By R. Boudarel, J. Delmas and P. Guichet (Eds.)
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Additional info for Dynamic Programming and Its Application to Optimal Control
In addition, the control law is linear in x, which facilitates its storage, and its calculation by the controller. Thus, in this special case, which is the problem of classical control theory, a simple solution can be obtained. We shall now develop these results. 1 De3nitions and Notation Suppose the process terminates at time N. The optimal recurrence equation evolves backward in time, so that it is convenient to introduce the time variable m, related to real time n by n = N - m. With this convention, a linear process is described by an equation of the form X, - = F, X, + H , U, .
9). With the following formula, an integral can be replaced by a sum, in which only a set of values of the function to be integrated, taken at particular points, appears : j1f(x) dx = -1 Hif(xi) i= 1 +E. 11) Here xi is the ith root of P,(x): P,(Xi) = 0. 12) The error E is zero iff(x) is a polynomial of degree at most 2r - 1, otherwise it is of order 2r in x. 1 1) is an approximate formula. 2 Application to First-Order Processes With a first-order process we wish to solve CP,,+~(X)= max[r(x, UER 4 + cp,(f(x, u)>l, u, x E R .
If P I-[ Calculate: 1T - u=u+6 FIG. 1. Flow chart for maximum search by simple comparison. $ ( u ) is sufficiently smooth, this procedure locates the optimum with a precision of the order of h/2, using ( h - a)/h evaluations of $(u). For the case of a unimodal function, Boudarel ef af. [ 1, Vol. 2 THE PROBLEM OF DIMENSIONALITY 31 minimum number of evaluations of $(u). This method uses thepropertiesof the Fibonacci number sequence, and the reader is referred to Chapter 5 of the work mentioned for details about the use of this very efficient algorithm.