  The problem of distributed controller synthesis for formation control of
multi-agent systems is considered. The agents (single integrators) communicate
over a communication graph and a decentralized linear feedback structure is
assumed. One of the agents is designated as the leader. If the communication
graph contains a directed spanning tree with the leader node as the root, then
it is possible to place the poles of the ensemble system with purely local
feedback controller gains. Given a desired formation, first one of the poles is
placed at the origin. Then it is shown that the inter-agent weights can be
independently adjusted to assign an eigenvector corresponding to the formation
positions, to the zero eigenvalue. Then, only the leader input is enough to
bring the agents to the desired formation and keep it there with no further
inputs. Moreover, given a formation, the computation of the inter-agent weights
that encode the formation information, can be calculated in a decentralized
fashion using only local information.
