Active Control of a Forced Mindlin-type Beam

In this study, optimal dynamic response control of a forced Mindlin-type beam is studied. The beam under consideration, which consists of central host layer and two piezoelectric patch actuators bonded on perfectly to both sides of the beam. It is assumed that the beam is subject to the forcing function, initially at rest and undeformed. Hence, a forced Mindlin-type beam is considered for active vibration control. For this aim, well-posedness and controllability of the system are presented. Performance index functional to be minimized by using minimum level of control voltage consists of a weighted quadratic functions of displacement and velocity of the beam and also includes a quadratic functional of the control function as a penalty term. In order to obtain the optimal control function, an adjoint variable satisfying the adjoint equation corresponding to state equation is defined. A maximum principle is introduced and optimal control function is obtained by means of maximum principle. It is not sensible to use the Linear Quadratic Regulator and Linear Quadratic Gaussian methods to solve the control problem in this paper since the equation under consideration also includes Heaviside function and its spatial derivatives due to existence of piezoelectric patch actuators. Therefore, maximum principle is employed in the present paper Also, by using maximum principle, control problem is reduced to solving a system of partial differential equations including state, adjoint variables, which are linked by initial, boundary and terminal conditions. The solution of this system is obtained by using MATLAB. Numerical results are presented in tables and graphical forms to demonstrate the effectiveness and capability of the introduced control algorithm