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On the Internal Stability of Diffusively Coupled Multi-Agent Systems and the Dangers of Cancel Culture

Gal Barkai, Leonid Mirkin, Daniel Zelazo

TL;DR

This paper analyzes the internal stability of diffusively-coupled multi-agent systems with linear time-invariant agents, showing a fundamental no-go condition: if there exists $\lambda \in \bar{\mathbb C}_0$ such that $\bigcap_{i=1}^\nu \ker [M_i(\lambda)]^\top \neq \{0\}$ or $\bigcap_{i=1}^\nu \ker \tilde{M}_i(\lambda) \neq \{0\}$, then no LTI edge controller can internally stabilize the interconnection. The authors prove this using coprime factorizations over $H_\infty$ and the matrix corona theorem, and extend the result to directed graphs, arbitrary symmetric couplings, unstable systems without poles in $\bar{\mathbb C}_0$, and time-varying topologies. A key insight is that internal instability arises from intrinsic unstable cancellations between the agent dynamics and the diffusive coupling, which cannot be circumvented by typical diffusive controllers. The work implies that uniform instability in agents requires breaking uniformity (e.g., introducing leaders or non-relative feedback) to achieve robust coordination under disturbances, with broad implications for the design of distributed MAS protocols.

Abstract

We study internal stability in the context of diffusively-coupled control architectures, common in multi-agent systems (i.e. the celebrated consensus protocol), for linear time-invariant agents. We derive a condition under which the system can not be stabilized by any controller from that class. In the finite-dimensional case the condition states that diffusive controllers cannot stabilize agents that share common unstable dynamics, directions included. This class always contains the group of homogeneous unstable agents, like integrators. We argue that the underlying reason is intrinsic cancellations of unstable agent dynamics by such controllers, even static ones, where directional properties play a key role. The intrinsic lack of internal stability explains the notorious behavior of some distributed control protocols when affected by measurement noise or exogenous disturbances.

On the Internal Stability of Diffusively Coupled Multi-Agent Systems and the Dangers of Cancel Culture

TL;DR

This paper analyzes the internal stability of diffusively-coupled multi-agent systems with linear time-invariant agents, showing a fundamental no-go condition: if there exists such that or , then no LTI edge controller can internally stabilize the interconnection. The authors prove this using coprime factorizations over and the matrix corona theorem, and extend the result to directed graphs, arbitrary symmetric couplings, unstable systems without poles in , and time-varying topologies. A key insight is that internal instability arises from intrinsic unstable cancellations between the agent dynamics and the diffusive coupling, which cannot be circumvented by typical diffusive controllers. The work implies that uniform instability in agents requires breaking uniformity (e.g., introducing leaders or non-relative feedback) to achieve robust coordination under disturbances, with broad implications for the design of distributed MAS protocols.

Abstract

We study internal stability in the context of diffusively-coupled control architectures, common in multi-agent systems (i.e. the celebrated consensus protocol), for linear time-invariant agents. We derive a condition under which the system can not be stabilized by any controller from that class. In the finite-dimensional case the condition states that diffusive controllers cannot stabilize agents that share common unstable dynamics, directions included. This class always contains the group of homogeneous unstable agents, like integrators. We argue that the underlying reason is intrinsic cancellations of unstable agent dynamics by such controllers, even static ones, where directional properties play a key role. The intrinsic lack of internal stability explains the notorious behavior of some distributed control protocols when affected by measurement noise or exogenous disturbances.
Paper Structure (17 sections, 12 theorems, 58 equations, 3 figures)

This paper contains 17 sections, 12 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivating example
  3. Contribution
  4. Problem formulation
  5. The main result
  6. Proof of Theorem \ref{['th:mainr']}
  7. Generalizations
  8. Asymmetric coupling
  9. Arbitrary symmetric coupling
  10. Unstable systems with no poles in $\bar{\mathbb C}_0$
  11. Time-varying $K$
  12. Finite-dimensional agents
  13. Diffusive control laws and unstable cancellations
  14. Proof of Proposition \ref{['pr:pcanc']}
  15. Concluding remarks
  16. ...and 2 more sections

Key Result

Theorem 3.1

No LTI $K_{\text{e},j}$ can internally stabilize the diffusively-coupled system in Fig. fig:BD_cl if there is $\lambda\in\bar{\mathbb C}_0$, common to all agents, such that where $M_i$ and $\tilde{M}_i$ are denominators in the coprime factorizations of $P_i$ under $\boldsymbol{\mathcal{A}}_{\text{\space\bf}ass:Picf}$.

Figures (3)

  • Figure 1: Simulation of protocol \ref{['eq:conprot']} perturbed by a step at $t=t_d$.
  • Figure 2: Diffusively-coupled feedback setup ($E$ is the incidence matrix of the connectivity graph $\mathcal{G}$)
  • Figure 3: Diffusively-coupled feedback setup as edge stabilization

Theorems & Definitions (24)

  • remark 1
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • remark 2
  • Proposition 4.1
  • proof
  • ...and 14 more