A Data-Driven ${\cal H}_{2}$ -Optimal Control Approach for Adaptive Optics

Adaptive optics (AO) is used in ground-based astronomical telescopes to improve the resolution by counteracting the effects of atmospheric turbulence. Most AO systems are based on a simple control law that neglects the temporal evolution of the distortions introduced by the atmosphere. This paper presents a data-driven control design approach that is able to exploit the spatio-temporal correlation in the wavefront, without assuming any form of decoupling. The approach consists of a dedicated subspace-identification algorithm to identify an atmospheric disturbance model from open-loop wavefront sensor data, followed by ${\cal H}_{2}$-optimal control design. It is shown that in the case that the deformable mirror and wavefront sensor dynamics can be represented by a delay and a two taps impulse response, it is possible to derive an analytical expression for the ${\cal H}_{2}$-optimal controller. Numerical simulations on AO test bench data demonstrate a performance improvement with respect to the common AO control approach.