Random walk theory probabilities that a radar servo will fail to lock on

The error detector in a certain sampled-data servomechanism, whose controller computes two simple, coupled, and first-order difference equations, periodically estimates the sign of the difference between the output of the difference equations and the input to the servo. Noise occasionally causes the servo's error detector to misestimate the sign of this error. If the error detector misestimates the sign of the error too often, the error can randomly walk into a region in which the error detector makes completely unreliable reports, thereby preventing the servo from ever locking onto its input and nulling the error. This paper shows how to calculate the probability that the servo will fail to lock onto its input, for any pair of initial values of the error and its first time derivative and for any finite number of iterations of the two difference equations, provided that certain probability distributions of the error detector's sign estimates are known.