Modelling techniques for optimal control of distributed parameter systems

Abstract

A direct method for solving optimal control problems by parameterizing control variables is considered. This control parameterization technique is formulated for controlling a class of self-adjoint distributed parameter systems using open-closed loop forces applied at discrete points in space. In particular, optimal control of flexible continuous structures is studied with the objective of minimizing a given performance index over a prescribed time interval. The performance index of the problem is taken as a combination of the total energy of the structure and the penalty terms on the expenditure of the open-closed loop forces used in the control process. The control is to be implemented by discrete sets of displacement sensors (closed-loop forces) and force actuators (open-loop forces) that monitor the response of the structure. In contrast to variational methods, a direct application of control parameterization is used in this study. The method is based on polynomial (or trigonometric function) approximation of each open-loop control variable that converts the linear quadratic problem into a mathematical programming problem, where the necessary condition of optimality is derived as a system of linear algebraic equations. Furthermore, the optimal feedback parameters of the closed-loop control forces are numerically determined from the solution of the total energy minimization problem. The effectiveness of the method is demonstrated by applying it to an example of a simply supported beam subject to initial disturbances.