Energy-optimal disturbance feedforward control for constrained deformable mirrors with thermoelastic actuation

This contribution focuses on designing a disturbance feedforward control for optically addressed deformable mirrors (OADMs). These mirrors can be used as intra-cavity actuators in high-power lasers to compensate for wavefront perturbations and for beam shaping within the framework of adaptive optics. For achieving the required deformations with high precision, the thermoelastic transfer behavior of the control intensity as well as disturbances due to the primary intensity have to be taken into account. The proposed control approach aims at minimizing the control effort in a spatial sense. Moreover, input constraints are considered explicitly, as the control intensity acts as a volumetric heat source which cannot cool the mirror. Furthermore, the 3D heat transfer in the cylindrical mirror is captured by a modal decomposition in the plate-direction in conjunction with a closed-form solution in the normal direction. By applying a similar approach to the mechanical equations, we are in the position to compute the equilibrium solution of the overall system in an efficient way. Based on the optimal solution, we subsequently derive an approximate solution by virtue of heuristics towards an online-capable algorithm. Finally, the performance is illustrated by means of simulations and a benchmark optimization.