A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control

In this paper, popular model reduction techniques from the fields of structural dynamics, numerical mathematics and systems and control are reviewed and compared. The motivation for such a comparison stems from the fact that the model reduction techniques in these fields have been developed fairly independently. In addition, the insight obtained by the comparison allows for making a motivated choice for a particular model reduction technique, on the basis of the desired objectives and properties of the model reduction problem. In particular, a detailed review is given on mode displacement techniques, moment matching methods and balanced truncation, whereas important extensions are outlined briefly. In addition, a qualitative comparison of these methods is presented, hereby focusing both on theoretical and computational aspects. Finally, the differences are illustrated on a quantitative level by means of application of the model reduction techniques to a common example.

[1]  Clifford T. Mullis,et al.  Synthesis of minimum roundoff noise fixed point digital filters , 1976 .

[2]  G. Kerschen,et al.  The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview , 2005 .

[3]  Peter Benner,et al.  Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems , 2008, Numer. Linear Algebra Appl..

[4]  E. Wilson,et al.  Dynamic analysis by direct superposition of Ritz vectors , 1982 .

[5]  Jr. Roy Craig,et al.  Coupling of substructures for dynamic analyses - An overview , 2000 .

[6]  Manolis Papadrakakis.  Solving Large-scale Problems in Mechanics , 1993 .

[7]  Paul Van Dooren,et al.  Model Reduction of Interconnected Systems , 2008 .

[8]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .

[9]  D. Enns.  Model reduction with balanced realizations: An error bound and a frequency weighted generalization , 1984, The 23rd IEEE Conference on Decision and Control.

[10]  P. Dooren,et al.  Model reduction of second order systems , 2005 .

[11]  A. G. Madievski,et al.  Error bound for transfer function order reduction using frequency weighted balanced truncation , 1995 .

[12]  H. P. Lee,et al.  PROPER ORTHOGONAL DECOMPOSITION AND ITS APPLICATIONS—PART I: THEORY , 2002 .

[13]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[14]  R. Ober.  Balanced parametrization of classes of linear systems , 1991 .

[15]  Roland W. Freund,et al.  Efficient linear circuit analysis by Pade´ approximation via the Lanczos process , 1994, EURO-DAC '94.

[16]  D. Rixen,et al.  General Framework for Dynamic Substructuring: History, Review and Classification of Techniques , 2008 .

[17]  Eric James Grimme,et al.  Krylov Projection Methods for Model Reduction , 1997 .

[18]  Duncan McFarlane,et al.  Balanced canonical forms for minimal systems: A normalized coprime factor approach , 1989 .

[19]  K. Fernando,et al.  Singular perturbational model reduction of balanced systems , 1982 .

[20]  Roland W. Freund,et al.  SPRIM: structure-preserving reduced-order interconnect macromodeling , 2004, IEEE/ACM International Conference on Computer Aided Design, 2004. ICCAD-2004..

[21]  L. W.,et al.  The Theory of Sound , 1898, Nature.

[22]  Zhong-Sheng Liu,et al.  An accurate modal method for computing response to periodic excitation , 1997 .

[23]  U. Desai,et al.  A transformation approach to stochastic model reduction , 1984 .

[24]  You-qun Zhao,et al.  An improved modal truncation method for responses to harmonic excitation , 2002 .

[25]  S. Rubin.  Improved Component-Mode Representation for Structural Dynamic Analysis , 1975 .

[26]  Daniel J. Rixen,et al.  Dual Craig-Bampton with enrichment to avoid spurious modes , 2008 .

[27]  Andrew J. Kurdila,et al.  『Fundamentals of Structural Dynamics』(私の一冊) , 2019, Journal of the Society of Mechanical Engineers.

[28]  Peter M. Pinsky,et al.  Matrix‐Padé via Lanczos solutions for vibrations of fluid–structure interaction , 2010 .

[29]  Danny C. Sorensen,et al.  A Modified Low-Rank Smith Method for Large-Scale Lyapunov Equations , 2004, Numerical Algorithms.

[30]  Athanasios C. Antoulas,et al.  An overview of approximation methods for large-scale dynamical systems , 2005, Annu. Rev. Control..

[31]  Boris Lohmann,et al.  Introduction to Krylov Subspace Methods in Model Order Reduction , 2004 .

[32]  J. M. A. Scherpen,et al.  Balancing for nonlinear systems , 1993 .

[33]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[34]  R. Freund.  Model reduction methods based on Krylov subspaces , 2003, Acta Numerica.

[35]  L. Sirovich.  Turbulence and the dynamics of coherent structures. II. Symmetries and transformations , 1987 .

[36]  Y. T. Leung,et al.  Fast response method for undamped structures , 1983 .

[37]  John Dickens,et al.  Modal truncation vectors for reduced dynamic substructure models , 2000 .

[38]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[39]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[40]  Jacob K. White,et al.  A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[41]  W. Hurty.  Dynamic Analysis of Structural Systems Using Component Modes , 1965 .

[42]  Richard B. Lehoucq,et al.  An Automated Multilevel Substructuring Method for Eigenspace Computation in Linear Elastodynamics , 2004, SIAM J. Sci. Comput..

[43]  Roland W. Freund,et al.  Reduced-order modeling of large passive linear circuits by means of the SyPVL algorithm , 1996, Proceedings of International Conference on Computer Aided Design.

[44]  H. V. D. Vorst,et al.  Model Order Reduction: Theory, Research Aspects and Applications , 2008 .

[45]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

[46]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[47]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[48]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[49]  Daniel Rixen,et al.  Generalized mode acceleration methods and modal truncation augmentation , 2001 .

[50]  Lawrence T. Pileggi,et al.  PRIMA: passive reduced-order interconnect macromodeling algorithm , 1998, 1997 Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[51]  J. Béliveau.  Introduction to finite element vibration analysis , 1992 .

[52]  R. Macneal.  A hybrid method of component mode synthesis , 1971 .

[53]  Daniel Rixen.  A Lanczos Procedure for Efficient Mode Superposition in Dynamic Analysis , 2002 .

[54]  R. Freund.  Krylov-subspace methods for reduced-order modeling in circuit simulation , 2000 .

[55]  Lawrence T. Pileggi,et al.  Asymptotic waveform evaluation for timing analysis , 1990, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[56]  Brian D. O. Anderson,et al.  Singular perturbation approximation of balanced systems , 1989 .

[57]  Mi-Ching Tsai,et al.  Robust and Optimal Control , 2014 .

[58]  M. J. Wittbrodt,et al.  A critique of mode acceleration and modal truncation augmentation methods for modal response analysis , 1997 .

[59]  Zhaojun Bai,et al.  A Unified Krylov Projection Framework for Structure-Preserving Model Reduction , 2008 .

[60]  Walter C. Hurty,et al.  Vibrations of Structural Systems by Component Mode Synthesis , 1960 .

[61]  Paul Van Dooren,et al.  Model Reduction of Large-Scale Systems Rational Krylov Versus Balancing Techniques , 1999 .

[62]  PJ Pieter Heres,et al.  Robust and Efficient Krylov Subspace Methods for Model Order Reduction , 2005 .

[63]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[64]  M. Géradin,et al.  Mechanical Vibrations: Theory and Application to Structural Dynamics , 1994 .

[65]  D. Meyer.  A Fractional Approach to Model Reduction , 1988, 1988 American Control Conference.

[66]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[67]  M. Akgün,et al.  A New Family Of Mode-Superposition Methods For Response Calculations , 1993 .