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Consensus of A Class of Nonlinear Systems with Varying Topology: A Hilbert Metric Approach

Dongjun Wu

TL;DR

This work addresses consensus in continuous-time nonlinear multi-agent systems with time-varying topology by developing a Hilbert-metric framework. By analyzing contraction on cones and using accumulated/averaging graphs, it establishes that a quasi-strongly connected accumulated graph guarantees asymptotic consensus, and continuity of the limiting graph yields exponential convergence. The key contributions include relaxing key technical assumptions from prior nonlinear results and achieving exponential consensus under mild measurability of switching, with a simpler, non-Lyapunov proof structure. The approach broadens nonlinear consensus theory to more general time-varying networks and offers a robust tool for applications in engineered multi-agent systems.

Abstract

In this technical note, we introduce a novel approach to studying consensus of continuous-time nonlinear systems with varying topology based on Hilbert metric. We demonstrate that this metric offers significant flexibility in analyzing consensus properties, while effectively handling nonlinearities and time dependencies. Notably, our approach relaxes key technical assumptions from some standard results while yielding stronger conclusions with shorter proofs. This framework provides new insights into nonlinear consensus under varying topology.

Consensus of A Class of Nonlinear Systems with Varying Topology: A Hilbert Metric Approach

TL;DR

This work addresses consensus in continuous-time nonlinear multi-agent systems with time-varying topology by developing a Hilbert-metric framework. By analyzing contraction on cones and using accumulated/averaging graphs, it establishes that a quasi-strongly connected accumulated graph guarantees asymptotic consensus, and continuity of the limiting graph yields exponential convergence. The key contributions include relaxing key technical assumptions from prior nonlinear results and achieving exponential consensus under mild measurability of switching, with a simpler, non-Lyapunov proof structure. The approach broadens nonlinear consensus theory to more general time-varying networks and offers a robust tool for applications in engineered multi-agent systems.

Abstract

In this technical note, we introduce a novel approach to studying consensus of continuous-time nonlinear systems with varying topology based on Hilbert metric. We demonstrate that this metric offers significant flexibility in analyzing consensus properties, while effectively handling nonlinearities and time dependencies. Notably, our approach relaxes key technical assumptions from some standard results while yielding stronger conclusions with shorter proofs. This framework provides new insights into nonlinear consensus under varying topology.
Paper Structure (9 sections, 9 theorems, 36 equations, 5 figures)

This paper contains 9 sections, 9 theorems, 36 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Problem setting
  4. The Hilbert metric
  5. Accumulated graph
  6. Main results
  7. Simulation Result
  8. Conclusion
  9. Appendix

Key Result

Theorem 1

Consider the system sys:lin. Assume that for each $p\in\{1,\cdots, {N}\}$: 1) $f_{p}^{i}$ is locally Lipschitz and $f_{p}^{i}$ is in the relative interior of the cone $T_{x_{i}}\mathcal{C}_{i}$, i.e., $f_{p}^{i}\in{\rm ri}(T_{x_{i}}\mathcal{C}_{i})$; 2) the switching signal is piece-wise constant wi

Figures (5)

  • Figure 1: The red arrows represent the vector fields of a system. The $x_{2}$-axis is invariant and hence cannot be mapped into the interior of the positive orthant. However, the system contracts the smaller cone painted in gray.
  • Figure 2: Cone $\mathcal{K}(\gamma)$ when $n=2$.
  • Figure 3: Asymptotic contraction.
  • Figure 4: A chain of coupled oscillators.
  • Figure 5: Simulation result. Left: the signal $a_{1,1+1}(t)$. Right: the state components.

Theorems & Definitions (25)

  • Theorem 1: Lin et al. Lin2007
  • Theorem 2: Moreau Moreau2004
  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Lemma 2
  • ...and 15 more