  When designing large-scale distributed controllers, the information-sharing
constraints between sub-controllers, as defined by a communication topology
interconnecting them, are as important as the controller itself. Controllers
implemented using dense topologies typically outperform those implemented using
sparse topologies, but it is also desirable to minimize the cost of controller
deployment. Motivated by the above, we introduce a compact but expressive graph
recurrent neural network (GRNN) parameterization of distributed controllers
that is well suited for distributed controller and communication topology
co-design. Our proposed parameterization enjoys a local and distributed
architecture, similar to previous Graph Neural Network (GNN)-based
parameterizations, while further naturally allowing for joint optimization of
the distributed controller and communication topology needed to implement it.
We show that the distributed controller/communication topology co-design task
can be posed as an $\ell_1$-regularized empirical risk minimization problem
that can be efficiently solved using stochastic gradient methods. We run
extensive simulations to study the performance of GRNN-based distributed
controllers and show that (a) they achieve performance comparable to GNN-based
controllers while having fewer free parameters, and (b) our method allows for
performance/communication density tradeoff curves to be efficiently
approximated.
