Backstepping Boundary Controllers for Tip-Force Induced Flexible Beam Instabilities Arising in AFM

We consider the undamped version of the "shear beam" model and design stabilizing output feedbacks which employ measurements only at the free end (tip) of the beam and apply actuation only at the beam base. Our control design is a novel combination of the classical "damping boundary feedback" ideas with backstepping. Our observer design is a dual of the similar ideas, combining the damping feedback with backstepping, adapted to the observer error system. The "anti-collocated" architecture that we pursue makes active control more readily implementable to several applications, including atomic force microscopy (AFM) where the piezo actuation is applied at the beam base. Besides vibration suppression, our design extends to some situations where the beam dynamics become unstable. For example, in AFM the beam tip is subjected to a boundary condition that may assume destabilizing character, which arises due to Van der Vaals forces acting between the atoms on the material surface and the beam tip. Backstepping yields controllers that eliminate this instability. Our backstepping boundary control is physically intuitive. A change of variable is constructed which converts the beam model into a wave equation (for a very short string) with boundary damping. This allows both an easy stability analysis and an easy selection of design parameters for achieving desired performance