Structure and approximations of LQG controllers based on a hybrid AO system model

The effects of a controller on the residual wavefront variance in an adaptive optics system can be represented by a discrete-time system. Consequently, the controller design is optimized by the solution of a discrete-time Linear-Quadratic-Gaussian (LQG) problem. The purpose of this paper is to analyze the structure of the LQG controller that minimizes the residual wavefront variance. It is shown that the LQG controller is an integral controller when the DM has no dynamics, there is no loop delay, and the PSD of the incident wavefront decreases by 40 db/decade at all frequencies. Nonzero loop delays result in a lead element being added to the controller with the zero of the lead element is at the origin. The dependence of the pole of the lead element on the loop delay is analyzed. Asymptotic approximations to the design are analyzed for fast frame rates and for incident wavefront dynamics.