Active control of Timoshenko beam vibrations using piezoelectric material

H_∞ robust control and filter design for controlling the beam vibrations are presented in this paper. The standard Timoshenko beam (TB) is employed as the nominal structural model. To stabilize the vibrations of flexible beams, active control using piezoelectric (PZT) material as actuator is introduced. The distributed sensor using the PZT material is employed to estimate the state variables of the system. The TB model governed by a partial differential equation (PDE) is discretized by a central difference scheme to generate a finite dimensional model. The variation in material properties, the discretization errors, and the modeling errors are considered as the uncertainties of the system. The H_∞ control theory is employed to determine the control voltages of the actuator such that the dynamic motion of the beam with uncertainties under disturbance is stabilized. Furthermore, considering the unknown noise and errors existing in the measurement process, the state estimator is realized by the robust H_∞ filter. The effectiveness of this approach is illustrated by a simulation example.