Optimal Control of a Class of Time-delayed Distributed Systems by Orthogonal Functions

A class of optimal control problems for multi-dimensional structures described by a linear partial differential-difference equation is considered. The control mechanism involves the simultaneous application of time-delayed feedback and open-loop controllers which provides an effective means of suppressing vibrations in the structures. A computationally attractive method for determining the optimal open-loop control of an optimization time-delay problem with quadratic performance index is presented. The method is based on using finite orthogonal expansions to approximate state and open-loop control variables. The representation leads to a system of linear algebraic equations in terms of feedback parameters as the necessary condition of optimality. This method provides a straightforward and convenient approach for digital computation. Thus the difficulty in obtaining the solution of the coupled initial-boundary-terminal-value problem with both delayed and advanced arguments, which is always required in applying the Pontryagins maximum principle to optimization of delay systems, is avoided. The unknown feedback parameters are numerically evaluated from the solution of the energy minimization problem. A numerical example is given to demonstrate the applicability and effectiveness of the proposed method.