Control of Resonant Disturbances on an Euler–Bernoulli Beam Using a Robust Partial Differential Equation Boundary Control Technique

This article discusses and analyzes the capabilities and limitations of a series of related controllers for Euler–Bernoulli beam vibration, and the powerful capabilities of a robust second-order sliding mode backstepping control method are exhibited. Motivated by the open-loop unstable response to harmonic excitations at resonant frequencies, specific attention is given to disturbances at system resonant frequencies. It is shown that the second-order sliding mode backstepping controller provides arbitrary exponential stability of the beam position where other similar controllers cannot. Furthermore, it is shown that other controllers exhibit large (relative to the disturbance) steady-state harmonic vibrations, or otherwise do not return the system to the origin. This article is an extension of the Dynamic Systems and Control Division Vibrations Technical Committee “Best Vibrations Paper Award”-winning conference paper (Karagiannis and Radisavlejevic-Gajic, 2017, “Robust Boundary Control for an Euler Bernoulli Beam Subject to Unknown Harmonic Disturbances With a Focus on Resonance”). The previous work is significantly extended to include an exponentially stabilizing, second-order sliding mode controller and discusses several boundary conditions.