Exponential stability for a class of boundary conditions on a Euler-Bernoulli beam subject to disturbances via boundary control

This paper outlines the procedure for applying a sliding mode backstepping boundary control technique to a Euler-Bernoulli beam subject to unknown bounded disturbances on the boundary for four different types of boundary conditions, referred to in this paper as a ‘class’ of boundary conditions. These boundary conditions include all combinations of ‘pinned’ and ‘sliding’ types with control and disturbances on one boundary, i.e. pinned-pinned, pinned-sliding, sliding-pinned, and sliding-sliding with input on the second boundary. The technique was developed for the specific case of a pinned-pinned beam in the literature, and in this paper the technique is generalized to the degree that it is possible to cover all four of the aforementioned cases. Furthermore, a proof of the fully arbitrary, exponential stability of the closed-loop system is provided in this paper; to this point only asymptotic stability was shown for the closed-loop system for one case. The technique outlined in this paper provides, for a class of boundary conditions, a controller that exponentially stabilizes the vibrations in a beam in the presence of bounded unknown disturbances. The beam displacement returns to the origin and its decay is bounded by a known exponential decay function after a known finite-time reaching phase.