Convergence Analysis of Signed Nonlinear Networks

Abstract

This work analyzes the convergence properties of signed networks with nonlinear edge functions. We consider diffusively coupled networks comprised of maximal equilibriumindependent passive (MEIP) dynamics on the nodes, and a general class of nonlinear coupling functions on the edges. The first contribution of this work is to generalize the classical notion of signed networks for graphs with scalar weights to graphs with nonlinear edge functions using notions from passivity theory. We show that the output of the network can finally form one or several steady-state clusters if all edges are positive, and in particular, all nodes can reach an output agreement if there is a connected subnetwork spanning all nodes and strictly positive edges. When there are non-positive edges added to the network, we show that the tension of the network still converges to the equilibria of the edge functions if the relative outputs of the nodes connected by non-positive edges converge to their equilibria. Furthermore, we establish the equivalent circuit models for signed nonlinear networks, and define the concept of equivalent edge functions which is a generalization of the notion of effective resistance. We finally characterize the relationship between the convergence property and the equivalent edge function, when a non-positive edge is added to a strictly positive network comprised of nonlinear integrators. We show that the convergence of the network is always guaranteed, if the sum of the equivalent edge function of the previous network and the new edge function is passive.

Synchronization in Electrical Circuits With Memristors and Grounded Capacitors

“…Proof: By Theorem 1 there exists φ such that (19) holds. Lemma 3 implies that the solution ϕ to ( 16) satisfies (20). Finally, Lemma 1 gives the desired result (35).…”

“…Note that the matrix D M ∂g(ϕ) ∂ϕ D M can be viewed as a statedependent Laplacian, such that (18) can be regarded as a consensus network with state-dependent weights. In other words, (18) represents a network dynamical system with dynamics on both the nodes (capacitors) and edges (memristors), as studied in, e.g., [19], [20].…”

“…This letter considers a specific class of circuits comprising memristors and grounded capacitors. Circuits of this class have dynamics that can be related to the literature on consensus networks (including dynamics on the edges) for which synchronisation is studied regularly [18], [19], [20].…”

Speeding Up Velocity Consensus Control with Small World Communication Topology for Unmanned Aerial Vehicle Swarms

“…Consensus-based approaches [1] realize the consensus of swarm decision making based on dynamic functions. In [2], Hao proved the convergence of a connected graph of the multiagent cooperative decision problem and designed the invulnerability topology generation algorithm to guarantee the connectivity when k edges or k nodes are removed from the graph. However, the problem of rapid consensus in decision making is not solved.…”