Control of a Tip-Force Destabilized Shear Beam by Observer-Based Boundary Feedback

We consider a model of the undamped shear beam with a destabilizing boundary condition. The motivation for this model comes from atomic force microscopy, where the tip of the cantilever beam is destabilized by van der Waals forces acting between the tip and the material surface. Previous research efforts relied on collocated actuation and sensing at the tip, exploiting the passivity property between the corresponding input and output in the beam model. In this paper we design a stabilizing output-feedback controller in a noncollocated setting, with measurements at the free end (tip) of the beam and actuation at the beam base. Our control design is a novel combination of the classical "damping boundary feedback" idea with a recently developed backstepping approach. A change of variables is constructed which converts the beam model into a wave equation (for a very short string) with boundary damping. This approach is physically intuitive and allows both an elegant stability analysis and an easy selection of design parameters for achieving desired performance. Our observer design is a dual of the similar ideas, combining the damping feedback with backstepping, adapted to the observer error system. Both stability and well-posedness of the closed-loop system are proved. The simulation results are presented.