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Scalable second-order consensus of hierarchical groups

Jiamin Wang, Jian Liu, Feng Xiao, Ning Xi, Yuanshi Zheng

TL;DR

This work studies scalable second-order consensus for groups of double-integrator agents on directed acyclic graphs with reverse edges, focusing on how fixed control gains perform as the network grows. It compares absolute velocity and relative velocity protocols, proving that the absolute velocity scheme achieves completely scalable consensus on general mixed graphs, while the relative velocity scheme fails to do so in general (though it can in special graph families). A hierarchical directed-star structure is proposed to guarantee scalability for networks of arbitrary size and reverse-edge counts under both protocols. The findings highlight that the choice of local interaction rules critically impacts scalability and offer design guidance for large-scale hierarchical multi-agent systems without retuning gains.

Abstract

Motivated by widespread dominance hierarchy, growth of group sizes, and feedback mechanisms in social species, we are devoted to exploring the scalable second-order consensus of hierarchical groups. More specifically, a hierarchical group consists of a collection of agents with double-integrator dynamics on a directed acyclic graph with additional reverse edges, which characterize feedback mechanisms across hierarchical layers. As the group size grows and the reverse edges appear, we investigate whether the absolute velocity protocol and the relative velocity protocol can preserve the system consensus property without tuning the control gains. It is rigorously proved that the absolute velocity protocol is able to achieve completely scalable second-order consensus but the relative velocity protocol cannot. This result theoretically reveals how the scalable coordination behavior in hierarchical groups is determined by local interaction rules. Moreover, we develop a hierarchical structure in order to achieve scalable second-order consensus for networks of any size and with any number of reverse edges.

Scalable second-order consensus of hierarchical groups

TL;DR

This work studies scalable second-order consensus for groups of double-integrator agents on directed acyclic graphs with reverse edges, focusing on how fixed control gains perform as the network grows. It compares absolute velocity and relative velocity protocols, proving that the absolute velocity scheme achieves completely scalable consensus on general mixed graphs, while the relative velocity scheme fails to do so in general (though it can in special graph families). A hierarchical directed-star structure is proposed to guarantee scalability for networks of arbitrary size and reverse-edge counts under both protocols. The findings highlight that the choice of local interaction rules critically impacts scalability and offer design guidance for large-scale hierarchical multi-agent systems without retuning gains.

Abstract

Motivated by widespread dominance hierarchy, growth of group sizes, and feedback mechanisms in social species, we are devoted to exploring the scalable second-order consensus of hierarchical groups. More specifically, a hierarchical group consists of a collection of agents with double-integrator dynamics on a directed acyclic graph with additional reverse edges, which characterize feedback mechanisms across hierarchical layers. As the group size grows and the reverse edges appear, we investigate whether the absolute velocity protocol and the relative velocity protocol can preserve the system consensus property without tuning the control gains. It is rigorously proved that the absolute velocity protocol is able to achieve completely scalable second-order consensus but the relative velocity protocol cannot. This result theoretically reveals how the scalable coordination behavior in hierarchical groups is determined by local interaction rules. Moreover, we develop a hierarchical structure in order to achieve scalable second-order consensus for networks of any size and with any number of reverse edges.
Paper Structure (5 sections, 3 theorems, 16 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem statement
  4. Main results
  5. Conclusion

Key Result

Theorem 1

Hierarchical group eq:2.2.1 can achieve completely scalable second-order consensus with the absolute velocity protocol ab_pro under Assumption assumption:2.

Figures (1)

  • Figure 1: Diagram for the case $0<\text{Re}(\omega)<\mu$.

Theorems & Definitions (12)

  • Definition 1: Gross2003
  • Definition 2: Zhang2017
  • Definition 3
  • Remark 1
  • Remark 2
  • Definition 4: Scalable second-order consensus
  • Remark 3
  • Theorem 1
  • Theorem 2
  • Remark 4
  • ...and 2 more