Acta Mechanica

, Volume 229, Issue 6, pp 2619–2629 | Cite as

Partial pole assignment with time delays for asymmetric systems

  • Rittirong Ariyatanapol
  • Y.P. XiongEmail author
  • Huajiang Ouyang
Open Access
Original Paper


Considering both single and multiple time delays, partial pole assignment for stabilising asymmetric systems is exemplified by friction-induced vibration and aerodynamic flutter. The control strategy is a single-input state feedback including constant time delays in the feedback loop. An unobservability condition is considered to assign some poles while keeping others unchanged. The receptance method is applied to avoid modelling errors from evaluating mass, damping and stiffness matrices by the finite element method. The solution is formulated in linear equations which allow determination of control gains. The stability of the closed-loop system is analysed by evaluating the first few dominant poles and determining a critical time delay. The numerical study shows that the proposed method is capable of making partial pole assignment with time delays. Since many structures and systems with non-conservative forces can be represented by asymmetric systems, this approach is widely applicable for vibration control of engineering structures.


  1. 1.
    Andry, A.N., Shapiro, E.Y., Chung, J.C.: Eigenstructure assignment for linear systems. IEEE Trans. Aerosp. Electron. Syst. 19(5), 711–729 (1983)CrossRefGoogle Scholar
  2. 2.
    Bai, Z.J., Chen, M.X., Yang, J.K.: A multi-step hybrid method for multi-input partial quadratic eigenvalue assignment with time delay. Linear Algebra Appl. 437(7), 1658–1669 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Breda, D., Maset, S., Vermiglio, R.: TRACE-DDE: a tool for robust analysis and characteristic equations for delay differential equations. In: Topics in Time Delay Systems, pp. 145–155. Springer (2009)Google Scholar
  4. 4.
    Chu, E.K.: Pole assignment for second-order systems. Mech. Syst. Signal Process. 16(1), 39–59 (2002)CrossRefGoogle Scholar
  5. 5.
    Chu, E.K., Datta, B.N.: Numerically robust pole assignment for second-order systems. Int. J. Control 64(6), 1113–1127 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Datta, B.N., Elhay, S., Ram, Y.M.: Orthogonality and partial pole assignment for the symmetric definite quadratic pencil. Linear Algebra Appl. 257, 29–48 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Datta, B.N., Elhay, S., Ram, Y.M., Sarkissian, D.R.: Partial eigenstructure assignment for the quadratic pencil. J. Sound Vib. 230(1), 101–110 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Datta, B.N., Sarkissian, D.R.: Multi-input partial eigenvalue assignment for the symmetric quadratic pencil. In: Proceedings of American Control Conference, vol. 4, pp. 2244–2247. IEEE (1999)Google Scholar
  9. 9.
    Gu, K., Chen, J., Kharitonov, V.L.: Stability of Time-Delay Systems. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Juang, J.N., Maghami, P.G.: Robust eigensystem assignment for state estimators using second-order models. J. Guid. Control Dyn. 15(4), 920–927 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kautsky, J., Nichols, N.K., Van Dooren, P.: Robust pole assignment in linear state feedback. Int. J. Control 41(5), 1129–1155 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liang, Y., Ouyang, H.J., Yamaura, H.: Active partial eigenvalue assignment for friction-induced vibration using receptance method. In: Journal of Physics: Conference Series, vol. 744, p. 12008. IOP Publishing (2016)Google Scholar
  13. 13.
    Liang, Y., Yamaura, H., Ouyang, H.: Active assignment of eigenvalues and eigen-sensitivities for robust stabilization of friction-induced vibration. Mech. Syst. Signal Process. 90, 254–267 (2017)CrossRefGoogle Scholar
  14. 14.
    Liu, Z., Li, W., Ouyang, H., Wang, D.: Eigenstructure assignment in vibrating systems based on receptances. Arch. Appl. Mech. 85(6), 713–724 (2015)CrossRefGoogle Scholar
  15. 15.
    Mottershead, J.E., Ram, Y.M.: Receptance method in active vibration control. AIAA J. 45(3), 562–567 (2007)CrossRefGoogle Scholar
  16. 16.
    Mottershead, J.E., Tehrani, M.G., James, S., Ram, Y.M.: Active vibration suppression by pole-zero placement using measured receptances. J. Sound Vib. 311(3), 1391–1408 (2008)CrossRefGoogle Scholar
  17. 17.
    Mottershead, J.E., Tehrani, M.G., Ram, Y.M.: Assignment of eigenvalue sensitivities from receptance measurements. Mech. Syst. Signal Process. 23(6), 1931–1939 (2009)CrossRefGoogle Scholar
  18. 18.
    Olgac, N., Sipahi, R.: An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Trans. Autom. Control 47(5), 793–797 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ouyang, H.: Prediction and assignment of latent roots of damped asymmetric systems by structural modifications. Mech. Syst. Signal Process. 23(6), 1920–1930 (2009)CrossRefGoogle Scholar
  20. 20.
    Ouyang, H.: Pole assignment of friction-induced vibration for stabilisation through state-feedback control. J. Sound Vib. 329(11), 1985–1991 (2010)CrossRefGoogle Scholar
  21. 21.
    Ouyang, H.: A hybrid control approach for pole assignment to second-order asymmetric systems. Mech. Syst. Signal Process. 25(1), 123–132 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pratt, J.M., Singh, K.V., Datta, B.N.: Quadratic partial eigenvalue assignment problem with time delay for active vibration control. In: Journal of Physics: Conference Series, vol. 181, p. 12092. IOP Publishing (2009)Google Scholar
  23. 23.
    Ram, Y.M.: Pole assignment for the vibrating rod. Q. J. Mech. Appl. Math. 51(3), 461–476 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ram, Y.M., Elhay, S.: Pole assignment in vibratory systems by multi-input control. J. Sound Vib. 230(2), 309–321 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ram, Y.M., Mottershead, J.E.: Multiple-input active vibration control by partial pole placement using the method of receptances. Mech. Syst. Signal Process. 40(2), 727–735 (2013)CrossRefGoogle Scholar
  26. 26.
    Ram, Y.M., Mottershead, J.E., Tehrani, M.G.: Partial pole placement with time delay in structures using the receptance and the system matrices. Linear Algebra Appl. 434(7), 1689–1696 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ram, Y.M., Singh, A., Mottershead, J.E.: State feedback control with time delay. Mech. Syst. Signal Process. 23(6), 1940–1945 (2009)CrossRefGoogle Scholar
  28. 28.
    Singh, K.V., Dey, R., Datta, B.N.: Partial eigenvalue assignment and its stability in a time delayed system. Mech. Syst. Signal Process. 42(1), 247–257 (2014)CrossRefGoogle Scholar
  29. 29.
    Singh, K.V., Ouyang, H.: Pole assignment using state feedback with time delay in friction-induced vibration problems. Acta Mech. 224(3), 645 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Singh, K.V., Ram, Y.M.: Transcendental eigenvalue problem and its applications. AIAA J. 40(7), 1402–1407 (2002)CrossRefGoogle Scholar
  31. 31.
    Tehrani, M.G., Elliott, R.N.R., Mottershead, J.E.: Partial pole placement in structures by the method of receptances: theory and experiments. J. Sound Vib. 329(24), 5017–5035 (2010)CrossRefGoogle Scholar
  32. 32.
    Tehrani, M.G., Mottershead, J.E., Shenton, A.T., Ram, Y.M.: Robust pole placement in structures by the method of receptances. Mech. Syst. Signal Process. 25(1), 112–122 (2011)CrossRefGoogle Scholar
  33. 33.
    Tehrani, M.G., Ouyang, H.: Receptance-based partial pole assignment for asymmetric systems using state-feedback. Shock Vib. 19(5), 1135–1142 (2012)CrossRefGoogle Scholar
  34. 34.
    Vyhlidal, T., Zitek, P.: Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Trans. Autom. Control 54(1), 171–177 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wonham, W.: On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom. Control 12(6), 660–665 (1967)CrossRefGoogle Scholar
  36. 36.
    Wright, J.R., Cooper, J.E.: Introduction to Aircraft Aeroelasticity and Loads, vol. 20. Wiley, New York (2008)Google Scholar
  37. 37.
    Xu, S., Qian, J.: Orthogonal basis selection method for robust partial eigenvalue assignment problem in second-order control systems. J. Sound Vib. 317(1), 1–19 (2008)CrossRefGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Engineering and the Environment, Fluid-Structure Interaction Research Group, Boldrewood Innovation CampusUniversity of SouthamptonSouthamptonUK
  2. 2.School of EngineeringUniversity of LiverpoolLiverpoolUK

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