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A comprehensive scheme for reduction of nonlinear dynamical systems

  • Danish RafiqEmail author
  • Mohammad Abid Bazaz
Article
  • 26 Downloads

Abstract

Model order reduction (MOR) also known as dimension reduction is a computational tool to obtain cost-effective lower order approximations of large scale dynamical systems. This paper presents a detailed yet simplified MOR approach using nonlinear moment matching (NLMM) in conjuncture with the Discrete Empirical Interpolation Method (DEIM). NLMM avoids the expensive simulation of the underlying nonlinear Sylvester partial differential equation by reducing it to a system of nonlinear algebraic equations using proper step-by-step simplifications. This reduces the offline computational cost of generating the orthonormal projection basis substantially. This is followed by the DEIM algorithm, resulting in comprehensive savings in computational resources. The proposed algorithms are tested on two benchmark problems and the results so obtained are compared with proper orthogonal decomposition for different test inputs.

Keywords

Model order reduction Nonlinear moment matching Steady state response 

Notes

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest regarding the publication of this paper.

References

  1. 1.
    Antoulas A C (2005) Approximation of large-scale dynamical systems, vol 6. SIAM, PhiladelphiaCrossRefGoogle Scholar
  2. 2.
    Benner P, Mehrmann V, Sorensen D C (2005) Dimension reduction of large-scale systems, vol 45. Springer, BerlinCrossRefGoogle Scholar
  3. 3.
    Moore B (1981) Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans Autom Control 26(1):17–32MathSciNetCrossRefGoogle Scholar
  4. 4.
    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 Circ Syst 22(2):155–170CrossRefGoogle Scholar
  5. 5.
    Rewienski M, White J (2002) Improving trajectory piecewise-linear approach to nonlinear model order reduction for micromachined devices using an aggregated projection basisGoogle Scholar
  6. 6.
    Lall S, Marsden JE, Glavaški S (2002) A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int J Robust Nonlinear Control IFAC Affil J 12(6):519–535MathSciNetCrossRefGoogle Scholar
  7. 7.
    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):2237–2251MathSciNetCrossRefGoogle Scholar
  8. 8.
    Willcox K, Peraire J (2002) Balanced model reduction via the proper orthogonal decomposition. AIAA J 40(11):2323–2330CrossRefGoogle Scholar
  9. 9.
    Beattie CA, Gugercin S (2014) Model reduction by rational interpolation. In: Benner P, Cohen A, Ohlberger M, Willcox K (eds) Model reduction and algorithms: theory and applications. Computational science and engineering, vol 15. SIAM, Philadelphia, pp 297–334 Google Scholar
  10. 10.
    Astolfi A (2010) Model reduction by moment matching for linear and nonlinear systems. IEEE Trans Autom Control 55(10):2321–2336MathSciNetCrossRefGoogle Scholar
  11. 11.
    Astolfi A (2010) Model reduction by moment matching, steady-state response and projections. In: 49th IEEE conference on decision and control (CDC). IEEE, pp 5344–5349Google Scholar
  12. 12.
    Ionescu TC, Astolfi A (2013) Families of reduced order models that achieve nonlinear moment matching. In: 2013 American control conference. IEEE, pp 5518–5523Google Scholar
  13. 13.
    Ionescu T, Astolfi A (2016) Nonlinear moment matching-based model order reduction. IEEE Trans Autom Control 61(10):2837–2847MathSciNetCrossRefGoogle Scholar
  14. 14.
    Scarciotti A (2016) Model reduction of neutral linear and nonlinear time-invariant time-delay systems with discrete and distributed delays. IEEE Trans Autom Control 61(6):1438–1451MathSciNetCrossRefGoogle Scholar
  15. 15.
    Giordano Scarciotti A (2017) Data-driven model reduction by moment matching for linear and nonlinear systems. Automatica 79:340–351MathSciNetCrossRefGoogle Scholar
  16. 16.
    Isidori A (2013) Nonlinear control systems. Springer, BerlinzbMATHGoogle Scholar
  17. 17.
    Krener AJ (1992) The construction of optimal linear and nonlinear regulators. In: Isidori A, Tarn TJ (eds) Systems, models and feedback: theory and applications. Springer, Berlin, pp 301–322 CrossRefGoogle Scholar
  18. 18.
    Huang J (2004) Nonlinear output regulation: theory and applications, vol 8. SIAM, PhiladelphiaCrossRefGoogle Scholar
  19. 19.
    Cruz Verona L, Gebhart R Practical simulation-free model order reduction by nonlinear moment matching. arXiv:1901.10750
  20. 20.
    Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764MathSciNetCrossRefGoogle Scholar
  21. 21.
    Grimme E (1997) Krylov projection methods for model reduction. Ph.D. dissertation, University of Illinois at Urbana ChampaignGoogle Scholar
  22. 22.
    Ames W F (1965) Nonlinear partial differential equations in engineering, vol 18. Academic Press, LondonCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical EngineeringNational Institute of TechnologySrinagarIndia

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