Control and dynamicsystems : advances in theory and by Cornelius T Leondes

By Cornelius T Leondes

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Basic Eng. (i960). 11. R. K. MEHRA, "Identification of Linear Dynamic Systems," paper presented at the IEEE Symposium on Adaptive Processes (8th), Pennsylvania (17-19 November I969). 12. T. KAIIATH, "An Innovation Approach to Least Squares Estimation, Part I," IEEE Trans, on Automatic Control, 13: 646-655 (December I96Ö). 13. P. L. SMITH, Estmation of Covariance Parameters in Time Discrete Linear Systems with Applications to Adaptive Filtering, Aerospace Corporation Report TOR-0059 (63ll)-23 (1971).

Let us first consider a way to determine this interval of un­ certainty, (x Q ,x^). Start at some point along the line, say the function at this point, F(x Q ). xn· Evaluate Move some distance, d, along the line and again evaluate the function at the new point F(x n ), χη = x Q + d. , ), is less than we are moving in the correct direction. , let x = x 0 - d, F(x-. ) or we should decrease is less than F(x Q ). d. Let us assume that Continue moving in this direction, evaluating the function, until it has a larger value than at the previous point.

This converges rapidly near the minimum, while converging very well farther away. For a quadratic function of n variables, Fletcher and Powell [15] prove that the procedure will converge to the 45 J. A. PAGE AND E. B. STEAR minimum in exactly n steps. For non-quadratic functions, convergence will require more steps depending upon the complexity of the problem. It has been shown by Rosen [2k] that this procedure is a special case of so called "Quasi-Newton" techniques, and also Pearson [25] has shown that the Fletcher-Powell procedure is a special case of the so called "Variable Metric" methods.

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