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Minimum‐variance control of astronomical adaptive optic systems with actuator dynamics under synchronous and asynchronous sampling

Adaptive optic (AO) systems are now routinely used in ground‐based telescopes to counter the effects of atmospheric turbulence. A deformable mirror (DM) generates a correction wavefront, which is subtracted from the turbulent wavefront using measurements of the residual phase provided by a wavefront sensor (WFS). Minimizing the variance of the residual phase defines a sampled data control problem combining a continuous time minimum‐variance (MV) performance criterion with a discrete‐time controller. For a fairly general class of linear time‐invariant DM and turbulence WFS models, this control problem can be transformed into an equivalent discrete‐time LQ optimization problem involving a set of (discrete‐time) control‐sufficient statistics of the incoming continuous‐time turbulence. This paper shows how to constructively solve this MV problem in the presence of DM's dynamics, starting from continuous‐time models of DM and turbulence. This result is extended to the case of asynchronous DM/WFS sampling. An illustrative application to optimal control of tip‐tilt turbulent modes for the European extremely large telescope in the presence of first‐order DM's dynamics is presented. Copyright © 2010 John Wiley & Sons, Ltd.