On the optimality of localized distributed controllers

Design of optimal distributed controllers with a priori assigned localization constraints is, in general, a difficult problem. Optimality of a closed-loop system is desirable because it guarantees, among other properties, favorable gain and phase margins. These margins provide robustness to different types of uncertainty. Alternatively, one can ask a following question: given a localized distributed exponentially stabilizing controller, is it inversely optimal with respect to some physically meaningful cost functional? We study this problem for linear spatially invariant (LSI) systems and establish a frequency domain criterion for inverse optimality (in the LQR sense). We utilize this criterion to separate localized distributed controllers that are never optimal from localized distributed controllers that are optimal. In the latter case, we provide examples to demonstrate optimality with respect to physically appealing cost functionals. These cost functionals are characterized by state penalties that are not fully decentralized. Our results can be used to motivate design of both optimal and inversely optimal controllers for other classes of distributed control problems.