Modeling and control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance

In this paper, modeling and control of a vibrating Euler-Bernoulli beam is considered under the unknown external disturbances. The dynamics of the beam derived based on Hamilton´s principle is represented by a partial differential equation (PDE) and four ordinary differential equations (ODEs) involving functions of both space and time. To deal with the system uncertainties and stabilize the beam, robust adaptive boundary control is developed at the tip of the beam based on Lyapunov´s direct method. With the proposed boundary control, all the signals in the closed loop system are guaranteed to be uniformly bounded. The state of the system is proven to converge to a small neighborhood of zero by appropriately choosing the design parameters. The simulations are provided to illustrate the effectiveness of the proposed control.