Skip to main content
Log in

Model Order Reduction via Moment-Matching: A State of the Art Review

  • Review article
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

The past few decades have seen a significant spurt in developing lower-order, parsimonious models of large-scale dynamical systems used for design and control. These surrogate models effectively capture the most interesting dynamic features of the full-order models (FOMs) while preserving the input–output relation. Model order reduction (MOR) techniques have intensively been further developed to treat increasingly complex, multi-resolution models spanning a thousand degrees of freedom. This manuscript presents a state-of-the-art review of the moment-matching based order reduction methods for linear and nonlinear dynamical systems. We track the progress of moment-matching methods from their inception to how they have emerged as the most commonly adopted platform for reducing systems in large-scale settings. We discuss the frequency and time-domain notions of moment-matching between the original and reduced models. Moreover, we also provide some new results highlighting the extensive applications of this technique in reducing micro-electro-mechanical systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Availability of data and material

Not applicable.

Code availability

Not applicable.

References

  1. Akram N, Alam M, Hussain R, Ali A, Muhammad S, Malik R, Haq AU (2020) Passivity preserving model order reduction using the reduce norm method. Electronics 9(6):964

    Article  Google Scholar 

  2. Al-Baiyat SA, Bettayeb M (1993) A new model reduction scheme for k-power bilinear systems. In: Proceedings of 32nd IEEE conference on decision and control, IEEE, pp 22–27

  3. Al-Baiyat SA, Beyttayeb M, Al-Saggaf UM (1994) New model reduction scheme for bilinear systems. Int J Syst Sci 25(10):1631–1642

    Article  MathSciNet  MATH  Google Scholar 

  4. Aliaga J, Boley D, Freund R, Hernández V (2000) A Lanczos-type method for multiple starting vectors. Math Comput 69(232):1577–1601

    Article  MathSciNet  MATH  Google Scholar 

  5. Alla A, Haasdonk B, Schmidt A (2020) Feedback control of parametrized PDE’s via model order reduction and dynamic programming principle. Adv Comput Math 46(1):9

    Article  MathSciNet  MATH  Google Scholar 

  6. Anić B, Beattie C, Gugercin S, Antoulas AC (2013) Interpolatory weighted-\(\cal{H}_{2}\) model reduction. Automatica 49(5):1275–1280

    Article  MathSciNet  MATH  Google Scholar 

  7. Antoulas AC (2005) Approximation of large-scale dynamical systems. SIAM, Philadelphia

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Antoulas AC (2005) An overview of approximation methods for large-scale dynamical systems. Annu Rev Control 29(2):181–190

    Article  Google Scholar 

  10. Antoulas AC, Sorensen DC, Gugercin S (2001) A survey of model reduction methods for large-scale systems. ContempMath 280:193–219

    MathSciNet  MATH  Google Scholar 

  11. Antoulas AC, Beattie CA, Gugercin S (2010) Interpolatory model reduction of large-scale dynamical systems. In: Efficient modeling and control of large-scale systems. Springer, pp 3–58

  12. Antoulas AC, Beattie CA, Gugercin S (2020) Interpolatory methods for model reduction. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  13. Arnoldi WE (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q Appl Math 9(1):17–29

    Article  MathSciNet  MATH  Google Scholar 

  14. Astolfi A (2007) Model reduction by moment matching. In: 7th IFAC symposium on nonlinear control systems. Elsevier, pp 577–584

  15. Astolfi A (2007) A new look at model reduction by moment matching for linear systems. In: 2007 46th IEEE conference on decision and control. IEEE, pp 4361–4366

  16. Astolfi A (2008) Model reduction by moment matching for nonlinear systems. In: 2008 47th IEEE conference on decision and control. IEEE, pp 4873–4878

  17. Astolfi A (2010) Model reduction by moment matching for linear and nonlinear systems. IEEE Trans Autom Control 55(10):2321–2336

    Article  MathSciNet  MATH  Google Scholar 

  18. Astolfi A (2010) Model reduction by moment matching, steady-state response and projections. In: 49th IEEE conference on decision and control (CDC). IEEE, pp 5344–5349

  19. Astrid P, Weiland S, Willcox K, Backx T (2008) Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control 53(10):2237–2251

    Article  MathSciNet  MATH  Google Scholar 

  20. Åström KJ, Wittenmark B (1994) Adaptive control, 2nd edn. Addison-Wesley Longman Publishing Company, London

    MATH  Google Scholar 

  21. Bai Z (2002) Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl Numer Math 43(1–2):9–44

    Article  MathSciNet  MATH  Google Scholar 

  22. Bai Z, Skoogh D (2006) A projection method for model reduction of bilinear dynamical systems. Linear Algebra Appl 415(2–3):406–425

    Article  MathSciNet  MATH  Google Scholar 

  23. Bai Z, Su Y (2005) Dimension reduction of large-scale second-order dynamical systems via a second-order Arnoldi method. SIAM J Sci Comput 26(5):1692–1709

    Article  MathSciNet  MATH  Google Scholar 

  24. Bai Z, Feldmann P, Freund RW (1997) Stable and passive reduced-order models based on partial Padé approximation via the Lanczos process. Numer Anal Manuscr 97(3):10

    Google Scholar 

  25. Bai Z, Slone RD, Smith WT, Ye Q (1999) Error bound for reduced system model by Padé approximation via the Lanczos process. IEEE Trans Comput Aided Des Integr Circuits Syst 18(2):133–141

    Article  Google Scholar 

  26. Barrault M, Maday Y, Nguyen NC, Patera AT (2004) An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. CR Math 339(9):667–672

    MathSciNet  MATH  Google Scholar 

  27. Bastian J, Haase J et al. (2003) Order reduction of second order systems. In: In Proceedings of 4th Mathmod, Citeseer

  28. Baur U, Benner P, Feng L (2014) Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Arch Comput Methods Eng 21(4):331–358

    Article  MathSciNet  MATH  Google Scholar 

  29. Beattie C, Gugercin S (2012) Realization-independent \(\cal{H}_{2}\)-approximation. In: 2012 IEEE 51st IEEE conference on decision and control (CDC). IEEE, pp 4953–4958

  30. Beattie CA, Gugercin S (2007) Krylov-based minimization for optimal h 2 model reduction. In: 2007 46th IEEE conference on decision and control. IEEE, pp 4385–4390

  31. Beattie CA, Gugercin S (2009) A trust region method for optimal h 2 model reduction. In: Proceedings of the 48h IEEE conference on decision and control (CDC) held jointly with 2009 28th Chinese Control conference. IEEE, pp 5370–5375

  32. Bechtold T, Rudnyi EB, Korvink JG (2004) Error indicators for fully automatic extraction of heat-transfer macromodels for mems. J Micromech Microeng 15(3):430

    Article  Google Scholar 

  33. Benner P (2004) Solving large-scale control problems. IEEE Control Syst Mag 24(1):44–59

    Article  Google Scholar 

  34. Benner P, Breiten T (2012) Interpolation-based \(\cal{H}_{2}\) model reduction of bilinear control systems. SIAM J Matrix Anal Appl 33(3):859–885

    Article  MathSciNet  MATH  Google Scholar 

  35. Benner P, Breiten T (2012) Krylov-subspace based model reduction of nonlinear circuit models using bilinear and quadratic-linear approximations. In: Progress in industrial mathematics at ECMI 2010. Springer, pp 153–159

  36. Benner P, Breiten T (2015) Two-sided projection methods for nonlinear model order reduction. SIAM J Sci Comput 37(2):B239–B260

    Article  MathSciNet  MATH  Google Scholar 

  37. Benner P, Damm T (2009) Lyapunov equations, energy functionals and model order reduction. Preprint, TU Chemnitz

  38. Benner P, Køhler M, Saak J (2011) Sparse-dense sylvester equations in \(\mathcal{H} _{2}\)-model order reduction. Technical Report MPIMD

  39. Benner P, Gugercin S, Willcox K (2015) A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev 57(4):483–531

    Article  MathSciNet  MATH  Google Scholar 

  40. Bieker K, Peitz S, Brunton SL, Kutz JN, Dellnitz M (2020) Deep model predictive flow control with limited sensor data and online learning. In: Theoretical and computational fluid dynamics, pp 1–15

  41. Boley DL (1994) Krylov space methods on state-space control models. Circuits Syst Signal Process 13(6):733–758

    Article  MathSciNet  MATH  Google Scholar 

  42. Bollhöfer M, Bodendiek A (2012) Adaptive-order rational Arnoldi method for Maxwell’s equations. In: Scientific computing in electrical engineering (Abstracts), pp 77–78

  43. Breiten T (2013) Interpolatory methods for model reduction of large-scale dynamical systems. PhD thesis, Otto-von-Guericke Universität Magdeburg

  44. Breiten T, Beattie C, Gugercin S (2015) Near-optimal frequency-weighted interpolatory model reduction. Syst Control Lett 78:8–18

    Article  MathSciNet  MATH  Google Scholar 

  45. Bunse-Gerstner A, Kubalińska D, Vossen G, Wilczek D (2010) \(\cal{H}_{2}\)-norm optimal model reduction for large scale discrete dynamical MIMO systems. J Comput Appl Math 233(5):1202–1216

    Article  MathSciNet  MATH  Google Scholar 

  46. Byrnes C, Isidori A (1989) Steady state response, separation principle and the output regulation of nonlinear systems. In: Proceedings of the 28th IEEE conference on decision and control. IEEE, pp 2247–2251

  47. Carr J (1982) Applications of center manifold theory, vol 35. Applied Mathematical Sciences, Providence

    Google Scholar 

  48. Chahlaoui Y, Lemonnier D, Vandendorpe A, Van Dooren P (2004) Second order structure preserving balanced truncation. In: Symposium on math theory of network and systems

  49. Chan J (2020) Entropy stable reduced order modeling of nonlinear conservation laws. J Comput Phys 423:109789

    Article  MathSciNet  Google Scholar 

  50. Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764

    Article  MathSciNet  MATH  Google Scholar 

  51. Chen Y (1999) Model order reduction for nonlinear systems. PhD thesis, Massachusetts Institute of Technology

  52. Chen Y, Balakrishnan V, Koh CK, Roy K (2002) Model reduction in the time-domain using Laguerre polynomials and Krylov methods. In: Proceedings 2002 design, automation and test in Europe conference and exhibition. IEEE, pp 931–935

  53. Chiprout E, Nakhla MS (1994) Asymptotic waveform evaluation. In: Asymptotic waveform evaluation. Springer, pp 15–39

  54. Cullum JK, Willoughby RA (2002) Lanczos algorithms for large symmetric eigenvalue computations: theory, vol 1. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  55. Dong N, Roychowdhury J (2003) Piecewise polynomial nonlinear model reduction. In: Proceedings 2003. Design automation conference (IEEE Cat. No. 03CH37451). IEEE, pp 484–489

  56. Druskin V, Simoncini V (2011) Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst Control Lett 60(8):546–560

    Article  MathSciNet  MATH  Google Scholar 

  57. Dziuk G, Elliott CM (2013) Finite element methods for surface PDEs. Acta Numer 22:289

    Article  MathSciNet  MATH  Google Scholar 

  58. Eid R (2009) Time domain model reduction by moment matching. PhD thesis, Technische Universität München

  59. Ern A, Guermond JL (2013) Theory and practice of finite elements, vol 159. Springer, Berlin

    MATH  Google Scholar 

  60. Faedo N, Piuma FJD, Giorgi G, Ringwood JV (2020) Nonlinear model reduction for wave energy systems: a moment-matching-based approach. Nonlinear Dyn 102(3):1215–1237

    Article  Google Scholar 

  61. Far MF, Martin F, Belahcen A, Rasilo P, Awan HAA (2020) Real-time control of an IPMSM using model order reduction. IEEE Trans Ind Electron

  62. Feldmann P, Freund RW (1995) Efficient linear circuit analysis by Padé approximation via the Lanczos process. IEEE Trans Comput Aided Des Integr Circuits Syst 14(5):639–649

    Article  Google Scholar 

  63. Feng L, Benner P (2012) Automatic model order reduction by moment-matching according to an efficient output error bound. In: Scientific computing in electrical engineering (Abstracts), pp 71–72

  64. Feng L, Zeng X, Chiang C, Zhou D, Fang Q (2004) Direct nonlinear order reduction with variational analysis. In: Proceedings design, automation and test in Europe conference and exhibition, vol 2. IEEE, pp 1316–1321

  65. Feng L, Benner P, Korvink JG (2013) System-level modeling of mems by means of model order reduction (mathematical approximations)—mathematical background. System-Level Modeling of MEMS, pp 53–93

  66. Feng L, Korvink JG, Benner P (2015) A fully adaptive scheme for model order reduction based on moment matching. IEEE Trans Compon Packag Manuf Technol 5(12):1872–1884

    Article  Google Scholar 

  67. Flagg G, Beattie C, Gugercin S (2012) Convergence of the iterative rational Krylov algorithm. Syst Control Lett 61(6):688–691

    Article  MathSciNet  MATH  Google Scholar 

  68. Freund RW (1999) Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation. In: Applied and computational control, signals, and circuits. Springer, pp 435–498

  69. Freund RW (2000) Krylov-subspace methods for reduced-order modeling in circuit simulation. J Comput Appl Math 123(1–2):395–421

    Article  MathSciNet  MATH  Google Scholar 

  70. Freund RW (2000) Passive reduced-order modeling via Krylov-subspace methods. In: CACSD. conference proceedings. IEEE international symposium on computer-aided control system design (Cat. No. 00TH8537). IEEE, pp 261–266

  71. Freund RW (2003) Model reduction methods based on Krylov subspaces. Acta Numer 12:267–319

    Article  MathSciNet  MATH  Google Scholar 

  72. Freund RW (2004) SPRIM: structure-preserving reduced-order interconnect macromodeling. In: IEEE/ACM international conference on computer aided design, 2004. ICCAD-2004. IEEE, pp 80–87

  73. Fujimoto K (2008) Balanced realization and model order reduction for port-Hamiltonian systems. J Syst Des Dyn 2(3):694–702

    Google Scholar 

  74. Fujimoto K, Scherpen JM (2005) Nonlinear input-normal realizations based on the differential eigenstructure of Hankel operators. IEEE Trans Autom Control 50(1):2–18

    Article  MathSciNet  MATH  Google Scholar 

  75. Fujimoto K, Scherpen JM (2010) Balanced realization and model order reduction for nonlinear systems based on singular value analysis. SIAM J Control Optim 48(7):4591–4623

    Article  MathSciNet  MATH  Google Scholar 

  76. Gallivan K, Grimme E, Dooren PV (1994) Asymptotic waveform evaluation via a Lanczos method. Appl Math Lett 7(5):75–80

    Article  MathSciNet  MATH  Google Scholar 

  77. Gallivan K, Vandendorpe A, Van Dooren P (2004) Model reduction of MIMO systems via tangential interpolation. SIAM J Matrix Anal Appl 26(2):328–349

    Article  MathSciNet  MATH  Google Scholar 

  78. Gallivan K, Vandendorpe A, Van Dooren P (2004) Sylvester equations and projection-based model reduction. J Comput Appl Math 162(1):213–229

    Article  MathSciNet  MATH  Google Scholar 

  79. Gallivan K, Vandendorpe A, Van Dooren P (2006) Model reduction and the solution of Sylvester equations. MTNS, Kyoto, p 50

    MATH  Google Scholar 

  80. Goyal PK (2018) System-theoretic model order reduction for bilinear and quadratic-bilinear systems. PhD thesis, Universitätsbibliothek

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

    Article  MathSciNet  MATH  Google Scholar 

  82. Gray WS, Mesko J (1997) General input balancing and model reduction for linear and nonlinear systems. In: 1997 European control conference (ECC). IEEE, pp 2862–2867

  83. Grepl MA, Maday Y, Nguyen NC, Patera AT (2007) Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM Math Model Numer Anal 41(3):575–605

    Article  MathSciNet  MATH  Google Scholar 

  84. Grimme E (1997) Krylov projection methods for model reduction. PhD thesis, University of Illinois at Urbana Champaign

  85. Grimme EJ, Sorensen DC, Van Dooren P (1996) Model reduction of state space systems via an implicitly restarted Lanczos method. Numer Algorithms 12(1):1–31

    Article  MathSciNet  MATH  Google Scholar 

  86. Gu C (2009) QLMOR: a new projection-based approach for nonlinear model order reduction. In: 2009 IEEE/ACM international conference on computer-aided design-digest of technical papers. IEEE, pp 389–396

  87. Gu C (2011) QLMOR: a projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems. IEEE Trans Comput Aided Des Integr Circuits Syst 30(9):1307–1320

    Article  Google Scholar 

  88. Gugercin S (2005) An iterative rational Krylov algorithm (IRKA) for optimal \(\mathcal{H}_{2}\) model reduction. In: Householder symposium XVI, Seven Springs Mountain Resort, PA, USA

  89. Gugercin S, Beattie C, Antoulas A (2006) Rational krylov methods for optimal \(\mathcal{H}_{2}\) model reduction. submitted for publication

  90. Gugercin S, Antoulas AC, Beattie C (2008) \(\mathcal{H}_{2}\) model reduction for large-scale linear dynamical systems. SIAM J Matrix Anal Appl 30(2):609–638

    Article  MathSciNet  MATH  Google Scholar 

  91. Gugercin S, Stykel T, Wyatt S (2013) Model reduction of descriptor systems by interpolatory projection methods. SIAM J Sci Comput 35:1010–1033

    Article  MathSciNet  MATH  Google Scholar 

  92. Gunupudi PK, Nakhla MS (1999) Model-reduction of nonlinear circuits using Krylov-space techniques. In: Proceedings of the 36th annual ACM/IEEE design automation conference, pp 13–16

  93. Halevi Y (1990) Frequency weighted model reduction via optimal projection. In: 29th IEEE conference on decision and control. IEEE, pp 2906–2911

  94. Hinze M, Volkwein S (2005) Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Dimension reduction of large-scale systems. Springer, pp 261–306

  95. Hochman A, Vasilyev DM, Rewienski MJ, White JK (2013) Projection-based nonlinear model order reduction. In: System-level modeling of MEMS, advanced micro and nanosystems. Wiley-VCH

  96. Ionescu TC, Astolfi A (2013) Families of reduced order models that achieve nonlinear moment matching. In: 2013 American control conference. IEEE, pp 5518–5523

  97. Ionescu TC, Astolfi A (2015) Nonlinear moment matching-based model order reduction. IEEE Trans Autom Control 61(10):2837–2847

    Article  MathSciNet  MATH  Google Scholar 

  98. Ionescu TC, Astolfi A, Colaneri P (2014) Families of moment matching based, low order approximations for linear systems. Syst Control Lett 64:47–56

    Article  MathSciNet  MATH  Google Scholar 

  99. Isidori A (1995) Nonlinear control systems, 3rd edn. Springer, Berlin

    Book  MATH  Google Scholar 

  100. Isidori A, Byrnes CI (1990) Output regulation of nonlinear systems. IEEE Trans Autom Control 35(2):131–140

    Article  MathSciNet  MATH  Google Scholar 

  101. Jaimoukha IM, Kasenally EM (1995) Oblique production methods for large scale model reduction. SIAM J Matrix Anal Appl 16(2):602–627

    Article  MathSciNet  MATH  Google Scholar 

  102. Jaimoukha IM, Kasenally EM (1997) Implicitly restarted Krylov subspace methods for stable partial realizations. SIAM J Matrix Anal Appl 18(3):633–652

    Article  MathSciNet  MATH  Google Scholar 

  103. Kaczynski J, Ranacher C, Fleury C (2020) Computationally efficient model for viscous damping in perforated mems structures. Sens Actuators A 314:112201

    Article  Google Scholar 

  104. Karatzas EN, Ballarin F, Rozza G (2020) Projection-based reduced order models for a cut finite element method in parametrized domains. Comput Math Appl 79(3):833–851

    Article  MathSciNet  MATH  Google Scholar 

  105. Kellems AR, Roos D, Xiao N, Cox SJ (2009) Low-dimensional, morphologically accurate models of subthreshold membrane potential. J Comput Neurosci 27(2):161

    Article  MathSciNet  Google Scholar 

  106. Kerns KJ, Yang AT (1998) Preservation of passivity during RLC network reduction via split congruence transformations. IEEE Trans Comput Aided Des Integr Circuits Syst 17(7):582–591

    Article  Google Scholar 

  107. Kim D, Bae Y, Yun S, Braun JE (2020) A methodology for generating reduced-order models for large-scale buildings using the Krylov subspace method. J Build Perform Simul 13(4):419–429

    Article  Google Scholar 

  108. Kim HM, Craig RR Jr (1988) Structural dynamics analysis using an unsymmetric block Lanczos algorithm. Int J Numer Methods Eng 26(10):2305–2318

    Article  MATH  Google Scholar 

  109. Kim HM, Craig RR Jr (1990) Computational enhancement of an unsymmetric block Lanczos algorithm. Int J Numer Methods Eng 30(5):1083–1089

    Article  MathSciNet  MATH  Google Scholar 

  110. Konkel Y, Farle O, Sommer A, Burgard S, Dyczij-Edlinger R (2014) A posteriori error bounds for Krylov-based fast frequency sweeps of finite-element systems. IEEE Trans Magn 50(2):441–444

    Article  Google Scholar 

  111. Krajewski W, Lepschy A, Redivo-Zaglia M, Viaro U (1995) A program for solving the l 2 reduced-order model problem with fixed denominator degree. Numer Algorithms 9(2):355–377

    Article  MathSciNet  MATH  Google Scholar 

  112. Krener AJ (1992) The construction of optimal linear and nonlinear regulators. In: Systems, models and feedback: theory and applications. Springer, pp 301–322

  113. Kudryavtsev M, Rudnyi EB, Korvink JG, Hohlfeld D, Bechtold T (2015) Computationally efficient and stable order reduction methods for a large-scale model of mems piezoelectric energy harvester. Microelectron Reliab 55(5):747–757

    Article  Google Scholar 

  114. Kunisch K, Volkwein S (1999) Control of the burgers equation by a reduced-order approach using proper orthogonal decomposition. J Optim Theory Appl 102(2):345–371

    Article  MathSciNet  MATH  Google Scholar 

  115. Kunisch K, Volkwein S (2008) Proper orthogonal decomposition for optimality systems. ESAIM Math Modell Numer Anal 42(1):1–23

    Article  MathSciNet  MATH  Google Scholar 

  116. Lall S, Krysl P, Marsden JE (2003) Structure-preserving model reduction for mechanical systems. Physica D 184(1–4):304–318

    Article  MathSciNet  MATH  Google Scholar 

  117. Lanczos C (1950) An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. United States Government Press Office, Los Angeles

    Book  MATH  Google Scholar 

  118. Lee HJ, Chu CC, Feng WS (2006) An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems. Linear Algebra Appl 415(2–3):235–261

    Article  MathSciNet  MATH  Google Scholar 

  119. Li H, Song Z, Zhang F (2020) A reduced-order modified finite difference method preserving unconditional energy-stability for the Allen–Cahn equation. Numer Methods Part Differ Equ 37:1869–1885

    Article  MathSciNet  Google Scholar 

  120. Li RC, Bai Z et al (2005) Structure-preserving model reduction using a Krylov subspace projection formulation. Commun Math Sci 3(2):179–199

    Article  MathSciNet  MATH  Google Scholar 

  121. Liu CC, Wang CC (2014) Numerical investigation into nonlinear dynamic behavior of electrically-actuated clamped-clamped micro-beam with squeeze-film damping effect. Appl Math Model 38(13):3269–3280

    Article  MathSciNet  Google Scholar 

  122. Ljung L (1999) System Identification: Theory for the User, Information and System Sciences Series, 2nd edn. Prentice Hall, Upper Saddle River

    Google Scholar 

  123. Lohmann B, Salimbahrami B (2004) Order reduction using Krylov subspace methods. Autom Technol 52(1):30–38

    MATH  Google Scholar 

  124. Lohmann B, Salimbahrami B (2005) Reduction of second order systems using second order Krylov subspaces. IFAC Proc Vol 38(1):614–619

    Article  MATH  Google Scholar 

  125. Maloberti F (2006) Analog design for CMOS VLSI systems, vol 646. Springer, Berlin

    Google Scholar 

  126. Martone R, Formisano A, Condon M, Ivanov R (2007) Krylov subspaces from bilinear representations of nonlinear systems. COMPEL Int J Comput Math Electr Electron Eng

  127. Mazumder S (2015) Numerical methods for partial differential equations: finite difference and finite methods. Academic Press, London

    MATH  Google Scholar 

  128. Meier L, Luenberger D (1967) Approximation of linear constant systems. IEEE Trans Autom Control 12(5):585–588

    Article  Google Scholar 

  129. Mendible A, Brunton SL, Aravkin AY, Lowrie W, Kutz JN (2020) Dimensionality reduction and reduced-order modeling for traveling wave physics. Theor Comput Fluid Dyn 34(4):385–400

    Article  MathSciNet  Google Scholar 

  130. Meyer DG, Srinivasan S (1996) Balancing and model reduction for second-order form linear systems. IEEE Trans Autom Control 41(11):1632–1644

    Article  MathSciNet  MATH  Google Scholar 

  131. Mohamed K (2019) Model order reduction method for large-scale RC interconnect and implementation of adaptive digital PI controller. IEEE Trans Very Large Scale Integr Syst 27(10):2447–2458

    Article  Google Scholar 

  132. Nayfeh AH, Younis MI, Abdel-Rahman EM (2005) Reduced-order models for mems applications. Nonlinear Dyn 41(1–3):211–236

    Article  MathSciNet  MATH  Google Scholar 

  133. Necoara I, Ionescu TC (2020) \(\cal{H}_{2}\) model reduction of linear network systems by moment matching and optimization. IEEE Trans Autom Control 65(12):5328–5335

    Article  MATH  MathSciNet  Google Scholar 

  134. Nguyen NC, Patera AT, Peraire J (2008) A ‘best points’ interpolation method for efficient approximation of parametrized functions. Int J Numer Methods Eng 73(4):521–543

    Article  MathSciNet  MATH  Google Scholar 

  135. Nguyen VB, Tran SBQ, Khan SA, Rong J, Lou J (2020) POD-DEIM model order reduction technique for model predictive control in continuous chemical processing. Comput Chem Eng 133:106638

    Article  Google Scholar 

  136. Nour-Omid B, Clough RW (1984) Dynamic analysis of structures using Lanczos co-ordinates. Earthq Eng Struct Dyn 12(4):565–577

    Article  Google Scholar 

  137. Odabasioglu A, Celik M, Pileggi LT (2003) PRIMA: passive reduced-order interconnect macromodeling algorithm. In: The best of ICCAD. Springer, pp 433–450

  138. Ouakad HM, Al-Qahtani HM, Hawwa MA (2016) Influence of squeeze-film damping on the dynamic behavior of a curved micro-beam. Adv Mech Eng 8(6):1687814016650120

    Article  Google Scholar 

  139. Benner MHP, Termaten E (2001) Model reduction for circuit simulation. Lecture Notes in Electrical Engineering, vol 74. Springer, Cham

    Google Scholar 

  140. Panzer HK, Jaensch S, Wolf T, Lohmann B (2013) A greedy rational Krylov method for \(\cal{H}_{2}\) pseudooptimal model order reduction with preservation of stability. In: 2013 American control conference. IEEE, pp 5512–5517

  141. Panzer HK, Wolf T, Lohmann B (2013) \(\cal{H}_{2}\) and \(\cal{H}_{\infty }\) error bounds for model order reduction of second order systems by Krylov subspace methods. In: 2013 European control conference (ECC). IEEE, pp 4484–4489

  142. Pelesko JA, Bernstein DH (2002) Modeling MEMS and NEMS. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  143. Phillips JR (2000) Projection frameworks for model reduction of weakly nonlinear systems. In: Proceedings of the 37th annual design automation conference, pp 184–189

  144. Phillips JR (2003) Projection-based approaches for model reduction of weakly nonlinear, time-varying systems. IEEE Trans Comput Aided Des Integr Circuits Syst 22(2):171–187

    Article  Google Scholar 

  145. Pillage LT, Rohrer RA (1990) Asymptotic waveform evaluation for timing analysis. IEEE Trans Comput Aided Des Integr Circuits Syst 9(4):352–366

    Article  Google Scholar 

  146. Pillai AG, Samuel ER (2020) Minimal realized power systems for load frequency control using optimal theory based PID controller. IETE J Res 1–13

  147. Proctor JL, Eckhoff PA (2015) Discovering dynamic patterns from infectious disease data using dynamic mode decomposition. Int Health 7(2):139–145

    Article  Google Scholar 

  148. Rafiq D, Bazaz MA (2019) A comprehensive scheme for fast simulation of Burgers’ equation. In: 2019 Sixth Indian control conference (ICC). IEEE, pp 397–402

  149. Rafiq D, Bazaz MA (2019) Model order reduction of non-linear transmission lines using non-linear moment matching. In: 2019 International conference on computing. Power and communication technologies (GUCON). IEEE, pp 394–399

  150. Rafiq D, Bazaz MA (2020) A comprehensive scheme for reduction of nonlinear dynamical systems. Int J Dyn Control 8(2):361–369

    Article  MathSciNet  Google Scholar 

  151. Rafiq D, Bazaz MA (2020) A framework for parametric reduction in large-scale nonlinear dynamical systems. Nonlinear Dyn 102(3):1897–1908

    Article  Google Scholar 

  152. Rafiq D, Bazaz MA (2020) Nonlinear model order reduction via nonlinear moment matching with dynamic mode decomposition. Int J Non-Linear Mech 128:103625

    Article  Google Scholar 

  153. Rewienski M, White J (2003) A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Trans Comput Aided Des Integr Circuits Syst 22(2):155–170

    Article  Google Scholar 

  154. Rewieński M, White J (2006) Model order reduction for nonlinear dynamical systems based on trajectory piecewise-linear approximations. Linear Algebra Appl 415(2–3):426–454

    Article  MathSciNet  MATH  Google Scholar 

  155. Saini P, Dixit A, Gupta A, Sharma K (2020) Modeling and control of load frequency control (LFC) using model order reduction (MOR) techniques. Math Eng Sci Aerospace 11(1):1–11

    Google Scholar 

  156. Salimbahrami B (2005) Structure preserving order reduction of large scale second order models. PhD thesis, Technical University of Munich, Germany

  157. Salimbahrami B, Lohmann B (2006) Order reduction of large scale second-order systems using Krylov subspace methods. Linear Algebra Appl 415(2–3):385–405

    Article  MathSciNet  MATH  Google Scholar 

  158. Salimbahrami SB (2005) Structure preserving order reduction of large scale second order models. PhD thesis, Technische Universität München

  159. Sastry S (2013) Nonlinear systems: analysis, stability, and control, vol 10. Springer, Berlin

    Google Scholar 

  160. Scarciotti G, Astolfi A (2015) Model reduction of neutral linear and nonlinear time-invariant time-delay systems with discrete and distributed delays. IEEE Trans Autom Control 61(6):1438–1451

    Article  MathSciNet  MATH  Google Scholar 

  161. Scarciotti G, Astolfi A (2017) Data-driven model reduction by moment matching for linear and nonlinear systems. Automatica 79:340–351

    Article  MathSciNet  MATH  Google Scholar 

  162. Scarciotti G, Astolfi A (2017) Nonlinear model reduction by moment matching. Found Trends Syst Control 4(3–4):224–409

    Article  MATH  Google Scholar 

  163. Scherpen JM, Gray WS (2000) Minimality and local state decompositions of a nonlinear state space realization using energy functions. IEEE Trans Autom Control 45(11):2079–2086

    Article  MathSciNet  MATH  Google Scholar 

  164. Scherpen JMA (1993) Balancing for nonlinear systems. Syst Control Lett 21(2):143–153

    Article  MathSciNet  MATH  Google Scholar 

  165. Scherpen JMA, Van der Schaft A (1994) Normalized coprime factorizations and balancing for unstable nonlinear systems. Int J Control 60(6):1193–1222

    Article  MathSciNet  MATH  Google Scholar 

  166. Sjöberg J, Fujimoto K, Glad T (2007) Model reduction of nonlinear differential-algebraic equations. IFAC Proc Vol 40(12):176–181

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  168. Spanos JT, Milman MH, Mingori DL (1992) A new algorithm for l2 optimal model reduction. Automatica 28(5):897–909

    Article  MathSciNet  MATH  Google Scholar 

  169. Su TJ, Craig RR Jr (1991) Model reduction and control of flexible structures using Krylov vectors. J Guid Control Dyn 14(2):260–267

    Article  Google Scholar 

  170. Taira K, Hemati MS, Brunton SL, Sun Y, Duraisamy K, Bagheri S, Dawson ST, Yeh CA (2020) Modal analysis of fluid flows: applications and outlook. AIAA J 58(3):998–1022

    Article  Google Scholar 

  171. Thomas D, Fabien C, Nissrine A, Ryckelynck D (2020) Model order reduction assisted by deep neural networks (rom-net). Adv Model Simul Eng Sci 7(1):1–27

    Google Scholar 

  172. Tian X, Sheng W, Tian F, Lu Y, Wang L (2020) Simulation study on squeeze film air damping. Micro Nano Lett 15(9):576–581

    Article  Google Scholar 

  173. Uyemura JP (2002) Introduction to VLSI circuits and systems. Wiley, Delhi

    Google Scholar 

  174. Van Dooren P (1992) Numerical linear algebra techniques for large scale matrix problems in systems and control. In: Proceedings of the 31st IEEE conference on decision and control. IEEE, pp 1933–1938

  175. Van Dooren P, Gallivan KA, Absil PA (2008) \(\cal{H}_{2}\)-optimal model reduction of mimo systems. Appl Math Lett 21(12):1267–1273

    Article  MathSciNet  MATH  Google Scholar 

  176. Varona MC, Nico S, Lohmann B (2019) Nonlinear moment matching for the simulation-free reduction of structural systems. In: IFAC Mechatronics and NolCoS, Vienna, Austria, IFAC, vol 52, pp 328–333

  177. Vasilyev D, Rewienski M, White J (2003) A TBR-based trajectory piecewise-linear algorithm for generating accurate low-order models for nonlinear analog circuits and mems. In: Proceedings 2003. Design automation conference (IEEE Cat. No. 03CH37451). IEEE, pp 490–495

  178. Vassilevski Y, Terekhov K, Nikitin K, Kapyrin I (2020) Parallel finite volume computation on general meshes. Springer, Berlin

  179. Wang JM, Kuh ES, Yu Q (2000) Passive model order reduction algorithm based on Chebyshev expansion of impulse response of interconnect networks. In: Design automation conference. IEEE Computer Society, pp 520–525

  180. Wang JM, Chu CC, Yu Q, Kuh ES (2002) On projection-based algorithms for model-order reduction of interconnects. IEEE Trans Circuits Syst I Fundam Theory Appl 49(11):1563–1585

    Article  MathSciNet  MATH  Google Scholar 

  181. Willcox K, Peraire J (2002) Balanced model reduction via the proper orthogonal decomposition. AIAA J 40(11):2323–2330

    Article  Google Scholar 

  182. Wolf T, Panzer H, Lohmann B (2011) Gramian-based error bound in model reduction by Krylov subspace methods. IFAC Proc Vol 44(1):3587–3592

    Article  Google Scholar 

  183. Yan WY, Lam J (1999) An approximate approach to \(\cal{H}_{2}\) optimal model reduction. IEEE Trans Autom Control 44(7):1341–1358

    MathSciNet  MATH  Google Scholar 

  184. Younis MI, Abdel-Rahman EM, Nayfeh A (2003) A reduced-order model for electrically actuated microbeam-based MEMS. J Microelectromech Syst 12(5):672–680

    Article  Google Scholar 

  185. Zamanzadeh M, Jafarsadeghi-Pournaki I, Ouakad HM (2020) A resonant pressure mems sensor based on levitation force excitation detection. Nonlinear Dyn 100:1105–1123

    Article  Google Scholar 

  186. Žigić D, Watson LT, Beattie C (1993) Contragredient transformations applied to the optimal projection equations. Linear Algebra Appl 188:665–676

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The Doctoral fellowship in favor of Danish Rafiq from Ministry of Human Resource Development (MHRD) India, via Grant No. 2017PHAELE006 is duly acknowledged.

Author information

Authors and Affiliations

Authors

Contributions

Danish Rafiq collected the data and wrote the manuscript. M. A. Bazaz helped in critical analysis and proof reading of the manuscript.

Corresponding author

Correspondence to Danish Rafiq.

Ethics declarations

Conflicts of interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rafiq, D., Bazaz, M.A. Model Order Reduction via Moment-Matching: A State of the Art Review. Arch Computat Methods Eng 29, 1463–1483 (2022). https://doi.org/10.1007/s11831-021-09618-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-021-09618-2

Navigation