Minimal interconnection topology in distributed control design

In this paper, we consider a distributed control design problem. Multiple agents (or subsystems) that are dynamically uncoupled need to be controlled to optimize a joint cost function. An interconnection graph specifies the topology according to which the agents can access information about each others' state. We propose and partially analyze a new model for determining the influence of the topology of the interconnection graph on the performance achieved by the subsystems. We consider the classical linear-quadratic regulator (LQR) cost function and propose making one of the weight matrices to be topology dependent to capture the extra cost incurred when more communication between the agents is allowed. We present results about optimal topologies for some models of the dependence of the weight matrix on the communication graph. We also give some results about the existence of "critical prices" at which adding supplementary edges becomes detrimental to closed-loop performance. One conclusion of the work is that if the communication between the agents comes at a cost, then adding communication edges may be harmful for the system performance.