Active control of a smart beam with time delay by Legendre wavelets

Active vibration control in a flexible beam is investigated by taking into account the time‐delay which may occur in voltage controlled piezoelectric actuators. The voltage applied to a piezoelectric patch actuator to suppress the dynamic responses of the beam at a specified terminal time is computed by solving an optimal control problem. The quadratic performance index function consists of related potential energy and kinetic energy at the terminal time as well as control effort as a penalty term. The solution method is based on a combination of spatial modal expansion and direct approximation approaches. The spatial modal expansion approach is used to transfer the optimal control problem of distributed parameter system (DPS) into the optimal control problem of a linear time‐invariant lumped parameter system (LPS). A direct state‐control parametrization approach is formulated where Legendre wavelets are employed to solve time delayed LPS. The operational matrices of integration and delay are utilized to reduce the solution of linear time‐varying delayed systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the applicability and the efficiency of the proposed method and the numerical results indicate the effectiveness of the proposed control mechanism.