Non common path aberration correction with non linear WFSs

Non Common Path Aberrations (NCPA) are a usual problem encountered when using an Adaptive Optics System (AO)to produce corrected images in an astronomical instrument. The usual way to correct for such NCPA is to introduce anoffset in the WaveFront Sensor (WFS) signals that corresponds to the aberration to correct. In such a way, when the AOloop is closed, the Deformable Mirror (DM) will converge to the shape required to null the NCPA. The method assumesthat the WFS operation is linear and completely described by some pre-calibrated interaction matrix. This is not the casefor some frequently used wavefront sensors like the Pyramid sensor or a quad-cell Shack-Hartmann sensor. Here wepresent a method to work in closed loop with a Pyramid Wavefront Sensor (PWS), or more generally a non linear WFS,while introducing a static offset on the DM. The DM shape is kept constant even if the AO residuals change over timebecause of variations of seeing, wind speed and so on. Results from numerical simulation and first on sky data from LBTFLAO system and LUCI2 facility instrument are presented.