Skip to main content
SpringerLink
Log in

Finite-time stabilization of an overhead crane with a flexible cable

  • Original Article
  • Published:
Mathematics of Control, Signals, and Systems Aims and scope Submit manuscript

Abstract

The paper is concerned with the finite-time stabilization of a hybrid PDE-ODE system which may serve as a model for the motion of an overhead crane with a flexible cable. The dynamics of the flexible cable is assumed to be described by the wave equation with constant coefficients. Using a nonlinear feedback law inspired by those given by Haimo (SIAM J Control Optim 24(4):760–770, 1986) for a second-order ODE, we prove that a finite-time stabilization occurs for the full system platform \(+\) cable. The global well-posedness of the system is also established by using the theory of nonlinear semigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Alabau-Boussouira F, Perrollaz V, Rosier L (2015) Finite-time stabilization of a network of strings. Math Control Relat Fields 5(4):721–742

    Article  MathSciNet  Google Scholar 

  2. Bacciotti A, Rosier L (2005) Liapunov functions and stability in control theory, 2nd edn. Springer, Berlin

    Book  Google Scholar 

  3. Bhat SP, Bernstein DS (2000) Finite-time stability of continuous autonomous systems. SIAM J Control Optim 38:751–766

    Article  MathSciNet  Google Scholar 

  4. Browder FE (1963) The solvability of non-linear functional equations. Duke Math J 30:557–566

    Article  MathSciNet  Google Scholar 

  5. Coron J-M, d’Andréa-Novel B (1998) Stabilization of a rotating body-beam without damping. IEEE Trans Autom Control 43(5):608–618

    Article  MathSciNet  Google Scholar 

  6. Coron J-M, d’Andréa-Novel B, Bastin G (2007) A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans Autom Control 52(1):2–11

    Article  MathSciNet  Google Scholar 

  7. Courant R, Hilbert D (1962) Methods of mathematical physics, vol II. Interscience Publishers, Geneva

    MATH  Google Scholar 

  8. d’Andréa-Novel B, Boustany F, Conrad F, Rao BP (1994) Feedback stabilization of a hybrid PDE-ODE system: application to an overhead crane. MCSS J 7:1–22

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. d’Andréa-Novel B, Coron J-M (2002) Stabilization of an overhead crane with a variable length flexible cable. Comput Appl Math 21(1):1–33

    MathSciNet  MATH  Google Scholar 

  11. Efimov D, Polyakov A, Levant A, Perruquetti W (2017) Realization and discretization of asymptotically stable homogeneous systems. IEEE Trans Autom Control 62(11):5962–5969

    Article  MathSciNet  Google Scholar 

  12. Evans LC (1998) Partial differential equations. Graduate studies in mathematics, 19. American Mathematical Society, Providence

  13. Haimo VT (1986) Finite time controllers. SIAM J Control Optim 24(4):760–770

    Article  MathSciNet  Google Scholar 

  14. Hong Y, Huang J, Xu Y (1999) On an output feedback finite-time stabilization problem. In: Proceedings of 38th IEEE conference on decision and control, Phoenix, pp 1302–1307

  15. Kato T (1967) Nonlinear semigroups and evolution equations. J Math Soc Jpn 19(4):508–520

    Article  MathSciNet  Google Scholar 

  16. Komornik V (1994) Exact controllability and stabilization. The multiplier method, Masson, Paris

    MATH  Google Scholar 

  17. Lions J-L (1988) Contrôlabilité exacte des systèmes distribués, vol 1. Masson, Paris

    MATH  Google Scholar 

  18. Mifdal A (1997) Stabilisation uniforme d’un système hybride. C R Acad Sci Paris 324:37–42

    Article  MathSciNet  Google Scholar 

  19. Minty GJ (1962) Monotone (nonlinear) operators in Hilbert spaces. Duke Math J 29:341–346

    Article  MathSciNet  Google Scholar 

  20. Morgül O, Rao BP, Conrad F (1994) On the stabilization of a cable with a tip mass. IEEE Trans Autom Control 39(10):440–454

    Article  MathSciNet  Google Scholar 

  21. Perrollaz V, Rosier L (2014) Finite-time stabilization of \(2\times 2\) hyperbolic systems on tree-shaped networks. SIAM J Control Optim 52(1):143–163

    Article  MathSciNet  Google Scholar 

  22. Reeken M (1977) The equation of motion of a chain, Mathematische Zeitschrift, 155. Springer, Berlin, pp 219–237

    MATH  Google Scholar 

  23. Sepulchre R, Janković M, Kokotović P (1997) Constructive nonlinear control. Springer, Berlin

    Book  Google Scholar 

Download references

Acknowledgements

This work was done when Iván Moyano was visiting Centre de Robotique, MINES ParisTech, as a postdoctoral student. Iván Moyano thanks Centre de Robotique, MINES ParisTech, for its hospitality. The authors were supported by the ANR project Finite4SoS (ANR-15-CE23-0007). LR was also partially supported by the MathAmSud project Icops.

Author information

Authors and Affiliations

  1. Centre de Robotique, MINES ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75272, Paris Cedex 06, France

    Brigitte d’Andréa-Novel

  2. STMS UMR9912, Ircam, CNRS, Sorbonne Université, Ministère de la culture, 1 place Igor Stravinsky, 75004, Paris, France

    Brigitte d’Andréa-Novel

  3. DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road CB3 OWA, Cambridge, United Kingdom

    Iván Moyano

  4. Centre Automatique et Systèmes (CAS) and Centre de Robotique, MINES ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75272, Paris Cedex 06, France

    Lionel Rosier

Authors
  1. Brigitte d’Andréa-Novel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iván Moyano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lionel Rosier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lionel Rosier.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

d’Andréa-Novel, B., Moyano, I. & Rosier, L. Finite-time stabilization of an overhead crane with a flexible cable. Math. Control Signals Syst. 31, 1–19 (2019). https://doi.org/10.1007/s00498-019-0235-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00498-019-0235-7

Keywords

Mathematics Subject Classification

Search

Navigation