Feedback Controller Stabilizing Vibrations of a Flexible Cable Related to an Overhead Crane

The problem of stabilizing vibrations of flexible cable related to an overhead crane is considered. The cable vibrations are described by a hyperbolic partial differential equation (HPDE) with an update boundary condition. We provide in this paper a systematic way to derive a boundary feedback law which restores in a closed form the cable vibrations to the desired zero equilibrium. Such a control law is explicitly constructed in terms of the solution of an appropriate kernel PDE. The pursued approach combines the “backstepping method” and “semigroup theory”. 1. Introduction In this paper, we are concerned with the problem of boundary feedback stabilization of a second-order HPDE describing vibrations of a flexible cable related to an overhead crane. As illustrated in Figure 1, the rigid load with mass is related to cart of the overhead crane by a flexible cable. Figure 1 The cable displacement , at time and height , is mathematically modeled by the following hyperbolic equation: coupled with the update boundary condition imposed at the level , The parameter denotes the tension force of the cable at the height , being the gravitational acceleration, the mass of the cart, and the mass of the rigid load. It is assumed that the line density of the cable is homogeneous and equal to . The vibrations in system (1.1) are not only being diffused and bifurcated but also a destabilizing displacement is generated. Here, , are two constants and is a control placed at the extremity . The boundary condition (1.2) corresponds to situations where the displacement is subject to a dispositive effect when the rigid load is arrived to the soil, that is, . Such effect arises in (1.2) as an external force which depends on the displacements. System (1.1)-(1.2) serves also as a linearized model of strings. Hereafter, we assume that the parameters , and the initial data , satisfy the regularity conditions where , are the usual Sobolev spaces on , see Section 2. The control objective that we are interested in, is to construct a feedback controller which restores the displacements to the equilibrium (as ). From a practical point of view, Rao [1] treated the stabilization problem of suppressing vibrations of the distributed overhead crane model with one rigid load, when . The exponential stability of the closed loop is proved by exploiting an energy functional. In the study by Rahn et al. in [2], a study has been conducted to develop control algorithms for flexible cable crane models. An appropriate coupling amplification controller which asymptotically stabilizes all modes of a