On Boundary Observation of a Simply Supported Beam via a Backstepping Design

In this paper the first part of a full-state boundary observer for a simply supported Euler-Bernoulli beam is developed using a backstepping method. The fourth order partial differential equation (PDE) system is first transformed into a second order coupled system using a variable transformation available in the literature. The feedback information available to the observer is limited to only two values related to the slope and the shear force of the beam at one boundary. Using the methods developed in this paper, an observer is derived such that the estimate of the full beam displacement converges exponentially to the actual beam displacement. This is a necessary complement to the feedback control problem, where full state knowledge is required but is practically impossible to measure directly. Due to format restrictions, the full observer will not be presented here, as a complete estimation must be coupled with a secondary observer that recovers information about the initial velocity of the beam. This information cannot be assumed to be known generally, but is taken as a given for the preliminary developments in this paper.