Controller Design for a Freeform Optical Surface Tracking System with a Non-linear Spring

Abstract

This paper presents a controller design approach aimed at optimizing the tracking capability of a freeform optical surface tracking system. It is shown that the second-order disturbance caused by the non-linearity in the system's spring and the changes in surface height is by far the biggest contributor to the tracking error. To reduce this error the paper proposes a position dependent spring force compensation, aiming to use the measurable non-linearity in the spring to effectively linearize the system. Thereafter, it is shown that the tracking error is mainly caused by the change in surface height and an unknown disturbance acting around the bandwidth. Since the change in surface height is known, a feedforward control structure is used to suppress the effect of this disturbance. Following this modification, the unknown disturbance becomes dominant. To suppress the effect of this disturbance a so called 'Constant in Gain Lead in Phase' controller is added that decreases the magnitude of the sensitivity function in the active frequency range of the disturbance, without increasing the sensitivity's magnitude at other frequencies. The proposed controller modifications are all experimentally validated on a freeform optical surface measurement machine through the evaluation of the tracking error during a tilted flat measurement.