On controlled sinusoidal phase coupling

The work in this paper considers a heterogeneous coordinated phase control problem in which most agents follow a classic sinusoidal coupling protocol, but a select few agents act as leaders. These leaders have the ability to report some value other than their current phase to their neighbors. This heterogeneous model is motivated by systems in biology and in engineering. In biological contexts, observations have been made that groups of trained animals can bias the behavior of a much larger groups of untrained animals. Translating these results to engineered contexts is of interest to, for example, reduce the number of human operators necessary to control a fleet of many autonomous agents. The contributions of this paper include a general reachability result, a proof that a chain of four or more agents is uncontrollable by a single leader, and a nonlinear controllability analysis of some example problems. An interesting result is that symmetry about the leader node is not sufficient to guarantee uncontrollability of the follower nodes, as it is in the related controlled linear agreement problem.