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Stability of distributed controllers with multiple delays

A model is presented of the dynamics of many controllers in a distributed system, each controlling one element of a vector and, as a system, trying to optimize a function of the vector. It is assumed that the controllers apply an adaptive method to achieve the optimal result. The optimization is challenging since the controllers have delayed information concerning the individual elements of the control vector. The potential exists for each element to be viewed with its own unique delay. Linear stability analysis provides an upper bound of the conditions for damped oscillations and a lower bound on the conditions for persistent oscillations. The results indicate that controllers can improve performance with respect to stability by: 1) communicating more frequently, 2) interacting in smaller groups, 3) slowing down the adaptive search process, and 4) creating more diversity in the delays.