OPTIMAL OPEN-LOOP CLOSED-LOOP CONTROL FOR ONE-DIMENSIONAL STRUCTURES WITH SEVERAL STATE VARIABLES

An optimal control problem for one-dimensional structures described by a system of partial differential equations is formulated. Open- and closed-loop control mechanisms are incorporated into the formulation and a general method of solution is presented in the context of a structural vibration problem. The expenditure of control force is restricted by penalizing a functional of the open-loop control force in the performance index and by imposing a constraint on the magnitude of the closed-loop control force. The optimality condition for the open-loop control is derived by using the methods of eigenfunction expansion and variational calculus. The closed-loop control parameters are obtained from the minimization of the energy of the system. The theory is illustrated by solving a vibration control problem for a beam undergoing torsional and flexural vibrations. Numerical results are presented in a graphical form in which the behaviours of the controlled and uncontrolled beams at different terminal times are compared. The numerical results indicate the effectiveness of the proposed control mechanism for reducing excessive vibrations as measured by the energy of the beam. The minimizing value of the closed-loop control parameter associated with displacement feedback is shown to be a function of the terminal time.