Stability of an N-loop ensemble-reference phase control system

The paper considers a control system for driving N sinusoidal wave forms with randomized time-varying phase shifts into phase coincidence with a reference which in the linearized problem is an average of the phase states of the remaining channels. A closed-form linearized solution is presented for the general N-element adaptive optics system in terms of the feedback parameters of the separate control loops. Attention is directed to whether a perturbation in any of the system inputs will generate growing or decaying exponential responses in the servo system. It is shown that adaptive optics systems which reference to the ensemble average phase of the remaining channels do not exhibit sufficient cross coupling to lead to a stability problem, for any order of an N-loop system. For almost any practical system (N greater than 10), the stability considerations and the system behavior are virtually determined by that of a single isolated loop.