Robust Boundary Control for an Euler Bernoulli Beam Subject to Unknown Harmonic Disturbances With a Focus on Resonance

In this paper, a sliding mode backstepping controller for a pinned-pinned Euler-Bernoulli beam is briefly reviewed and its efficacy in the presence of unknown bounded harmonic disturbances at arbitrary frequencies is analyzed. A brief discussion of the open-loop unstable response to harmonic excitations at resonant frequencies is provided. Motivated by this, particular attention is given to excitations at the natural frequencies of the system. It is shown that in the face of such resonant disturbances, the sliding mode backstepping controller is able to eliminate the vibrations in the beam system where backstepping control alone cannot. Indeed it is shown that if the disturbances are not accounted for, the closed loop system exhibits large (relative to the initial conditions) steady state harmonic vibrations. When the unknown resonant harmonic disturbances are accounted for via the sliding mode backstepping technique, the steady state position is constant and does not exhibit any vibrations, and furthermore it reaches this steady state exponentially at an arbitrarily selected rate.