The zero divisor problem of multivariable stochastic adaptive control

Abstract Stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time. In order that the analysis of these algorithms is valid, zero divisions must be events of probability zero. This property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to Lebesgue measure. This result is deduced from the following general result established in this paper: a non-constant rational function of a finite set of random variables { x 1 }, x n } is absolutely continuous with respect to Lebesgue measure if the joint distribution function of { x 1 ,…, x n } has this property.