Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andry, A.N., Shapiro, E.Y., Chung, J.C.: Eigenstructure assignment for linear systems. IEEE Trans. Aerosp. Electron. Syst. 19(5), 711–729 (1983)
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)
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)
Chu, E.K.: Pole assignment for second-order systems. Mech. Syst. Signal Process. 16(1), 39–59 (2002)
Chu, E.K., Datta, B.N.: Numerically robust pole assignment for second-order systems. Int. J. Control 64(6), 1113–1127 (1996)
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)
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)
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)
Gu, K., Chen, J., Kharitonov, V.L.: Stability of Time-Delay Systems. Springer, Berlin (2003)
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)
Kautsky, J., Nichols, N.K., Van Dooren, P.: Robust pole assignment in linear state feedback. Int. J. Control 41(5), 1129–1155 (1985)
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)
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)
Liu, Z., Li, W., Ouyang, H., Wang, D.: Eigenstructure assignment in vibrating systems based on receptances. Arch. Appl. Mech. 85(6), 713–724 (2015)
Mottershead, J.E., Ram, Y.M.: Receptance method in active vibration control. AIAA J. 45(3), 562–567 (2007)
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)
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)
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)
Ouyang, H.: Prediction and assignment of latent roots of damped asymmetric systems by structural modifications. Mech. Syst. Signal Process. 23(6), 1920–1930 (2009)
Ouyang, H.: Pole assignment of friction-induced vibration for stabilisation through state-feedback control. J. Sound Vib. 329(11), 1985–1991 (2010)
Ouyang, H.: A hybrid control approach for pole assignment to second-order asymmetric systems. Mech. Syst. Signal Process. 25(1), 123–132 (2011)
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)
Ram, Y.M.: Pole assignment for the vibrating rod. Q. J. Mech. Appl. Math. 51(3), 461–476 (1998)
Ram, Y.M., Elhay, S.: Pole assignment in vibratory systems by multi-input control. J. Sound Vib. 230(2), 309–321 (2000)
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)
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)
Ram, Y.M., Singh, A., Mottershead, J.E.: State feedback control with time delay. Mech. Syst. Signal Process. 23(6), 1940–1945 (2009)
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)
Singh, K.V., Ouyang, H.: Pole assignment using state feedback with time delay in friction-induced vibration problems. Acta Mech. 224(3), 645 (2013)
Singh, K.V., Ram, Y.M.: Transcendental eigenvalue problem and its applications. AIAA J. 40(7), 1402–1407 (2002)
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)
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)
Tehrani, M.G., Ouyang, H.: Receptance-based partial pole assignment for asymmetric systems using state-feedback. Shock Vib. 19(5), 1135–1142 (2012)
Vyhlidal, T., Zitek, P.: Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Trans. Autom. Control 54(1), 171–177 (2009)
Wonham, W.: On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom. Control 12(6), 660–665 (1967)
Wright, J.R., Cooper, J.E.: Introduction to Aircraft Aeroelasticity and Loads, vol. 20. Wiley, New York (2008)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
About this article
Cite this article
Ariyatanapol, R., Xiong, Y. & Ouyang, H. Partial pole assignment with time delays for asymmetric systems. Acta Mech 229, 2619–2629 (2018). https://doi.org/10.1007/s00707-018-2118-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-018-2118-2