Nonlinear Dynamics

, Volume 95, Issue 2, pp 1395–1413 | Cite as

Dynamic modeling and vibration control of a three-dimensional flexible string with variable length and spatiotemporally varying parameters subject to input constraints

  • Xueyan Xing
  • Jinkun LiuEmail author
  • Zhijie Liu
Original Paper


In this paper, a three-dimensional dynamic model is developed for a flexible string system with variable length as well as spatiotemporally varying parameters. The dynamic system model is described by coupled partial differential equations and ordinary differential equations. On the basis of the established model, a boundary control method is proposed via backstepping technology and Nussbaum functions to eliminate the vibration of the three-dimensional string with input constraints and disturbances. Deformations of the three-dimensional string system can be verified to converge to small neighborhoods of zero under the proposed control. Input constraints can also be guaranteed by applying smooth hyperbolic tangent function. Simulation results present that the vibration suppression of the flexible string can be achieved and input constraints can be ensured with the proposed control scheme.


Three-dimensional flexible string Variable length Dynamic modeling Vibration control Input constraints Nussbaum function 



This work was supported by the Research Fund for the National Natural Science Foundation of China [Grant Number 61873296].

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    He, W., Ge, S.S.: Cooperative control of a nonuniform gantry crane with constrained tension. Automatica 66(4), 146–154 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Suweken, G., van Horssen, W.T.: On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case. J. Sound. Vib. 264(1), 117–133 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Zhu, W.D., Ren, H.: An accurate spatial discretization and substructure method with application to moving elevator cable-car systems-Part I: methodogy. J. Vib. Acoust. 135(5), 051036 (2013)CrossRefGoogle Scholar
  4. 4.
    Zhu, W.D., Ren, H.: An accurate spatial discretization and substructure method with application to moving elevator cable-car systems-Part II: application. J. Vib. Acoust. 135(5), 051037 (2013)CrossRefGoogle Scholar
  5. 5.
    Sandilo, S.H., van Horssen, W.T.: On variable length induced vibrations of a vertical string. J. Sound Vib. 333(11), 2432–2449 (2014)CrossRefGoogle Scholar
  6. 6.
    Sandilo, S.H., van Horssen, W.T.: On a cascade of autoresonances in an elevator cable system. Nonlinear Dyn. 80(3), 1613–1630 (2015)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kucuk, I., Sadek, I.: Active vibration control of an elastically connected double-string continuous system. J. Frank. Inst. 344(5), 684–697 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Alsahlani, A., Mukherjee, R.: Vibration control of a string using a scabbard-like actuator. J. Sound Vib. 330(12), 2721–2732 (2011)CrossRefGoogle Scholar
  9. 9.
    Armaou, A., Christofides, P.D.: Wave suppression by nonlinear finite-dimensional control. Chem. Eng. Sci. 55(14), 2627–2640 (2000)CrossRefGoogle Scholar
  10. 10.
    Shahruz, S.M., Kurmaji, D.A.: Vibration suppression of a non-linear axially moving string by boundary control. J. Sound Vib. 201(1), 145–152 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lee, S.-Y., Mote, C.D.: Vibration control of an axially moving string by boundary control. J. Dyn. Syst. Meas. Control. 118(1), 66–74 (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Zhang, S., He, W., Huang, D.: Active vibration control for a flexible string system with input backlash. IET Control Theory Appl. 10(7), 800–805 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ge S.S., Zhang S., He W.: Vibration control of a coupled nonlinear string system in transverse and longitudinal directions. In: 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA, pp. 3742–3747 (2011)Google Scholar
  14. 14.
    Foda, M.A.: Vibration control and suppression of an axially moving string. J. Vib. Control. 18(1), 58–75 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nguyen Q.C., Hong K.-S.: Longitudinal and transverse vibration control of an axially moving string. In: 5th International Conference on Cybernetics and Intelligent Systems, Qingdao, China, pp. 24–29 (2011)Google Scholar
  16. 16.
    Sandilo S.H., van Horssen W.T.: On boundary damping for an axially moving beam and on the variable length induced vibrations of an elevator cable. In: European Nonlinear Dynamics Conference, Rome, Italy (2011)Google Scholar
  17. 17.
    Zhang, S., He, W., Ge, S.S.: Modeling and control of a nonuniform vibrating string under spatiotemporally varying tension and disturbance. IEEE/ASME Trans. Mech. 17(6), 1196–1203 (2012)CrossRefGoogle Scholar
  18. 18.
    Rahn, C.D., Zhang, F., Joshi, S., Dawson, D.: Asymptotically stabilizing angle feedback for a flexible cable gantry crane. J. Dyn. Syst. Meas. Control. 121(3), 563–565 (1999)CrossRefGoogle Scholar
  19. 19.
    Takagi, K., Nishimura, H.: Gain-scheduled control of a tower crane considering varying load-rope length. JSME Int. J. Ser. C. 42(4), 914–921 (1999)CrossRefGoogle Scholar
  20. 20.
    Tuan, L.A., Lee, S.-G., Dang, V.-H., Moon, S., Kim, B.S.: Partial feedback linearization control of a three-dimensional overhead crane. Int. J. Control Autom. Syst. 11(4), 718–727 (2013)CrossRefGoogle Scholar
  21. 21.
    He, W., Qin, H., Liu, J.-K.: Modelling and vibration control for a flexible string system in three-dimensional space. IET Control Theory Appl. 9(16), 1–9 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hamed, Y.S., Amer, Y.A.: Nonlinear saturation controller for vibration supersession of a nonlinear composite beam. J. Mech. Sci. Technol. 28(8), 2987–3002 (2014)CrossRefGoogle Scholar
  23. 23.
    Sun, N., Fang, Y., Zhang, X.: Energy coupling output feedback control of 4-DOF underactuated cranes with saturated inputs. Automatica 49(5), 1318–1325 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wen, C., Zhou, J., Liu, Z., Su, H.: Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance. IEEE Trans. Autom. Control. 56(7), 1672–1678 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    He, W., Meng, T., He, X., Ge, S.S.: Unified iterative learning control for flexible structures with input constraints. Automatica 86, 326–336 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, Z., Liu, J., He, W.: Modeling and vibration control of a flexible aerial refueling hose with variable lengths and input constraint. Automatica 77, 302–310 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nussbaum, R.D.: Nussbaum function and its application in system stabilization. Syst. Control Lett. 3(5), 243–246 (1983)CrossRefGoogle Scholar
  28. 28.
    Hong, K.-S., Park, H.: Boundary control of container cranes as an axially moving string system. IFAC Proc. 38(1), 132–137 (2005)CrossRefGoogle Scholar
  29. 29.
    Zhu, W.D., Ni, J., Huang, J.: Active control of translating media with arbitrarily varying length. J. Vib. Acoust. 123(3), 347–358 (2001)CrossRefGoogle Scholar
  30. 30.
    Rahn, C.D.: Mechatronic control of distributed noise and vibration: a Lyapunov approach. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  31. 31.
    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1959)zbMATHGoogle Scholar
  32. 32.
    Polycarpou M.M., Ioannou P.A.: A robust adaptive nonlinear control design. In: American Control Conference, San Francisco, USA (1993)Google Scholar
  33. 33.
    Zhou, J., Wen, C., Zhang, Y.: Adaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis. IEEE Trans. Autom. Control. 49(10), 1751–1757 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ge, S.S., Wang, C., Lee, T.H.: Adaptive backstepping control of a class of chaotic systems. Int. J. Bifurc. Chaos. 10(5), 1149–1156 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lee, H.: Robust Adaptive fuzzy control by backstepping for a class of MIMO nonlinear systems. IEEE Trans. Fuzzy Syst. 19(2), 265–275 (2011)CrossRefGoogle Scholar
  36. 36.
    Zhang, Y., Wen, C., Soh, Y.C.: Discrete-time robust backstepping adaptive control for nonlinear time-varying systems. IEEE Trans. Autom. Control. 45(9), 1749–1755 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhou, J., Wen, C., Zhang, Y.: Adaptive output control of a class of time-varying uncertain nonlinear systems. Nonlinear Dyn. Syst. Theory 5(3), 285–298 (2005)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Mudgett, D.R., Morse, A.S.: Adaptive stabilization of linear systems with unknown high frequency gains. IEEE Trans. Autom. Control. 30(6), 549–554 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Liu, Z., Liu, J., He, W.: Robust adaptive fault tolerant control for a linear cascaded ODE-beam systems. Automatica 98, 42–50 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Liu, Z., Liu, J., He, W.: Dynamic modeling and vibration control for a nonlinear three-dimensional flexible manipulator. Int. J. Robust and Nonlinear Control 28(13), 3927–3945 (2018)CrossRefzbMATHGoogle Scholar
  41. 41.
    Nguyen, Q.C., Hong, K.-S.: Simultaneous control of longitudinal and transverse vibrations of an axially moving string with velocity tracking. J. Sound Vib. 331(13), 3006–3019 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Automation Science and Electrical EngineeringBeihang UniversityBeijingPeople’s Republic of China

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