A comparison of different models for beam vibrations from the standpoint of control design

In this paper we analyze different models for beam vibrations from the standpoint of designing finite-dimensional controllers to stabilize the beam vibrations. We show that a distributed system described by an undamped Euler-Bernoulli equation cannot be stabilized by any finite-dimensional controller, i.e., any controller which can be described an ordinary differential equation with constant coefficients. If viscous damping is included, a similar problem occurs in that all the poles can't be moved to the left of a given vertical line. These negative results should be interpreted as a commentary on the limitations of these models, rather than on the control of real beams. We then show that if a Rayleigh damping model is used, a finite-dimensional controller may be designed to move the closed loop system poles essentially as far to the left in the complex plane as desired. This result will also hold for certain hysteresis damping models. This has implications for the settling time of the vibrations