Maximally Reliable Exponential Prediction Equations for Data-Rate-Limited Tracking Servomechanisms*

Abstract

Certain radars, sonars, and other sampling instruments periodically measure a variable but can measure accurately only if prediction equations provide the instrument with an accurate prediction of the next value of this variable. For these instruments, it is appropriate to define reliability as the probability that the error in this prediction does not exceed some limit. In choosing the form and parameters of the prediction equations, it is reasonable to attempt to maximize this reliability. Assumptions that the prediction equations utilize linear error measurements, are recursive, and provide least-squares smoothing with an exponential weighting function establish a realistic basis for calculating the reliability. The first through the third orders of these equations predict the variable as reliably as possible in the presence of large initial errors in the variable and its velocity, provided that the smoothing interval of the prediction equations is sufficiently short. The dynamic error component of the prediction error of these equations is proportional to a smoothed version of the qth time-derivative of the variable, where q is the order of the prediction equations. The assumption that the measurement errors are uncorrelated and stationary makes it possible to calculate the standard deviation of the random component of the prediction error. On the assumption that the random component of the prediction error has a normal (or Gaussian) probability distribution, there exists a safety factor which is monotonically related to the reliability. The choice of the smoothing interval of the prediction equations which maximizes this safety factor can be found, which in turn permits the optimum safety factor to be calculated. The ratio of pairs of these optimum safety factors determines which order