| Issue |
ESAIM: COCV
Volume 27, 2021
|
|
|---|---|---|
| Article Number | 94 | |
| Number of page(s) | 30 | |
| DOI | https://doi.org/10.1051/cocv/2021090 | |
| Published online | 06 October 2021 | |
Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension
1
Sorbonne Université, UMR 9912 STMS (Ircam – CNRS – Sorbonne Université),
1 place Igor Stravinsky,
75004
Paris, France.
2
Université du Littoral Côte d’Opale, Laboratoire de Mathématiques Pures et Appliquées J. Liouville,
50 Rue Ferdinand Buisson,
62100
Calais, France.
* Corresponding author: marc.wijnand@ircam.fr
Received:
15
February
2021
Accepted:
10
September
2021
The paper is concerned with the finite-time stabilization of a coupled PDE–ODE system describing the motion of an overhead crane with a flexible cable. The dynamics of the flexible cable is described by the wave equation with a variable coefficient which is an affine function of the curvilinear abscissa along the cable. Using several changes of variables, a backstepping transformation, and a finite-time stable second-order ODE for the dynamics of a conveniently chosen variable, we prove that a global finite-time stabilization occurs for the full system constituted of the platform and the cable. The kernel equations and the finite-time stable ODE are numerically solved in order to compute the nonlinear feedback law, and numerical simulations validating our finite-time stabilization approach are presented.
Mathematics Subject Classification: 93C20 / 93D15
Key words: Finite-time stability / PDE–ODE system / Volterra integral transformation / cascade systems / nonlinear feedback law / transparent boundary conditions / wave equation
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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