 1401–1418 (1999/2000)

    Article  MathSciNet  Google Scholar 

  58. Penzl, T.: Algorithms for model reduction of large dynamical systems. Linear Algebra Appl. 415(2–3), 322–343 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  59. Perev, K., Shafai, B.: Balanced realization and model reduction of singular systems. Int. J. Syst. Sci. 25(6), 1039–1052 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  60. Phillips, J., Daniel, L., Silveira, L.: Guaranteed passive balancing transformations for model order reduction. IEEE Trans. Comput. Aided Design Integr. Circuits Syst. 22(8), 1027–1041 (2003)

    Article  Google Scholar 

  61. Reis, T.: Circuit synthesis of passive descriptor systems—a modified nodal approach. Int. J. Circuit Theory Appl. 38(1), 44–68 (2010). doi:10.1002/cta.532

    Article  MATH  Google Scholar 

  62. Reis, T.: Lur’e equations and even matrix pencils. Linear Algebra Appl. 434(4),152–173 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  63. Reis, T., Stykel, T.: Positive real and bounded real balancing for model reduction of descriptor systems. Int. J. Control 83(1), 74– 88 (2009) doi:10.1016/j.lao.2010.09.005

    Article  MathSciNet  Google Scholar 

  64. Reis, T., Stykel, T.: Passivity-preserving balanced truncation for electrical circuits. IEEE Trans. Comput. Aided Design Integr. Circuits Syst. 29(9), 1354–1367 (2010)

    Article  Google Scholar 

  65. Reis, T., Stykel, T.: Lyapunov balancing for passivity-preserving model reduction of RC circuits. SIAM J. Appl. Dyn. Syst. 10(1),1–34 (2011)

    Google Scholar 

  66. Riaza, R., Tischendorf, C.: Qualitative features of matrix pencils and DAEs arising in circuit dynamics. Dyn. Syst. 22, 107–131 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  67. Riaza, R.: Differential-Algebraic Systems: Analytical Aspects and Circuit Applications. World Scientific Publishing Co. Pte. Ltd, Hackensack (2008)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  69. Sabino, J.: Solution of large-scale Lyapunov equations via the block modified Smith method. Ph.D. thesis, Rice University, Houston (2006)

    Google Scholar 

  70. Schilders, W., van der Vorst, H., Rommes, J. (eds.): Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13. Springer, Berlin (2008)

    Google Scholar 

  71. Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29(3), 1268–1288 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  72. Sorensen, D.: Passivity preserving model reduction via interpolation of spectral zeros. Syst. Control Lett. 54(4), 347–360 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  73. Stykel, T.: Stability and inertia theorems for generalized Lyapunov equations. Linear Algebra Appl. 355, 297–314 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  75. Stykel, T.: Low-rank iterative methods for projected generalized Lyapunov equations. Electron. Trans. Numer. Anal. 30, 187–202 (2008)

    MATH  MathSciNet  Google Scholar 

  76. Takaba, K., Morihira, N., Katayama, T.: A generalized Lyapunov theorem for descriptor system. Syst. Control Lett. 24, 49–51 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  78. Varga, A.: On computing high accuracy solutions of a class of Riccati equations. Control Theory Adv. Technol. 10, 2005–2016 (1995)

    Google Scholar 

  79. Vlach, J., Singhal, K.: Computer Methods for Circuit Analysis and Design. Van Nostrand Reinhold, New York (1994)

    Google Scholar 

  80. Wachspress, E.: Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 1, 87–90 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  81. Wachspress, E.: The ADI minimax problem for complex spectra. In: Kincaid, D., Hayes, L. (eds.) Iterative Methods for Large Linear Systems, pp. 251–271. Academic Press, San Diego (1990)

    Google Scholar 

  82. Weiss, H., Wang, Q., Speyer, J.: System characterization of positive real conditions. IEEE Trans. Automat. Contr. 39(3), 540–544 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  83. Yan, B., Tan, S.D., McGaughy, B.: Second-order balanced truncation for passive order reduction of RLCK circuits. IEEE Trans. Circuits Syst. II 55(9), 942–946 (2008)

    Article  Google Scholar 

  84. Yang, F., Zeng, X., Su, Y., Zhou, D.: RLC equivalent circuit synthesis method for structure-preserved reduced-order model of interconnect in VLSI. Commun. Comput. Phys. 3(2), 376–396 (2008)

    MATH  Google Scholar 

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Acknowledgements

This work was supported by the Research Network SyreNe—System Reduction for Nanoscale IC Design funded by the German Federal Ministry of Education and Science (BMBF), Grant No. 03STPAE3. Responsibility for the contents of this publication rests with the author. The author would like to thank Achim Basermann and Carsten Neff from NEC Laboratories Europe, IT Research Division for providing the circuit examples.

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

    Tatjana Stykel

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  1. Tatjana Stykel
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  1. Inst. Dynamics of Complex Techn. Systems, Max Planck Institute, Sandtorstrasse 1, Magdeburg, 39106, Germany

    Peter Benner

  2. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  3. , Dept. Mathematics & Comp. Sci., Eindhoven University of Technology, P.O.Box 513, Den Dolech 2, HG 8.02, Eindhoven, 5600 MB, Netherlands

    E. Jan W. ter Maten

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Stykel, T. (2011). Balancing-Related Model Reduction of Circuit Equations Using Topological Structure. In: Benner, P., Hinze, M., ter Maten, E. (eds) Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol 74. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0089-5_3

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