Design of surface shape control for large two-dimensional arrays

This paper develops control design technology for active shape control of reflective surfaces using large spatially distributed actuator arrays. The potential applications are in astronomy, adaptive optic, beam control, space-based imaging, and other optics and imaging applications. In a very large lightweight active reflector, surface shape (figure) might be controlled by an array of actuators and sensors that counts millions of cells. The control technology discussed in this paper is scalable to these large array dimensions. This paper develops a classically motivated design methodology for distributed localized control laws of very large actuator/sensor arrays. The methodology uses standard PI-compensation, plus lags and/or notch-filters, to deal with temporal dynamics in each actuator channel. It achieves scalability to very large array sizes by imposing spatially localized fixed-form constraints on the control law structure. In this setup, the entire spatial-temporal design model can be transformed, via Laplace transforms in time and two-dimensional (2-D) discrete Fourier transforms in space, to produce a family of dynamic systems whose closed-loop characteristics can be subjected to standard classical control-engineering specifications, including stability, performance, and robustness. These specifications can be satisfied for all members of the family by solving linear programs (LPs) to find parameters of the fixed-form structure. The veracity of this methodology is illustrated with a design example loosely resembling an actively controlled reflector whose local deformations are controlled by a hexagonal array of actuator/sensor cells.