Optimal time-delayed boundary control of beams using wavelets

Time delays exist in active control systems. Although a time delay in most cases is small, it results in the actuator applying energy to the control system with a time delay. This might cause instability of the dynamic systems and degrade the performance of the control systems. The objective of the present work is to propose a feasible methodology that can achieve good control performance for a dynamic beam structure system by time-delayed boundary torque actuators. The control problem is to determine optimal time-delayed boundary control by minimizing a given performance criterion. The performance criterion is specified as a weighted quadratic functional of the dynamic responses of the beam which is to be minimized at a specified terminal time using continuous time-delayed torque actuators. The modal expansion approach is used to convert the optimal control problem of a distributed parameter system into the optimal control problem of a linear lumped parameter system (LPS). A direct state-control parameterization 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. A numerical simulation is presented to illustrate the efficiency of the proposed control algorithm.