Global synchronization of multi-agent systems with nonlinear interactions

TL;DR

The paper tackles global synchronization of multi-agent systems where interactions are mediated by a general monotonic signal $s:[-1,1]\to[-1,1]$, capturing nonlinear and potentially discontinuous communication. It analyzes the continuous-time dynamics $\dot{\boldsymbol{x}}=\boldsymbol{D}^{-1}\boldsymbol{A}s(\boldsymbol{x})-\boldsymbol{x}$ on a connected graph, proving that synchronization equilibria correspond exactly to the fixed points of $s$ and establishing stability criteria based on local estimation properties around those fixed points. A Lyapunov-based LaSalle argument shows convergence to the synchronization set under global underestimation, and the authors demonstrate that graph symmetry can enforce synchronization between agents with similar neighborhoods, with all-to-all and complete bipartite topologies guaranteeing full synchronization. The results illuminate how communication nonlinearity and network connectivity jointly shape coordination, providing a framework for designing advanced synchronization strategies in systems with nonlinear or quantized communications.

Abstract

The paper addresses the synchronization of multi-agent systems with continuous-time dynamics interacting through a very general class of monotonic continuous signal functions that covers estimation biases, approximation of discrete quantization, or state-dependent estimation. Our analysis reveals that, in the setup under consideration, synchronization equilibria are exactly the fixed points of the signal function. We also derive intuitive stability conditions based on whether the signal underestimates or overestimates the state of the agents around these fixed points. Moreover, we show that network topology plays a crucial role in asymptotic synchronization. These results provide interesting insights into the interplay between communication nonlinearity and network connectivity, paving the way for advanced coordination strategies in complex systems.

Figures (4)

• Figure 1: State evolution for agents with global underestimation signal over a connected graph (Erdős-Rényi with $N=100$ and $p=0.1$) with random uniform initial conditions and $s(x) = \tanh(x)$.
• Figure 2: Illustration of the local stability conditions for synchronization equilibria as described in Theorem \ref{['thm:stability_consensus']}.
• Figure 3: State evolution for agents with a globally overestimating signal under the same conditions as in Figure \ref{['fig:opinion_underestimation']}, $s(x) = 0.8 \sin(2x)/\sin(1.6)$ for $x\in [-0.8,0.8]$ and $s(x) = x$ otherwise.
• Figure 4: State evolution for agents with a globally overestimating signal ($s(x) = \tanh(20 x)$) over a line graph of 6 agents. The agents do not converge to a synchronization equilibrium for these random initial conditions.

Theorems & Definitions (23)

• Remark 1
• Lemma 1: Perron-Frobenius
• Proposition 1
• proof
• Definition 1
• Lemma 2
• proof
• Proposition 2
• proof
• Definition 2