Control of Strings and Flexible Beams by Backstepping Boundary Control

We present the controller and observer designs for hyperbolic PDE systems. The main ideas are introduced on the example of a wave equation - a model of an undamped, vibrating string. Both a full state feedback boundary controller and a boundary sensor based observer are introduced. The string results are followed by a presentation of controller and observer designs for shear beam and Timoshenko beam models. The flexible systems considered in this paper allow the presence of a source of instability at the uncontrolled end of the string/beam. This result is of relevance to control of cantilevered beams in atomic force microscopy where the piezo actuator and the van der Waals forces act on the opposite ends of the beam.