Abstract
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.
Highlights
IntroductionAssuming that the transversal and angular displacements were small
Stabilization of coupled PDE–ODE systems The stabilization of coupled PDE–ODE systems has attracted the attention of the control community since several decades
The settling-time function corresponding to the finite-time stabilization of the system (2.2)–(2.4) with the feedback law (4.27) as obtained by the Theorem 4.5 is composed of two contributions: a first contribution corresponding to the finite-time stabilization of the variable φ (described by (4.26), with a settling-time for which no analytic expression is known but which depends on the control parameters ν1 and ν2 and the initial state), and a second contribution corresponding to the finite-time stabilization of the variables β and α (described by (3.3)–(3.4), with a settling-time of 2
Summary
Assuming that the transversal and angular displacements were small. Where the angular deviation θ(t) of the cable with respect to the vertical axis, at the curvilinear abscissa s = 1 (i.e. at the connection point to the platform), is supposed to be measured (see Fig. 1), we obtain the following system: ytt − (d(s)ys)s = 0, (s, t) ∈ (0, 1) × (0, +∞), ys(0, t) = 0, t ∈ (0, +∞), y(1, t) = Xp(t), t ∈ (0, +∞), Xp(t) = U (t), t ∈ (0, +∞).
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