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Boundary control for flexible mechanical systems with input dead-zone

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Abstract

In this paper, the control problem for flexible mechanical systems with input dead-zone nonlinearity is investigated. We employ the boundary control technique aiming at regulating the deformation of flexible mechanical systems (including both the string and the Euler–Bernoulli beam) in the presence of various external disturbances. With the proposed control method, we prove that all the states of the closed-loop flexible mechanical systems are uniformly ultimately bounded, and the input dead-zone nonlinearity and boundary external disturbance are handled. Numerical simulation studies are carried out to verify the effectiveness of the developed boundary control.

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Abbreviations

L :

Length of the flexible mechanical systems

\(M_{s}\) :

Mass of the tip payload

\(\rho \) :

Uniform mass per unit length of the flexible mechanical systems

EI :

Bending stiffness of the flexible beam system

T :

Tension of the flexible mechanical systems

w(x, t):

Elastic deflection at position x for time t

D(v):

Control input dead-zone nonlinearity

v(t):

Desired control input

\({\bar{d}}(t)\) :

Unknown time-varying boundary disturbance on the tip payload

References

  1. Wang, X.-S., Su, C.-Y., Hong, H.: Robust adaptive control of a class of nonlinear systems with unknown dead-zone. Automatica 40(3), 407–413 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Wang, X.-S., Hong, H., Su, C.-Y.: Model reference adaptive control of continuous-time systems with an unknown input dead-zone. IEE Proc. Control Theory Appl. 150(3), 261–266 (2003)

  3. Chen, M., Ge, S.S., How, B.: Robust adaptive neural network control for a class of uncertain mimo nonlinear systems with input nonlinearities. IEEE Trans. Neural Netw. 21(5), 796–812 (2010)

    Article  Google Scholar 

  4. Wang, J.-M., Ren, B., Krstic, M.: Stabilization and Gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation. IEEE Trans. Autom. Control 57(1), 179–185 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. Feng, H., Guo, B.-Z.: Output feedback stabilization of an unstable wave equation with general corrupted boundary observation. Automatica 50(12), 3164–3172 (2014)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. He, W., Zhang, S., Ge, S.S.: Adaptive control of a flexible crane system with the boundary output constraint. IEEE Trans. Ind. Electron. 61(8), 4126–4133 (2014)

    Article  Google Scholar 

  9. d’Andrea Novel, B., Coron, J.M.: Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach. Automatica 36, 587–593 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kim, C.-S., Hong, K.-S.: Boundary control of container cranes from the perspective of controlling an axially moving string system. Int. J. Control Autom. Syst. 7(3), 437–445 (2009)

    Article  Google Scholar 

  11. He, W., He, X., Ge, S.S.: Boundary output feedback control of a flexible string system with input saturation. Nonlinear Dyn. 80(2), 871–888 (2015)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Guo, B.-Z., Jin, F.-F.: The active disturbance rejection control to stabilization for multi-dimensional wave equation with boundary control matched disturbance. IEEE Trans. Autom. Control 60(1), 143–157 (2015)

    Article  MathSciNet  Google Scholar 

  14. Zulli, D., Luongo, A.: Nonlinear energy sink to control vibrations of an internally nonresonant elastic string. Meccanica 50(3), 781–794 (2015)

    Article  MathSciNet  Google Scholar 

  15. 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)

    Article  MATH  MathSciNet  Google Scholar 

  16. 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)

    Article  MATH  Google Scholar 

  17. He, W., Sun, C., Ge, S.S.: Top tension control of a flexible marine riser by using integral-barrier lyapunov function. IEEE/ASME Trans. Mechatronics. 20(2), 497–505 (2015)

    Article  Google Scholar 

  18. Do, K., Pan, J.: Boundary control of transverse motion of marine risers with actuator dynamics. J. Sound Vib. 318, 768–791 (2008)

    Article  Google Scholar 

  19. Do, K., Pan, J.: Boundary control of three-dimensional inextensible marine risers. J. Sound Vib. 327(3–5), 299–321 (2009)

    Article  Google Scholar 

  20. Xu, G.Q., Wang, H.: Stabilisation of Timoshenko beam system with delay in the boundary control. Int. J. Control 86(6), 1165–1178 (2013)

    Article  MATH  Google Scholar 

  21. Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, Englewood Cliffs, NJ (1991)

    MATH  Google Scholar 

  22. Guerra, T.M., Vermeiren, L.: Lmi-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno’s form. Automatica 40(5), 823–829 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lam, H., Li, H.: Output-feedback tracking control for polynomial fuzzy-model-based control systems. IEEE Trans. Ind. Electron. 60(12), 5830–5840 (2011)

    Article  Google Scholar 

  24. Labiod, S., Boucherit, M.S., Guerra, T.M.: Adaptive fuzzy control of a class of mimo nonlinear systems. Fuzzy Sets Syst. 151(1), 59–77 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. 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)

    Article  MathSciNet  Google Scholar 

  26. Wang, J.-W., Li, H.-X., Wu, H.-N.: Distributed proportional plus second-order spatial derivative control for distributed parameter systems subject to spatiotemporal uncertainties. Nonlinear Dyn. 76(4), 2041–2058 (2014)

    Article  MathSciNet  Google Scholar 

  27. Bhikkaji, B., Moheimani, S.O.R., Petersen, I.: A negative imaginary approach to modeling and control of a collocated structure. IEEE/ASME Trans. Mechatron. 17(4), 717–727 (2012)

    Article  Google Scholar 

  28. Balas, M.J.: Active control of flexible systems. J. Optim. Theory Appl. 25, 415–436 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  29. Huang, D., Xu, J.-X., Li, X., Xu, C., Yu, M.: D-type anticipatory iterative learning control for a class of inhomogeneous heat equations. Automatica 49(8), 2397–2408 (2013)

    Article  MathSciNet  Google Scholar 

  30. 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)

    Article  MathSciNet  Google Scholar 

  31. 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. Auotmatica 47(4), 722–732 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. He, W., Ge, S.S., How, B.V.E., Choo, Y.S.: Dynamics and Control of Mechanical Systems in Offshore Engineering. Springer, London, UK (2013)

    Google Scholar 

  33. Zhang, T., Ge, S.S.: Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form. Automatica 44, 1895–1903 (2008)

  34. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge, UK (1959)

    Google Scholar 

  35. Queiroz, M.S., Dawson, D.M., Nagarkatti, S.P., Zhang, F.: Lyapunov Based Control of Mechanical Systems. Birkhauser, Boston (2000)

    Book  MATH  Google Scholar 

  36. He, W., Ge, S.S.: Robust adaptive boundary control of a vibrating string under unknown time-varying disturbance. IEEE Trans. Control Syst. Technol. 20, 48–58 (2012)

    Google Scholar 

  37. He, W., Ge, S.S.: Vibration control of a flexible beam with output constraint. IEEE Trans. Ind. Electron. 62(8), 5023–5030 (2015)

    Article  Google Scholar 

  38. He, W., Ge, S.S., Huang, D.Q.: Modeling and vibration control for a nonlinear moving string with output constraint. IEEE/ASME Trans. Mechatron. (2014). doi:10.1109/TMECH.2014.2358500

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Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61403063 and 61203057, and the National Basic Research Program of China (973 Program) under Grant 2014CB744206.

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Authors and Affiliations

  1. School of Aeronautics and Astronautics, University of Electronic and Technology of China, Chengdu, 611731, China

    Shuang Zhang

  2. School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Wei He

  3. School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Shuangxi Nie & Chang Liu

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  1. Shuang Zhang
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  2. Wei He
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  3. Shuangxi Nie
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  4. Chang Liu
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Correspondence to Wei He.

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Zhang, S., He, W., Nie, S. et al. Boundary control for flexible mechanical systems with input dead-zone. Nonlinear Dyn 82, 1763–1774 (2015). https://doi.org/10.1007/s11071-015-2275-y

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  • DOI: https://doi.org/10.1007/s11071-015-2275-y

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