Optimum Design of a Feedback Controller for a Phase-Locked Loop

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 filter
G
is 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 filter
G
to 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.