Topology design for distributed formation control towards optimal convergence rate

This paper considers the optimal local leader selection for a leader-first follower minimally persistent formation system. The objective is to maximize the local convergence rate of the followers to the unique equilibrium under small perturbations. Except for the leader and the first follower, every other agent in the system follows exactly two agents called the local leaders and is responsible for maintaining a pair of desired distances from two local leaders. The control algorithm is the linearized form of the decentralized nonlinear control law proposed in our previous work. When the agents are distributed over a rectangular area, the selection of the optimal local leader for each follower is discussed, and it is discovered that the boundary optimality rule applies. The general case when agents distributed over an arbitrary convex domain is further considered based on the matrix perturbation theory. Information of agents in its sensing range is enough for the agent to pick up its optimal local leaders, which allows a distributed implementation of the proposed algorithm.