Minimum-Variance Control for Astronomical Adaptive Optics with Resonant Deformable Mirrors

Adaptive Optics (AO) systems use a Deformable Mirror (DM) to counter in real-time the nefarious effects of atmospheric turbulence on ground-based telescope imaging. This article presents a brief historical overview of AO design, seen as a strongly multi-variable Minimum-Variance (MV) disturbance rejection problem associated with a hybrid continuous/discrete time MV control problem. It is shown that for a wide class of LTI DM and turbulence models, this hybrid MV problem can be transformed into an equivalent discrete-time LQG formulation. A discrete-time stochastic model enables to compute the optimal control in standard reconstructed feedback form and to evaluate performance degradation for simpler sub-optimal solutions. An example of tip-tilt DM control for the European Extremely Large Telescope (E-ELT) is presented.