  In this paper, we introduce a nonlinear time-varying coupling law, which can
be designed in a fully decentralized manner and achieves approximate
synchronization with arbitrary precision, under only mild assumptions on the
individual vector fields and the underlying graph structure. Meanwhile, we
consider undirected graphs and scalar input affine systems for simplicity. The
proposed coupling law is motivated by the so called funnel control method
studied in adaptive control under the observation that arbitrary precision
synchronization can be achieved for heterogeneous multi-agent systems by the
high-gain coupling, and thus, we follow to call our coupling law as
`(node-wise) funnel coupling.' By getting out of the conventional proof
technique in the funnel control study, we now can obtain even asymptotic
synchronization with the same funnel coupling law. Moreover, the emergent
collective behavior that arises for a heterogeneous multi-agent system when
enforcing arbitrary precision synchronization by the proposed funnel coupling
law, has been analyzed in this paper. In particular, we introduce a single
scalar dynamics called `emergent dynamics' that is capable of illustrating the
emergent synchronized behavior by its solution trajectory. Characterization of
the emergent dynamics is important because, for instance, one can design the
emergent dynamics first such that the solution trajectory behaves as desired,
and then, provide a design guideline to each agent so that the constructed
vector fields yield the desired emergent dynamics. A particular example
illustrating the utility of the emergent dynamics is given also in the paper as
a distributed median solver.
