  We present the results obtained with an end-to-end simulator of an Extreme
Adaptive Optics (XAO) system control loop. It is used to predict its on-sky
performances and to optimise the AO loop algorithms. It was first used to
validate a novel analytical model of the fitting error, a limit due to the
Deformable Mirror (DM) shape. Standard analytical models assume a sharp
correction under the DM cutoff frequency, disregarding the transition between
the AO corrected and turbulence dominated domains. Our model account for the
influence function shape in this smooth transition. Then, it is well-known that
Shack-Hartmann wavefront sensors (SH-WFS) have a limited spatial bandwidth, the
high frequencies of the wavefront being seen as low frequencies. We show that
this aliasing error can be partially compensated (both in terms of Strehl ratio
and contrast) by adding priors on the turbulence statistics in the framework of
an inverse problem approach. This represents an alternative to the standard
additional optical filter used in XAO systems. In parallel to this numerical
work, a bench was aligned to experimentally test the AO system and these new
algorithms comprising a DM192 ALPAO deformable mirror and a 15x15 SH-WFS. We
present the predicted performances of the AO loop based on end-to-end
simulations.
