Approximate optimal control of distributed parameter systems.

Abstract

An approximation technique for determining the optimal control of distributed parameter systems is developed. Treated explicitly, to illustrate the basic algorithm, are systems with a single state variable and control variables which appear linearly. The system partial differential equation may be nonlinear. Extension can be made to systems with more than one state variable and/or nonlinear control variables. The basis of the approach involves the intro- duction of a finite Chebyshev series approximation for the state variable. State and control variable constraints are treated using penalty functions. Numerical results are obtained for a representative example involving the optimal heating of a metal slab. The computational results obtained compare favorably with results presented by other authors. vI approximation technique for determining the optimal control of distributed parameter systems is given in this paper. The technique represents an extension of one used by Johnson1 for a finite-thrust trajectory problem involving the solution of ordinary differential equations. The basis of the approach developed here involves the introduction of a finite Chebyshev series approximation for the state variable. The coefficients of the approximation are determined by insisting that the state variable approximation be equal to specified values of the true state variable at the finite set of nonuniformly distributed Chebyshev points. The values of the state variable at the Chebyshev points become the free parameters of the problem. Since a closed form analytic expression for the state variable is available it is possible to eliminate all numerical differentiation and integration. Consequently a problem with a functional performance index is transformed into one involving minimization of an ordinary function of many variables. Optimization of the performance index takes place when the values of the state variable are manipulated using one of the well known mathematical programming algorithms. Thus for the sake of computation, the roles of the state and control variable are reversed, from that prevalent in most common optimization techniques. To illustrate the basic algorithm to be developed, a one- dimensional heat conduction problem is considered. 2-3 It should be noted that this problem was chosen for illustrative purposes only and does not represent a limitation on the type of distributed parameter system for which the technique is applicable.