Skip to main content
SpringerLink
Log in

Reduced ordered modelling of time delay systems using galerkin approximations and eigenvalue decomposition

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

In this paper, an r-dimensional reduced-order model (ROM) for infinite-dimensional delay differential equations (DDEs) is developed. The eigenvalues of the ROM match the r rightmost characteristic roots of the DDE with a user-specified tolerance of \(\varepsilon \). Initially, the DDE is approximated by an N-dimensional set of ordinary differential equations using Galerkin approximations. However, only \(N_{c}\)\((< N)\) eigenvalues of this N-dimensional model match (with a tolerance of \(\varepsilon \)) the rightmost characteristic roots of the DDEs. By performing numerical simulations, an empirical relationship for \(N_{c}\) is obtained as a function of N and \(\varepsilon \) for a scalar DDE with multiple delays. Using eigenvalue decomposition, an r\((= N_{c})\) dimensional model is constructed. First, an appropriate r is chosen, and then the minimum value of N at which at least r roots converge is selected. For each of the test cases considered, the time and frequency responses of the original DDE obtained using direct numerical simulations are compared with the corresponding r- and N-dimensional systems. By judiciously selecting r, solutions of the ROM and DDE match closely. Next, an r-dimensional model is developed for an experimental 3D hovercraft in the presence of delay. The time responses of the r-dimensional model compared favorably with the experimental results.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  1. Chakraborty S, Kandala SS, Vyasarayani CP (2018) Reduced ordered modelling of time delay systems using Galerkin approximations and eigenvalue decomposition. IFAC-PapersOnLine 51(1):566–571

    Article  Google Scholar 

  2. Thomson W (1949) Delay networks having maximally flat frequency characteristics. Proc IEE-Part III: Radio Commun Eng 96(44):487–490

    Google Scholar 

  3. Storch L (1954) Synthesis of constant-time-delay ladder networks using Bessel polynomials. Proc IRE 42(11):1666–1675

    Article  MathSciNet  Google Scholar 

  4. Glader C, Hognas G, Makila P, Toivonen H (1991) Approximation of delay systems-a case study. Int J Control 53(2):369–390

    Article  MathSciNet  MATH  Google Scholar 

  5. Gu G, Khargonekar PP, Lee EB (1989) Approximation of infinite-dimensional systems. IEEE Trans Automat Contr 34(6):610–618

    Article  MathSciNet  MATH  Google Scholar 

  6. Lam J (1990) Convergence of a class of Padé approximations for delay systems. Int J Control 52(4):989–1008

    Article  MATH  Google Scholar 

  7. Lam J (1993) Model reduction of delay systems using Padé approximants. Int J Control 57(2):377–391

    Article  MathSciNet  MATH  Google Scholar 

  8. Mäkilä PM (1990) Approximation of stable systems by Laguerre filters. Automatica 26(2):333–345

    Article  MathSciNet  MATH  Google Scholar 

  9. Mäkilä PM (1990) Laguerre series approximation of infinite dimensional systems. Automatica 26(6):985–995

    Article  MathSciNet  MATH  Google Scholar 

  10. Partington JR (1991) Approximation of delay systems by Fourier-Laguerre series. Automatica 27(3):569–572

    Article  MathSciNet  MATH  Google Scholar 

  11. Lam J (1994) Analysis on the Laguerre formula for approximating delay systems. IEEE Trans Automat Contr 39(7):1517–1521

    Article  MathSciNet  MATH  Google Scholar 

  12. Yoon M, Lee B (1997) A new approximation method for time-delay systems. IEEE Trans Automat Contr 42(7):1008–1012

    Article  MathSciNet  MATH  Google Scholar 

  13. Mäkilä PM, Partington JR (1999) Laguerre and Kautz shift approximations of delay systems. Int J Contr 72(10):932–946

    Article  MathSciNet  MATH  Google Scholar 

  14. Mäkilä PM, Partington JR (1999) Shift operator induced approximations of delay systems. SIAM J Control Optim 37(6):1897–1912

    Article  MathSciNet  MATH  Google Scholar 

  15. Harkort C, Deutscher J (2011) Krylov subspace methods for linear infinite-dimensional systems. IEEE Trans Automat Contr 56(2):441–447

    Article  MathSciNet  MATH  Google Scholar 

  16. Michiels W, Jarlebring E, Meerbergen K (2011) Krylov-based model order reduction of time-delay systems. SIAM J Matrix Anal Appl 32(4):1399–1421

    Article  MathSciNet  MATH  Google Scholar 

  17. Michiels W, Jarlebring E, Meerbergen K (2011) A projection approach for model reduction of large-scale time-delay systems, with application to a boundary controlled PDE. In: 50th conference on decision and control and European control conference. IEEE, pp 3482–3489

  18. Michiels W, Hilhorst G, Pipeleers G, Swevers J (2016) Model order reduction for time-delay systems, with application to fixed-order \(h_{2}\) optimal controller design. In: Recent results on time-delay systems. Springer, pp 45–66

  19. Rommes J, Martins N (2006) Efficient computation of multivariable transfer function dominant poles using subspace acceleration. IEEE Trans Power Syst 21(4):1471

    Article  Google Scholar 

  20. Rommes J, Martins N (2006) Efficient computation of transfer function dominant poles using subspace acceleration. IEEE Trans Power Syst 21(3):1218–1226

    Article  Google Scholar 

  21. Rommes J, Martins N (2008) Computing transfer function dominant poles of large-scale second-order dynamical systems. SIAM J Sci Comput 30(4):2137–2157

    Article  MathSciNet  MATH  Google Scholar 

  22. Rommes J, Sleijpen GL (2008) Convergence of the dominant pole algorithm and Rayleigh quotient iteration. SIAM J Matrix Anal Appl 30(1):346–363

    Article  MathSciNet  MATH  Google Scholar 

  23. Saadvandi M, Meerbergen K, Jarlebring E (2012) On dominant poles and model reduction of second order time-delay systems. Appl Numer Math 62(1):21–34

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang Y, Su Y (2013) A memory-efficient model order reduction for time-delay systems. BIT Numer Math 53(4):1047–1073

    Article  MathSciNet  MATH  Google Scholar 

  25. Cullum J, Zhang T (2002) Two-sided Arnoldi and nonsymmetric Lanczos algorithms. SIAM J Matrix Anal Appl 24(2):303–319

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang Q, Wang Y, Lam EY, Wong N (2013) Model order reduction for neutral systems by moment matching. Circuits Syst Signal Process 32(3):1039–1063

    Article  MathSciNet  Google Scholar 

  27. Scarciotti G, Astolfi A (2016) Model reduction of neutral linear and nonlinear time-invariant time-delay systems with discrete and distributed delays. IEEE Trans Automat Contr 61(6):1438–1451

    Article  MathSciNet  MATH  Google Scholar 

  28. Wahi P, Chatterjee A (2005) Galerkin projections for delay differential equations. J Dyn Syst Meas Contr 127(1):80–87

    Article  Google Scholar 

  29. de Jesus Kozakevicius A, Kalmár-Nagy T (2010) Weak formulation for delay equations. In: 9th Brazilian conference on dynamics, control and their applications

  30. Vyasarayani CP (2012) Galerkin approximations for higher order delay differential equations. J Comput Nonlinear Dyn 7(3):031004

    Article  Google Scholar 

  31. Vyasarayani CP, Subhash S, Kalmár-Nagy T (2014) Spectral approximations for characteristic roots of delay differential equations. Int J Dyn Contr 2(2):126–132

    Article  Google Scholar 

  32. Radke RJ (1996) A Matlab implementation of the implicitly restarted Arnoldi method for solving large-scale eigenvalue problems. Ph.D. thesis, Rice University

  33. Michiels W, Engelborghs K, Vansevenant P, Roose D (2002) Continuous pole placement for delay equations. Automatica 38(5):747–761

    Article  MathSciNet  MATH  Google Scholar 

  34. Lehotzky D, Insperger T (2016) A pseudospectral tau approximation for time delay systems and its comparison with other weighted-residual-type methods. Int J Numer Methods Eng 108(6):588–613

    Article  MathSciNet  Google Scholar 

  35. Ahsan Z, Uchida TK, Vyasarayani CP (2015) Galerkin approximations with embedded boundary conditions for retarded delay differential equations. Math Comput Model Dyn Syst 21(6):560–572

    Article  MathSciNet  MATH  Google Scholar 

  36. Heizer B, Kalmár-Nagy T (2018) Proper orthogonal decomposition and dynamic mode decomposition of delay-differential equations. IFAC-PapersOnLine 51(14):254–258

    Article  Google Scholar 

  37. Vyhlídal T, Zítek P (2014) QPmR-Quasi-polynomial root-finder: algorithm update and examples. In: Delay systems. Springer, pp 299–312

  38. Michiels W, Hilhorst G, Pipeleers G, Vyhlídal T, Swevers J (2017) Reduced modelling and fixed-order control of delay systems applied to a heat exchanger. IET Control Theory Appl 11(18):3341–3352

    Article  MathSciNet  Google Scholar 

  39. Apkarian J, Lévis M (2013) Laboratory guide, 3 DOF Hover experiment for Matlab/Simulink users. Quanser Inc

Download references

Acknowledgements

CPV gratefully acknowledges the Department of Science and Technology for funding this research through the Inspire fellowship (DST/INSPIRE/04/2014/000972). We thank Dr. Ketan P. Detroja for providing laboratory working space for the 3D hovercraft used in Section 5 The authors thank K. Subhash Babu for assistance in performing the experiments.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana, 502285, India

    Sayan Chakraborty

  2. Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana, 502285, India

    Shanti Swaroop Kandala & C. P. Vyasarayani

Authors
  1. Sayan Chakraborty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shanti Swaroop Kandala
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. P. Vyasarayani
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

CPV conceived the idea. SC and SSK generated the results and conducted the experiments. All the authors equally contributed to writing the manuscript.

Corresponding author

Correspondence to Shanti Swaroop Kandala.

Additional information

A preliminary version of this paper was presented at the 5th IFAC Conference on Advances in Control and Optimization of Dynamical Systems, Hyderabad, India, 2018 [1].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chakraborty, S., Kandala, S.S. & Vyasarayani, C.P. Reduced ordered modelling of time delay systems using galerkin approximations and eigenvalue decomposition. Int. J. Dynam. Control 7, 1065–1083 (2019). https://doi.org/10.1007/s40435-019-00510-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-019-00510-3

Keywords

Search

Navigation