Vibration control of a flexible timoshenko beam under unknown external disturbances

In this paper, we present the vibration control of a Timoshenko beam under the external unknown spatiotemporally varying distributed disturbance and time-varying boundary disturbances. The model of the Timoshenko beam extracted from a class of flexible systems is represented by a hybrid model including both nonhomogeneous partial differential equations (PDEs) and ordinary differential equations (ODEs). To suppress the vibration of the system and reduce the effects of the external disturbances, adaptive boundary control is proposed based on Lyapunov´s direct method. With the proposed boundary control strategy, the states of the system are proven to be uniformly ultimately bounded and converge to a small neighborhood of zero by choosing the design parameters. Several simulations are shown to illustrate the effectiveness of the proposed adaptive boundary control.