  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 (undirected) graph structure. 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 a
high-gain coupling; consequently we call our novel synchronization method
`(node-wise) funnel coupling.' By adjusting the conventional proof technique in
the funnel control study, we are even able to obtain 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, is analyzed in
this paper. In particular, we introduce a single scalar dynamics called
`emergent dynamics' which describes the emergent synchronized behavior of the
multi-agent system under funnel coupling. 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. We illustrate this idea via the example of
a distributed median solver based on funnel coupling.
