H 2 model reduction for large-scale linear dynamical systems

SIAM Journal on Matrix Analysis and Applications

Volumn 30, Issue 2, 2008, Pages 609-638

  Gugercin, S ^a   Antoulas, A C ^b   Beattie, C ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^b Rice University   (United States)

Author keywords

H 2 approximation; Model Reduction; Rational Krylov

Indexed keywords

APPROXIMATION PROBLEMS; BEST APPROXIMATIONS; ELEMENTARY PROOF; EXISTING METHOD; FIRST-ORDER NECESSARY CONDITIONS; H 2 APPROXIMATION; LARGE-SCALE PROBLEM; LINEAR DYNAMICAL SYSTEMS; LOCAL OPTIMALITY CONDITIONS; LYAPUNOV EQUATION; MIRROR IMAGES; MODEL REDUCTION; MODEL REDUCTION PROBLEMS; NUMERICAL EXPERIMENTS; OPTIMALITY; ORTHOGONALITY CONDITIONS; RATIONAL KRYLOV; REDUCED ORDER MODELS; REDUCED SYSTEMS;

DYNAMICAL SYSTEMS; INTERPOLATION; LINEAR CONTROL SYSTEMS; LYAPUNOV FUNCTIONS; NUMERICAL METHODS;

MATHEMATICAL MODELS;

EID: 56749147915     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/060666123     Document Type: Article

References (40)

1. A.C. ANTOULAS, Recursive modeling of discrete-time time series, in Linear Algebra for Control Theory, P. Van Dooren and B. W. Wyman, eds., IMA Vol. Math. Appl. 62, Springer-Verlag, New York, 1993, pp. 1-20.
2. A.C. ANTOULAS, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM, Philadelphia, 2005.
3. A.C. ANTOULAS and J.C. WILLEMS, A behavioral approach to linear exact modeling, IEEE Trans. Automat. Control, 38 (1993), pp. 1776-1802.
4. A.C. ANTOULAS, D.C. SORENSEN, and S. GUGERCIN, A survey of model reduction methods for large scale systems, in Structured Matrices in Mathematics, Computer Science, and Engineering, I (Boulder, CO, 1999), Contemp. Math. 280, AMS, Providence, RI, 2001, pp. 193-219.
5. J. BADIA, P. BENNER, R. MAYO, E.S. QUINTANA-ORTÍ, G. QUINTANA-ORTI, and J. SAAK, Parallel order reduction via balanced truncation for optimal cooling of steel profiles, in Proceedings of the 1 lth International European Conference on Parallel Processing, EuroPar 2005, Lisbon, J. C. Cunha and P. D. Medeiros, eds., Lecture Notes in Comput. Sci. 3648, Springer-Verlag, Berlin, 2005, pp. 857-866.
6. 2 approximation: A gradient algorithm, Automat., 27 (1991), pp. 413-418.
7. P. BENNER, Solving large-scale control problems, IEEE Control Systems Mag., 24 (2004), pp. 44-59.
8. P. BENNER and J. SAAK, Efficient numerical solution of the LQR-problem for the heat equation, Proc. Appl. Math. Mech., 4 (2004), pp. 648-649.
9. A.E. BRYSON and A. CARRIER, Second-order algorithm for optimal model order reduction, J. Guidance Control Dynam., 13 (1990), pp. 887-892.
10. 2-optimal Model Reduction for Large Scale Discrete Dynamical MIMO Systems, ZeTeM Technical Report 07-04, University of Bremen, 2007; available online from http://www.math.uni-bremen.de/zetem/reports/reports-liste.htmlμreports2007.
11. C. DE VILLEMAGNE and R. SKELTON, Model reduction using a projection formulation, Internat. J. Control, 40 (1987), pp. 2141-2169.
12. P. FELDMAN and R.W. FREUND, Efficient linear circuit analysis by Padé approximation via a Lanczos method, IEEE Trans. Computer-Aided Design, 14 (1995), pp. 639-649.
13. 2 approximation: A gradient algorithm based on Schur analysis, SIAM J. Control Optim., 36 (1998), pp. 2103-2127.
14. D. GAIER, Lectures on Complex Approximation, Birkhäuser, Boston, 1987.
15. K. GALLIVAN, E. GRIMME, and P. VAN DOOREN, A rational Lanczos algorithm for model reduction, Numer. Algorithms, 2 (1996), pp. 33-63.
16. K. GALLIVAN, P. VAN DOOREN, and E. GRIMME, On some recent developments in projectionbased model reduction, in ENUMATH 97 (Heidelberg), World Scientific, River Edge, NJ, 1998, pp. 98-113.
17. E.J. GRIMME, Krylov Projection Methods for Model Reduction, Ph.D. thesis, University of Illinois, Urbana-Champaign, Urbana, IL, 1997.
18. S. GUGERCIN, Projection Methods for Model Reduction of Large-Scale Dynamical Systems, Ph.D. thesis, Rice University, Houston, TX, 2002.
19. S. GUGERCIN and A.C. ANTOULAS, A comparative study of 7 model reduction algorithms, in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, 2000, pp. 2367-2372.
20. 2 error expression for the Lanczos procedure, in Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, pp. 1869-1872.
21. Y. HALEVI, Frequency weighted model reduction via optimal projection, in Proceedings of the 29th IEEE Conference on Decision and Control, 1990, pp. 2906-2911.
22. D.C. HYLAND and D.S. BERNSTEIN, The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore, IEEE Trans. Automat. Control, 30 (1985), pp. 1201-1211.
23. J.G. KORVINK and E.B. RUDYNI, Oberwolfach benchmark collection, in Dimension Reduction of Large-Scale Systems, P. Benner, G. Golub, V. Mehrmann, and D. Sorensen, eds., Lecture Notes in Comput. Sci. Engrg. 45, Springer-Verlag, New York, 2005.
24. 2-optimal interpolation based model reduction for large-scale systems, in Proceedings of the 16th International Conference on System Science, Wroclaw, Poland, 2007.
25. 2 approximation: A nongradient algorithm, in Proceedings of the 30th IEEE Conference on Decision and Control, 1991, pp. 2321-2323.
26. L. MEIER and D.G. LUENBERGER, Approximation of linear constant systems, IEEE Trans. Automat. Control, 12 (1967), pp. 585-588.
27. B.C. MOORE, Principal component analysis in linear system: Controllability, observability and model reduction, IEEE Trans. Automat. Control, 26 (1981), pp. 17-32.
28. C.T. MULLIS and R.A. ROBERTS, Synthesis of minimum roundoff noise fixed point digital filters, IEEE Trans. Circuits Systems, CAS-23 (1976), pp. 551-562.
29. T. PENZL, Algorithms for model reduction of large dynamical systems, Linear Algebra Appl., 415 (2006), pp. 322-343.
30. L.T. PILLAGE and R.A. ROHRER, Asymptotic waveform evaluation for timing analysis, IEEE Trans. Computer-Aided Design, 9 (1990), pp. 352-366.
31. V. RAGHAVAN, R.A. ROHRER, L.T. PILLAGE, J.Y. LEE, J.E. BRACKEN, and M.M. ALAYBEYI, AWE inspired, in Proceedings of the IEEE Custom Integrated Circuits Conference, 1993, pp. 18.1.1-18.1.8.
32. F. RIESZ AND B. SZ.-NAGY, Functional Analysis, Ungar, New York, 1955.
33. A. RUHE, Rational Krylov algorithms for nonsymmetric eigenvalue problems II: Matrix pairs, Linear Algebra Appl., 197 (1994), pp. 283-295.
34. 2 optimal model reduction, Automat., 28 (1992), pp. 897-909.
35. P. VAN DOOREN, K.A. GALLIVAN, and P.-A. ABSIL, H 2 optimal model reduction of MIMO systems, Appl. Math. Lett., to appear; doi:10.1016/j.aml.2007.09.015.
36. D.A. WILSON, Optimum solution of model reduction problem, Proc. IEE-D, 117 (1970), pp. 1161-1165.
37. 2 optimal model reduction, IEEE Trans. Automat. Control, 44 (1999), pp. 1341-1358.
38. A. YOUSOUFF and R.E. SKELTON, Covariance equivalent realizations with applications to model reduction of large-scale systems, in Control and Dynamic Systems, Vol. 22, C. T. Leondes, ed., Academic Press, New York, 1985, pp. 273-348.
39. A. YOUSOUFF, D.A. WAGIE. and R.E. SKELTON, Linear system approximation via covariance equivalent realizations, J. Math. Anal. Appl., 196 (1985), pp. 91-115.
40. D. ZIGIC, L. WATSON, AND C. BEATTIE, Contragredient transformations applied to the optimal projection equations, Linear Algebra Appl., 188/189 (1993), pp. 665-676.