Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Extended arXiv version

Abstract

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 highgain 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.

Edge-wise funnel output synchronization of heterogeneous agents with relative degree one

“…Under a set of mild assumptions, it was shown in [1] that (3) holds and that this implies the desired synchronization performance…”

“…The emergent dynamics was shown to be determined by an algebraic equation in [1], and under a contractivity assumption each of the agents synchronizes to its solution. It was further studied how networks can be synthesized which exhibit these emergent dynamics.…”

“…Furthermore, the task of designing a heterogeneous network with a desired collective behavior is much simpler, since it is not necessary to solve a complicated algebraic equation. Additional extensions to the results of [1], which are also applicable to, but not studied in [1], are as follows:…”

“…In the recent work [1], arbitrary precision approximate synchronization for scalar heterogeneous multi-agent systems ẋi (t) = f i (t, x i (t)) + u i (t), i ∈ N ,…”

“…The present brief paper is devoted to an extension of the results of [1] and to overcome some of its limits. In this spirit, we skip an extensive literature review on synchronization of multi-agent systems and simply refer to the precursor [1].…”