This work presents the dynamics and active vibration control of a non-uniform functionally graded beam. Thanks to the strong use of beams and specifically with non-uniform section, in the different industrial applications, such as helicopter rotor blades, wind turbines, space and marine structures, we present in this article, a comparison of linearly and non-linearly tapering beams. The FGM beam is equipped with four layers of piezoelectric materials as sensors and actuators, bonded on the upper and lower surfaces of the main structure, on different finite elements to see the influence of its location on the dynamics and active control. In this study, the Timoshenko beam’s theory combined with FEM is applied to a beam divided into a finite number of elements. Hamilton's principle is applied to generate the equation of motion. The structure is modeled analytically and numerically and the simulation results are presented at the end. The optimal LQG control with Kalman filtering is applied.