Design of surface shape control for large two-dimensional arrays

Abstract

Abstract-This work 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 classicallymotivated 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 2-D discrete Fourier transforms in space, to produce a family of dynamic systems whose closed loop characteristics can be subjected to standard classical controlengineering specifications, including stability, performance, and robustness. These specifications can be satisfied for all members of the family by solving linear programs 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.

A PDE viewpoint on basic properties of coordination algorithms with symmetries

“…Phenomena related to the latter fact -e.g. bad system conditioning for robustness/performance (see [3], [24], [14], [8]), or slow consensus in non-"small-world" networks (see [21], [16] and references therein) -have been independently observed in several applications. The following reviews several examples with our common viewpoint.…”

“…For instance, distributed control is applied e.g. to stabilize a perfectly smooth configuration for a membrane or segmented telescope mirror (see [24], [13], [3]), or for cross-directional control of industrial paper machines (see [25], with additional applications); platoons of cars following each other are considered a key technology enabling automated highways (see [26]); the "consensus problem" where a set of communicating agents must reach agreement on a decision value has become standard (see [27] for an early discussion, [20] for a review).…”

“…phenomena due to very different system response to long-range and short-range effects (see e.g. [3], [24], [14], [8], [21], [16]). …”

Design of Low-Bandwidth Spatially Distributed Feedback

“…• the actuator limit (30) • the performance specification (31) • robustness to plant variation (34) • robustness to controller variation (36) • robustness to smoothing operator variation (37).…”

“…A deformable mirror control analysis closely related to formulation in this paper is presented in [31] where further references can be found. A related area is shape control of large scale deformable reflectors for ground and space applications, e.g., see [22].…”