Tractability Of Complex Control Systems

This thesis is divided into two main parts. In the first part, we consider the problem of efficiently computing wavefront estimates for use in adaptive optics hardware on ground-based telescopes. Our contribution is a warm-started single-iteration multigrid algorithm that performs as well as conventional vector-matrix-multiplication methods, but at a fraction of the computational cost. We used numerical simulations to compare our algorithm to a variety of other published methods, and validated our findings at the Palomar Observatory. In the second part, we consider feedback control subject to an information constraint. Using a novel algebraic framework, we are able to prove many structural results, including a new convexity result, in a natural and purely algebraic way. We also develop a new condition we call internal quadratic invariance, under which the controller synthesis can be cast as a convex optimization problem. This describes the most general class of tractable decentralized control problems known to date. Both parts of the thesis fit into the broader question of tractability of complex systems. In the first part we look at a practical example which is difficult because of the large number of sensors and actuators. In the second part, we look at decentralized control, which is difficult because of the non-classical information constraint.