Realizability of similar formation and local control of directed multi-agent networks in discrete-time

This paper introduces a new concept called similar formation and studies the realizability condition, concerning with the existence of a protocol that has the ability to drive a directed multi-agent network to a desired formation shape. With the help of complex Laplacian, it is shown that in order to uniquely realize a similar formation in the plane, a necessary and sufficient graphical condition is that the directed multi-agent network is 2-rooted, a new type of connectivity. Moreover, a distributed control law is also derived using local measurements for the purpose of stabilizing a directed multi-agent network to a desired formation shape in discrete time, which ensures globally asymptotic stability of the closed-loop system. Simulation results are provided as well to illustrate our results.