Optimal Actuator Placement for the High-Precision Control of Quasi-Static Elastic Plates

Abstract

Adaptive optical elements are often used to compensate for disturbances in the beam to enhance the image quality. If the element is thin, the force profile for its motion may lead to a significant unevenness of the optical surface impairing the image quality. A remedy falls back on the overactuation with a larger number of actuators. However, the question arises of what the minimal number of actuators for a given optical requirement is. Thus, we investigate the case of a low-frequent reference acceleration in the spatial degrees of freedom of a rigid body, where the elastic modes of a thin plate experience quasistatic amplification. Considering the information on the elastic plate modes, an optimal mapping of the demanded reference acceleration on the actuators leading to a minimal surface deviation from ideally flat is derived analytically. Furthermore, this mapping and the information of the nonsquare relative gain array (RGA) are exploited to obtain an actuator placement to further reduce the elastic plate deformation. Numerical results show a major improvement in flattening the optical surface profile compared to the case of neglecting the information about the elastic plate modes and lead to a Pareto front that supports the choice of a minimal number of actuators.