Optimization of the adaptive function by the -transform method

A study is made on the optimization of the adjustment process in a self-optimizing system under rather general assumptions. The system is designed to keep a performance parameter m either at a prescribed value or at an unknown extremal value. A direct measurement on m is made and the adjustment is based on measured m only. Factors considered are (1) the finite measuring interval, (2) the necessity of looking ahead one interval, (3) the probable error in measurement, and (4) the changing situation. A set of weighting factors on present and past data, and the proper value of test bias for extremal seeking systems, are determined by a least square optimization process. The criterion of optimization is least reduction in m for peak seeking systems and least square error in m for systems with prescribed value of m. Two types of extremal seeking systems are studied. The alternate biasing systems are found to be superior in performance compared to the derivative sensing systems. This paper has been published by the AIEE as Paper No. CP 59-1296.