To enable optical long baseline interferometry toward faint objects, long integrations are necessary despite atmospheric turbulence. Fringe trackers are needed to stabilize the fringes and thus increase the fringe visibility and phase signal-to-noise ratio (SNR), with efficient controllers robust to instrumental vibrations, and to subsequent path fluctuations and flux drop-outs. We report on simulations, analysis and comparison of the performances of a classical integrator controller and of a Kalman controller, both optimized to track fringes under realistic observing conditions for different source magnitudes, disturbance conditions, and sampling frequencies. The key parameters of our simulations (instrument photometric performance, detection noise, turbulence and vibrations statistics) are based on typical observing conditions at the Very Large Telescope observatory and on the design of the GRAVITY instrument, a 4-telescope single-mode long baseline interferometer in the near-infrared, next in line to be installed at VLT Interferometer. We find that both controller performances follow a two-regime law with the star magnitude, a constant disturbance limited regime, and a diverging detector and photon noise limited regime. Moreover, we find that the Kalman controller is optimal in the high and medium SNR regime due to its predictive commands based on an accurate disturbance model. In the low SNR regime, the model is not accurate enough to be more robust than an integrator controller. Identifying the disturbances from high SNR measurements improves the Kalman performances in case of strong optical path difference disturbances.
