Active control of smart beam with time delay

Active control of a flexible beam is investigated using time-delayed piezoelectric actuator. A minimization problem is formulated that consists of finding the control voltage applied to a piezoelectric patch actuator which suppresses the dynamic responses of the beam at a specified terminal time with least control effort. The solution method is based on a combination of modal expansion and direct approximation approaches. The modal expansion approach is used to convert 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 proposed where 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. An illustrative example is included to demonstrate the applicability and the efficiency of the proposed method and the results are quite satisfactory.