Optimal active pointwise control of thin plates via state-control parametrization

Abstract

A direct method to solve an optimal control problem by parametrizing both state and open-loop control variables is developed. This technique is designed to suppress the undesirable vibrations of a rectangular plate by open-closed loop control applied at discrete points in space. The closed-loop mechanism is assumed to be proportional to displacement. The measure of performance of the structure is taken as a combination of its total energy and the penalty terms describing the expenditure of the open-closed loop forces used in the control process. Using modal expansion and an appropriate transformation, the optimal control of a distributed parameter system is reduced to the optimal control of a linear time-varying lumped parameter system. Next, a computationally efficient formulation to evaluate the optimal control and trajectory of the structure, based on shifted Legendre polynomial approximations of the state variable and each open-loop control variable, is developed. This converts the linear quadratic problem into a mathematical programming problem, where a necessary condition for optimality is derived as a system of linear algebraic equations. A minimization algorithm is then used to determine numerically the feedback parameters of the closed-loop control. A numerical example involving a simply supported plate is provided to substantiate the theoretical results by demonstrating the effectiveness of the proposed method. The results of computer simulation studies compare favourably to optimal solutions obtained by the variational method (Blanton and Sadek 1992), indicating that the suggested approach provides better control of the vibrating plate