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The role of local bounds on neighborhoods in the network for scale-free state synchronization of multi-agent systems

Anton A. Stoorvogel, Ali Saberi, Zhenwei Liu

TL;DR

The paper investigates scale-free state synchronization for homogeneous MAS under linear dynamic non-collaborative protocols in both continuous- and discrete-time settings, considering scenarios with and without locally bounded neighborhood information. It derives necessary and sufficient conditions tied to agent dynamics (stabilizability/detectability, neutral stability, minimum phase, and relative degree) and network descriptions (Laplacian $L$ versus row-stochastic conversions), showing discrete-time synchronization is impossible without local bounds while continuous-time synchronization is achievable under milder assumptions when local bounds are available. A key contribution is clarifying how local neighborhood bounds enable network-agnostic protocol design and providing constructive criteria that scale with the number of agents $N$, independent of the specific graph in $ ext{G}^N$. The results bridge Laplacian-based continuous-time analysis and row-stochastic discrete-time analysis, offering robust, scalable guidelines for designing scale-free MAS controllers and highlighting avenues for nonlinear protocol extensions when strict neutral-stability assumptions are restrictive.

Abstract

This paper provides necessary and sufficient conditions for the existence of solutions to the state synchronization problem of homogeneous multi-agent systems (MAS) via scale-free linear dynamic non-collaborative protocol for both continuous- and discrete-time. These conditions guarantee for which class of MAS, one can achieve scale-free state synchronization. We investigate protocol design with and without utilizing local bounds on neighborhood. The results show that the availability of local bounds on neighborhoods plays a key role.

The role of local bounds on neighborhoods in the network for scale-free state synchronization of multi-agent systems

TL;DR

The paper investigates scale-free state synchronization for homogeneous MAS under linear dynamic non-collaborative protocols in both continuous- and discrete-time settings, considering scenarios with and without locally bounded neighborhood information. It derives necessary and sufficient conditions tied to agent dynamics (stabilizability/detectability, neutral stability, minimum phase, and relative degree) and network descriptions (Laplacian versus row-stochastic conversions), showing discrete-time synchronization is impossible without local bounds while continuous-time synchronization is achievable under milder assumptions when local bounds are available. A key contribution is clarifying how local neighborhood bounds enable network-agnostic protocol design and providing constructive criteria that scale with the number of agents , independent of the specific graph in . The results bridge Laplacian-based continuous-time analysis and row-stochastic discrete-time analysis, offering robust, scalable guidelines for designing scale-free MAS controllers and highlighting avenues for nonlinear protocol extensions when strict neutral-stability assumptions are restrictive.

Abstract

This paper provides necessary and sufficient conditions for the existence of solutions to the state synchronization problem of homogeneous multi-agent systems (MAS) via scale-free linear dynamic non-collaborative protocol for both continuous- and discrete-time. These conditions guarantee for which class of MAS, one can achieve scale-free state synchronization. We investigate protocol design with and without utilizing local bounds on neighborhood. The results show that the availability of local bounds on neighborhoods plays a key role.
Paper Structure (20 sections, 7 theorems, 53 equations, 4 figures)

This paper contains 20 sections, 7 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Multi-agent systems and local bounds on neighborhoods in the network
  3. Continuous-time results
  4. Scale-free synchronization without locally bounded neighborhoods
  5. Necessary conditions
  6. Sufficient conditions
  7. Scale-free synchronization with locally bounded neighborhoods
  8. Necessary conditions
  9. Sufficient conditions
  10. Discrete--time results
  11. Scale-free synchronization without locally bounded neighborhoods
  12. Scale-free synchronization with locally bounded neighborhoods
  13. Necessary conditions
  14. Sufficient conditions
  15. Numerical examples
  16. ...and 5 more sections

Key Result

Theorem 1

Consider the scale-free state synchronization problem without local bounds as formulated in Problem prob4 for continuous-time systems.

Figures (4)

  • Figure 1: Directed topology network with $4$ nodes
  • Figure 2: Directed loop topology network with $60$ nodes
  • Figure 3: State synchronization for continuous-time MAS with local bounded communication graph in Case I.
  • Figure 4: State synchronization for continuous-time MAS with local bounded communication graph in Case II. For clarity of the graph, we only show the synchronized trajectories for states $x_{i1}$ and $x_{i2}$

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2
  • Theorem 2
  • Remark 4
  • Theorem 3
  • ...and 7 more