Piezo‌electric materials are used as sensor and actuator in order to control the vibrations of structures. Geometry and location of the piezoelectric sensors and actuators have a substantial effect on the consumed electric energy and performance of the control system, therefore, in this study by defining an appropriate cost function, an optimum length and location of the piezoelectric actuator was determined in order to achieve a desirable decrease on vibration amplitude of a cantilever beam by using appropriate control energy. The standard quadratic function of beam displacement and control energy was used as the cost function. Mathematical modeling was based on Euler Bernoulli beam theory and Hamilton's principle was used in order to achieve the equations of motion. In this approach, the control voltage of actuator layer is emerged in the boundary conditions of the problem, which turns it to a time varying boundary condition problem. By defining special displacement functions and homogenizing the boundary conditions, control voltage of the actuator is appeared as external excitement in the equations of motion. In the current study, optimum LQR and LQG controllers were investigated and Kalman filter theory was used in order to estimate the state variables. In numerical simulations, by investigating the performance of optimized limited or unlimited patches in comparison with complete one, the effective role of the objective function and optimization have been shown in decreasing applied control voltage.

Keywords: active control, Piezoelectric, Optimization, Optimal Control

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