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8). 3 DYNAMIC PROGRAMMING SOLUTION T h e dynamic programming solution is based on principles that are a direct consequence of the structure of the problem. 1 The Principle of Causality T hi s principle is a fundamental property of deterministic multistage systems. It may be stated as follows. Principle of Causality: T h e state x ( k ) and control parameter 01 at the Kth stage, together with the sequence 38 3. OPTIMAL CONTROL OF DISCRETE SYSTEMS + of controls u[k, r - 11 = [u(k),u(k I), ... , u(r - l)] uniquely determine the state of the rth stage x ( r ) .

I n the gradient method, the simpler problem was obtained by a linearization of the original problem, whereas the Newton-Raphson algorithm was obtained by considering a second-order expansion of the original problem. Such techniques are called direct methods. Another class of methods for solving optimization problems is the class of indirect methods. Indirect methods are obtained from a consideration of the theoretical conditions for an optimum. Here the theory was used to obtain closed-form solutions to the problem.

PARAMETER OPTIMIZATION methods, and the interested reader should consult the references listed in the bibliography at the end of this chapter. However, one particular class of methods, the so-called conjugate direction or conjugate gradient methods, which were originally developed for minimizing quadratic forms, have recently been shown to be particularly useful and will be discussed briefly. These methods are claimed to exhibit a speed of convergence that approaches that of the Newton-Raphson algorithm without the need for the formation or inversion of the matrix of second partial derivatives L,, .

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