Control of the Optical Surface of a Thin Deformable Primary Mirror with Application to an Orbiting Astronomical Observatory

Abstract

One of the more significant technological problems associated with the orbital operation of large astronomical telescopes is the fabrication and maintenance of the primary mirror surface to the tolerance required for diffraction-limited performance. An interesting approach to the solution of this problem involves continuously measuring and automatically correcting the optical surface of a thin deformable mirror by means of discrete actuators located on its rear surface. The realization of diffraction-limited performance from a telescope in space by this method rests on the ability of the designer to achieve extremely accurate control of a highly complex, interacting, multivariable system. This paper presents the results of a detailed study of the discrete control of linear distributed systems with specific application to the design of a practical controller for a plant representative of a telescope primary mirror for an orbiting astronomical observatory.The problem of controlling the distributed plant is treated by employing modal techniques to represent variations in the optical figure. Distortion of the mirror surface, which arises primarily from thermal gradients, is countered by actuators working against a backing structure to apply a corrective force distribution to the controlled surface. Each displacement actuator is in series with a spring attached to the mirror by means of a pad intentionally introduced to restrict the excitation of high-order modes. Control is then exerted over a finite number (equal to the number of actuators) of the most significant modes.A quadratic performance index incorporating image quality and control effort is formulated which permits determination of the trade-off between the number of actuators and optical purity, A criterion for defining actuator placement and pad size is presented which minimizes the tendency of the controller to excite the unmonitored modes in a plant modeled by a flat plate. By the model expansion technique the mirror equation of motion is transformed to a set of uncoupled, linear, time-invariant, ordinary differential equations and the desired dynamic response and static accuracy is achieved by application of the classical single-variable control design techniques.