This paper is concerned with tracking a vector of signal process generated by a family of distributed (geographically separated) nonlinear noisy dynamic subsystems over the binary symmetric channel. Nonlinear subsystems are subject to bounded external disturbances. Measurements are also subject to bounded noises. For this system and channel, subject to constraints on transmission rates, cross over probabilities and Lipschitz constants, a simple methodology is presented ensuring tracking with bounded mean absolute error.

In communication engineering the problem of synchronization is important and is usually achieved by use of a phase-locked loop. If for any reason synchronization is lost, then it is not only necessary that it be restored but it be done so in the shortest possible time. This correspondence considers precisely this problem. It has been shown recently [2, pp. 253-259] that in case the loop filterGis ideal low-pass, the optimum nonlinear feedback controller that achieves synchronization in the shortest possible time is an on-off device. This correspondence considers the filterGto be nonideal and assumes that it is described by a differential equation of arbitrary order. For the class of admissible feedback controllers this correspondence considers polynomials of arbitrary but finite degree with coefficients chosen from a closed bounded subset of an Euclidean space of dimension equal to the degree of the polynomials. It is shown here that the optimal feedback controller is a polynomial of the highest degree admissible with coefficient-vector taking values at one of the vertices of the cube. The result is illustrated by a numerical example involving a second-order filter.

The results of the above paper,1 concerning a simulation study of an adaptive pattern classification procedure using a mean-square error performance criterion and using an estimated Bayes rule procedure, are compared with theoretical solutions obtained by using an exact Bayes rule procedure. An inherent defect of the mean-square error criterion is also pointed out, and a comparison between a system using this criterion and two other systems is made.