  The implementation of fringe tracking for optical interferometers is
inevitable when optimal exploitation of the instrumental capacities is desired.
Fringe tracking allows continuous fringe observation, considerably increasing
the sensitivity of the interferometric system. In addition to the correction of
atmospheric path-length differences, a decent control algorithm should correct
for disturbances introduced by instrumental vibrations, and deal with other
errors propagating in the optical trains. We attempt to construct control
schemes based on Kalman filters. Kalman filtering is an optimal data processing
algorithm for tracking and correcting a system on which observations are
performed. As a direct application, control schemes are designed for GRAVITY, a
future four-telescope near-infrared beam combiner for the Very Large Telescope
Interferometer (VLTI). We base our study on recent work in adaptive-optics
control. The technique is to describe perturbations of fringe phases in terms
of an a priori model. The model allows us to optimize the tracking of fringes,
in that it is adapted to the prevailing perturbations. Since the model is of a
parametric nature, a parameter identification needs to be included. Different
possibilities exist to generalize to the four-telescope fringe tracking that is
useful for GRAVITY. On the basis of a two-telescope Kalman-filtering control
algorithm, a set of two properly working control algorithms for four-telescope
fringe tracking is constructed. The control schemes are designed to take into
account flux problems and low-signal baselines. First simulations of the
fringe-tracking process indicate that the defined schemes meet the requirements
for GRAVITY and allow us to distinguish in performance. In a future paper, we
will compare the performances of classical fringe tracking to our Kalman-filter
control.
