Flatness-based control of a flexible beam in a gravitational field

This paper considers a flatness-based boundary control of a hub-beam system with tip payload moving in a vertical plane in the presence of gravity. The homogeneous flexible beam is modelled using an Euler-Bernoulli hypothesis which assumes no shear deformation or rotatory inertia. As well, small transverse deformations are assumed. A linearized system model involving a coupled PDE-ODE is derived and a change of coordinates is introduced to simplify this model. The method of control relies on a flatness property of the system; namely, that the system solution can be differentially parameterized in terms of a flat output. This parameterization allows for straightforward motion planning and computation of a control law. The approach is based on power series in the spatial variable, and the convergence of these series is ensured by choosing the flat output to be a nonanalytic, smooth function of appropriate Gevrey class.