Advertisement

Acta Mechanica Sinica

, Volume 34, Issue 4, pp 781–791 | Cite as

A frequency-domain method for solving linear time delay systems with constant coefficients

  • Mengshi Jin
  • Wei Chen
  • Hanwen Song
  • Jian Xu
Research Paper

Abstract

In an active control system, time delay will occur due to processes such as signal acquisition and transmission, calculation, and actuation. Time delay systems are usually described by delay differential equations (DDEs). Since it is hard to obtain an analytical solution to a DDE, numerical solution is of necessity. This paper presents a frequency-domain method that uses a truncated transfer function to solve a class of DDEs. The theoretical transfer function is the sum of infinite items expressed in terms of poles and residues. The basic idea is to select the dominant poles and residues to truncate the transfer function, thus ensuring the validity of the solution while improving the efficiency of calculation. Meanwhile, the guideline of selecting these poles and residues is provided. Numerical simulations of both stable and unstable delayed systems are given to verify the proposed method, and the results are presented and analysed in detail.

Keywords

Time delay system Delay differential equation Approximate solution Transfer function 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11272235).

References

  1. 1.
    Cai, G.P., Lim, C.W.: Optimal tracking control of a flexible hub-beam system with time delay. Multibody Syst. Dyn. 16, 331–350 (2006)CrossRefMATHGoogle Scholar
  2. 2.
    Xu, J., Sun, Y.X.: Experimental studies on active control of a dynamic system via a time-delayed absorber. Acta Mech. Sinica 31, 229–247 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Olgac, N., Holmhansen, B.T.: A novel active vibration absorption technique: delayed resonator. J. Sound Vib. 176, 93–104 (1994)CrossRefMATHGoogle Scholar
  4. 4.
    Wang, Q., Wang, Z.H.: Optimal feedback gains of a delayed proportional-derivative (PD) control for balancing an inverted pendulum. Acta Mech. Sinica 33, 635–645 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Sun, X.T., Xu, J., Fu, J.S.: The effect and design of time delay in feedback control for a nonlinear isolation system. Mech. Syst. Signal Process. 87, 206–217 (2017)CrossRefGoogle Scholar
  6. 6.
    Qin, Y.X.: The Motion Stability of Dynamical Systems with Time-delay, 2nd edn. Science Press, Beijing (1984). (In Chinese)Google Scholar
  7. 7.
    Jiang, S.Y., Xu, J., Yan, Y.: Stability and oscillations in a slow–fast flexible joint system with transformation delay. Acta Mech. Sinica 30, 727–738 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Wang, Z.H., Hu, H.Y.: A modified averaging scheme with application to the secondary hopf bifurcation of a delayed van der pol oscillator. Acta Mech. Sinica 24, 449–454 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chanda, A., Fischer, A., Eberhard, P., et al.: Stability analysis of a thin-walled cylinder in turning operation using the semi-discretization method. Acta Mech. Sinica 30, 214–222 (2014)CrossRefGoogle Scholar
  10. 10.
    Asl, F.M., Ulsoy, A.G.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. T. ASME 125, 215–223 (2003)CrossRefGoogle Scholar
  11. 11.
    Lam, J.: Balanced realization of pade approximants of e-st. IEEE Trans. Autom. Control 36, 1096–1100 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lam, J.: Model reduction of delay systems using pade approximants. Int. J. Control 57, 377–391 (1993)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Falbo, C.E.: Analytic and numerical solutions to the delay differential equations. In: Joint Meeting of the Northern and Southern California Sections of the MAA, San Luis Obispo (1995)Google Scholar
  14. 14.
    Wright, E.M.: The non-linear difference-differential equation. Q. J. Math. 17, 245–252 (1946)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic Press, New York (1963)MATHGoogle Scholar
  16. 16.
    Zwillinger, D.: Handbook of Differential Equations. Academic Press, Boston (1989)MATHGoogle Scholar
  17. 17.
    Erneux, T.: Applied Delay Differential Equations. Springer, New York (2009)MATHGoogle Scholar
  18. 18.
    Jin, M.S., Sun, Y.Q., Song, H.W., et al.: Experiment-based identification of time delays in linear systems. Acta Mech. Sinica 33, 429–439 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Caughey, T.K.: Classical normal modes in damped linear dynamic systems. J. Appl. Mech. T. ASME 32, 583–588 (1965)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. Siam, Philadelphia (2002)CrossRefMATHGoogle Scholar
  21. 21.
    Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Deliv. 14, 1052–1061 (1999)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina

Personalised recommendations