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.
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Keywords: Finite-time stability, PDE–ODE system, Volterra integral transformation, cascade systems, nonlinear feedback law, transparent boundary conditions, wave equation
@article{COCV_2021__27_1_A96_0,
author = {Wijnand, Marc and d{\textquoteright}Andr\'ea-Novel, Brigitte and Rosier, Lionel},
title = {Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021090},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021090/}
}
TY - JOUR AU - Wijnand, Marc AU - d’Andréa-Novel, Brigitte AU - Rosier, Lionel TI - Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021090/ DO - 10.1051/cocv/2021090 LA - en ID - COCV_2021__27_1_A96_0 ER -
%0 Journal Article %A Wijnand, Marc %A d’Andréa-Novel, Brigitte %A Rosier, Lionel %T Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021090/ %R 10.1051/cocv/2021090 %G en %F COCV_2021__27_1_A96_0
Wijnand, Marc; d’Andréa-Novel, Brigitte; Rosier, Lionel. Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 94. doi: 10.1051/cocv/2021090
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