Nonlinear Dynamics

, Volume 87, Issue 1, pp 617–632 | Cite as

Vibration control of a flexible marine riser with joint angle constraint

Original Paper


In this paper, the constrained problem of the joint angles for a flexible marine riser is investigated. Boundary control based on the integral-barrier Lyapunov function is achieved by three actuators equipped at the top boundary of the riser. Under the time-varying disturbances, the designed control can suppress the vibration of the riser and ensure the joint angles in the constrained ranges. The stability is proved under the designed control laws. Numerical simulations are given to illustrate the effectiveness of the designed control laws.


Nonlinear PDE Flexible marine riser Boundary control Constraints Distributed parameter system (DPS) 

List of symbols


Length of the riser


Mass of the vessel

\(\rho \)

Uniform mass per unit length of the riser


Bending stiffness of the riser


Axial stiffness of the riser


Tension of the riser


Constraints on \(x^{\prime }_L\), \(y^{\prime }_L\) and \(z^{\prime }_L\)


Boundary control inputs in XYZ directions


Distributed disturbances of the riser in XYZ directions


Boundary disturbances of the riser in XYZ directions

x(st), y(st), z(st)

Displacements in XYZ directions


  1. 1.
    Dai, S.-L., Wang, C., Luo, F.: Identification and learning control of ocean surface ship using neural networks. IEEE Trans. Ind. Inf. 8(4), 801–810 (2012)CrossRefGoogle Scholar
  2. 2.
    How, B.V.E., Ge, S.S., Choo, Y.S.: Active control of flexible marine risers. J. Sound Vib. 320, 758–776 (2009)CrossRefGoogle Scholar
  3. 3.
    Zulli, D., Luongo, A.: Nonlinear energy sink to control vibrations of an internally nonresonant elastic string. Meccanica 50(3), 781–794 (2015)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Luongo, A., Rega, G., Vestroni, F.: Planar non-linear free vibrations of an elastic cable. Int. J. Non-Linear Mech. 19(1), 39–52 (1984)MATHCrossRefGoogle Scholar
  5. 5.
    Oueini, S.S., Nayfeh, A.H., Pratt, J.R.: A nonlinear vibration absorber for flexible structures. Nonlinear Dyn. 15(3), 259–282 (1998)MATHCrossRefGoogle Scholar
  6. 6.
    Nayfeh, S.A., Nayfeh, A.H., Mook, D.T.: Nonlinear response of a taut string to longitudinal and transverse end excitation. J. Vib. Control 1(3), 307–334 (1995)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Do, K.D., Pan, J.: Boundary control of transverse motion of marine risers with actuator dynamics. J. Sound Vib. 318, 768–791 (2008)CrossRefGoogle Scholar
  8. 8.
    Do, K.D., Pan, J.: Boundary control of three-dimensional inextensible marine risers. J. Sound Vib. 327(3–5), 299–321 (2009)CrossRefGoogle Scholar
  9. 9.
    He, W., Sun, C., Ge, S.S.: Top tension control of a flexible marine riser by using integral-barrier Lyapunov function. IEEE/ASME Trans. Mechatron. 2(20), 497–505 (2015)CrossRefGoogle Scholar
  10. 10.
    Wu, H.-N., Wang, J.-W.: Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion PDE-governed sensor dynamics. Nonlinear Dyn. 72(3), 615–628 (2013)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Wang, J.-W., Wu, H.-N., Li, H.-X.: Fuzzy control design for nonlinear ODE-hyperbolic PDE cascaded systems: a fuzzy and entropy-like Lyapunov function approach. IEEE Trans. Fuzzy Syst. 22, 1313–1324 (2014)CrossRefGoogle Scholar
  12. 12.
    Wang, J.-W., Wu, H.-N., Li, H.-X.: Stochastically exponential stability and stabilization of uncertain linear hyperbolic pde systems with Markov jumping parameters. Automatica 48, 569–576 (2012)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Luo, B., Wu, H.-N., Li, H.-X.: Adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro-dynamic programming. IEEE Trans. Neural Netw. Learn. Syst. 26(4), 684–696 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wang, N., Wu, H.-N., Guo, L.: Coupling-observer-based nonlinear control for flexible air-breathing hypersonic vehicles. Nonlinear Dyn. 1(1), 1–24 (2014)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ge, S.S., Lee, T.H., Zhu, G.: A nonlinear feedback controller for a single-link flexible manipulator based on a finite element model. J. Robotic Syst. 14(3), 165–178 (1997)MATHCrossRefGoogle Scholar
  16. 16.
    He, W., Ouyang, Y., Hong, J.: Vibration control of a flexible robotic manipulator in the presence of input deadzone. IEEE Trans. Ind. Inform. (2016). doi: 10.1109/TII.2016.2608739
  17. 17.
    Armaou, A., Christofides, P.: Wave suppression by nonlinear finite-dimensional control. Chem. Eng. Sci. 55(14), 2627–2640 (2000)CrossRefGoogle Scholar
  18. 18.
    Chritofides, P., Armaou, A.: Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control. Syst. Control Lett. 39(4), 283–294 (2000)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Balas, M.J.: Feedback control of flexible systems. IEEE Trans. Autom. Control 23, 673–679 (1978)MATHCrossRefGoogle Scholar
  20. 20.
    Vandegrift, M.W., Lewis, F.L., Zhu, S.Q.: Flexible-link robot arm control by a feedback linearization/singular perturbation approach. J. Robotic Syst. 11(7), 591–603 (1994)Google Scholar
  21. 21.
    Sun, C., He, W., Hong, J.: Neural network control of a flexible robotic manipulator using the lumped spring-mass mode. IEEE Trans. Syst. Man Cybern. Syst. (2016). doi: 10.1109/TSMC.2016.2562506 Google Scholar
  22. 22.
    Balas, M.J.: Active control of flexible systems. J. Optim. Theory Appl. 23(3), 415–436 (1978)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Nguyen, Q.C., Hong, K.-S.: Transverse vibration control of axially moving membranes by regulation of axial velocity. IEEE Trans. Control Syst. Technol. 20(4), 1124–1131 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Guo, B.-Z., Jin, F.-F.: Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance. IEEE Trans. Autom. Control 60(3), 824–830 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jin, F.-F., Guo, B.-Z.: Lyapunov approach to output feedback stabilization for the Euler–Bernoulli beam equation with boundary input disturbance. Automatica 52(1), 95–102 (2015)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Yang, K.-J., Hong, K.-S., Matsuno, F.: Robust boundary control of an axially moving string by using a PR transfer function. IEEE Trans. Autom. Control 50(12), 2053–2058 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wu, Y., Xue, X., Shen, T.: Absolute stability of the Kirchhoff string with sector boundary control. Automatica 50(7), 1915–1921 (2014)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    He, W., Zhang, S., Ge, S.S.: Robust adaptive control of a thruster assisted position mooring system. Automatica 50(7), 1843–1851 (2014)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Guo, Q., Yu, T., Jiang, D.: Robust \(h_\infty \) positional control of 2-DOF robotic arm driven by electro-hydraulic servo system. ISA Trans. 59, 55–64 (2015)CrossRefGoogle Scholar
  31. 31.
    Kang, Y., Zhai, D.-H., Liu, G.-P., Zhao, Y.-B., Zhao, P.: Stability analysis of a class of hybrid stochastic retarded systems under asynchronous switching. IEEE Trans. Autom. Control 59(6), 1511–1523 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kang, Y., Zhai, D.-H., Liu, G.-P., Zhao, Y.-B.: On input-to-state stability of switched stochastic nonlinear systems under extended asynchronous switching. IEEE Trans. Cybern. 46(5), 1092–1105 (2016)CrossRefGoogle Scholar
  33. 33.
    Guo, Q., Yu, T., Jiang, D.: High-gain observer-based output feedback control of single-rod electro-hydraulic actuator. IET Control Theory Appl. 9(16), 2395–2404 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Li, Y., Ge, S.S.: Human-robot collaboration based on motion intention estimation. IEEE/ASME Trans. Mechatron. 19(3), 1007–1014 (2014)CrossRefGoogle Scholar
  35. 35.
    Yang, C., Li, Z., Cui, R., Xu, B.: Neural network-based motion control of an underactuated wheeled inverted pendulum model. IEEE Trans. Neural Netw. Learn. Syst. 25(11), 2004–2016 (2014)CrossRefGoogle Scholar
  36. 36.
    Gong, D., Lewis, F.L., Wang, L., Xu, K.: Synchronization for an array of neural networks with hybrid coupling by a novel pinning control strategy. Neural Netw. 77, 41–50 (2016)CrossRefGoogle Scholar
  37. 37.
    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
  38. 38.
    Wang, J.-M., Liu, J.-J., Ren, B., Chen, J.: Sliding mode control to stabilization of cascaded heat PDE–ODE systems subject to boundary control matched disturbance. Automatica 52, 23–34 (2015)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Paranjape, A.A., Guan, J., Chung, S.-J., Krstic, M.: PDE boundary control for flexible articulated wings on a robotic aircraft. IEEE Trans. Robotics 29(3), 625–640 (2013)CrossRefGoogle Scholar
  40. 40.
    Bernard, P., Krstic, M.: Adaptive output-feedback stabilization of non-local hyperbolic pdes. Automatica 50(10), 2692–2699 (2014)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Liu, Z., Liu, J.-K., He, W.: Adaptive boundary control of a flexible manipulator with input saturation. Int. J. Control 89(6), 1191–1202 (2016)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    He, W., Zhang, S.: Control design for nonlinear flexible wings of a robotic aircraft. IEEE Trans. Control Syst. Technol. (2016). doi: 10.1109/TCST.2016.2536708 Google Scholar
  43. 43.
    Zhao, Z., Liu, Y., He, W., Fei, L.: Adaptive boundary control of an axially moving belt system with high acceleration/deceleration. Int. J. Syst. Sci. 10(11), 1299–1306 (2016)MathSciNetGoogle Scholar
  44. 44.
    Wu, H.-N., Wang, J.-W.: Static output feedback control via pde boundary and ode measurements in linear cascaded ode-beam systems. Automatica 50(11), 2787–2798 (2014)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    He, W., Ge, S.S., How, B.V.E., Choo, Y.S., Hong, K.-S.: Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47(4), 722–732 (2011)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    He, W., He, X., Ge, S.S.: Vibration control of flexible marine riser systems with input saturation. IEEE/ASME Trans. Mechatron. 21(1), 254–265 (2016)Google Scholar
  47. 47.
    Gao, Y., Wu, H., Wang, J., Guo, L.: Feedback control design with vibration suppression for flexible air-breathing hypersonic vehicles. Sci. China Inf. Sci. 57(3), 1–14 (2014)MATHCrossRefGoogle Scholar
  48. 48.
    He, W., Nie, S., Meng, T., Liu, Y.-J.: Modeling and vibration control for a moving beam with application in a drilling riser. IEEE Trans. Control Syst. Technol. (2016). doi: 10.1109/TCST.2016.2577001
  49. 49.
    He, W., Ge, S.S.: Cooperative control of a nonuniform gantry crane with constrained tension. Automatica 66(4), 146–154 (2016)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    He, W., Ge, S.S., Huang, D.: Modeling and vibration control for a nonlinear moving string with output constraint. IEEE/ASME Trans. Mechatron. 20(4), 1886–1897 (2015)CrossRefGoogle Scholar
  51. 51.
    Tee, K.P., Ge, S.S., Tay, E.: Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4), 918–927 (2009)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    He, W., Yin, Z., Sun, C.: Adaptive neural network control of a marine vessel with constraints using the asymmetric barrier Lyapunov function. IEEE Trans. Cybern. (2016). doi: 10.1109/TCYB.2016.2554621
  53. 53.
    Tee, K.P., Ren, B., Ge, S.S.: Control of nonlinear systems with time-varying output constraints. Automatica 47(11), 2511–2516 (2011)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    He, W., Chen, Y., Yin, Z.: Adaptive neural network control of an uncertain robot with full-state constraints. IEEE Trans. Cybern. 46(3), 620–629 (2016)CrossRefGoogle Scholar
  55. 55.
    He, W., Ge, S.S.: Vibration control of a flexible beam with output constraint. IEEE Trans. Ind. Electron. 62(8), 5023–5030 (2015)CrossRefGoogle Scholar
  56. 56.
    Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading, Mass (1951)Google Scholar
  57. 57.
    Queiroz, M.S., Dawson, D.M., Nagarkatti, S.P., Zhang, F.: Lyapunov Based Control of Mechanical Systems. Birkhauser, Boston (2000)MATHCrossRefGoogle Scholar
  58. 58.
    Faltinsen, O.M.: Sea Loads on Ships and Offshore Structures. Cambridge University Press, New York (1990)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of Aeronautics and AstronauticUniversity of Electronic Science and Technology of ChinaChengduChina
  2. 2.School of Automation and Electrical EngineeringUniversity of Science and Technology BeijingBeijingChina
  3. 3.School of Automation Engineering and Center for RoboticsUniversity of Electronic Science and Technology of ChinaChengduChina

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