Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension

TL;DR

This paper develops a control method to achieve finite-time stabilization of an overhead crane system with a flexible cable modeled by a wave PDE with variable coefficients, validated through numerical simulations.

Contribution

It introduces a novel control strategy combining backstepping and finite-time stable ODEs for a hybrid PDE-ODE crane system with variable cable dynamics.

Findings

Successful finite-time stabilization demonstrated via numerical simulations.

The control law accounts for variable cable tension and dynamics.

The approach extends existing methods to more complex, realistic cable models.

Abstract

The paper is concerned with the finite-time stabilization of a hybrid 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.