Regret Analysis of Distributed Online Control for LTI Systems with
  Adversarial Disturbances
Ting-Jui Chang Shahin Shahrampour
This paper addresses the distributed online control problem over a network of
linear time-invariant (LTI) systems (with possibly unknown dynamics) in the
presence of adversarial perturbations. There exists a global network cost that
is characterized by a time-varying convex function, which evolves in an
adversarial manner and is sequentially and partially observed by local agents.
The goal of each agent is to generate a control sequence that can compete with
the best centralized control policy in hindsight, which has access to the
global cost. This problem is formulated as a regret minimization. For the case
of known dynamics, we propose a fully distributed disturbance feedback
controller that guarantees a regret bound of $O(\sqrt{T}\log T)$, where $T$ is
the time horizon. For the unknown dynamics case, we design a distributed
explore-then-commit approach, where in the exploration phase all agents jointly
learn the system dynamics, and in the learning phase our proposed control
algorithm is applied using each agent system estimate. We establish a regret
bound of $O(T^{2/3} \text{poly}(\log T))$ for this setting.