  Designing distributed optimal controllers subject to communication
constraints is a difficult problem unless structural assumptions are imposed on
the underlying dynamics and information exchange structure, e.g., sparsity,
delay, or spatial invariance. In this paper, we borrow ideas from graph signal
processing and define and analyze a class of Graph Symmetric Systems (GSSs),
which are systems that are symmetric with respect to an underlying graph
topology. We show that for linear quadratic problems subject to dynamics
defined by a GSS, the optimal centralized controller is given by a novel class
of graph filters with transfer function valued filter taps and can be
implemented via distributed message passing. We then propose several methods
for approximating the optimal centralized graph filter by a distributed
controller only requiring communication with a small subset of neighboring
subsystems. We further provide stability and suboptimality guarantees for the
resulting distributed controllers. Finally, we empirically demonstrate that our
approach allows for a principled tradeoff between communication cost and
performance while guaranteeing stability. Our results can be viewed as a first
step towards bridging the fields of distributed optimal control and graph
signal processing.
