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A Scalable Approach for Analysing Multi-Agent Systems with Heterogeneous Stochastic Packet Loss

Christian Hespe, Herbert Werner

TL;DR

This paper improves on a recently proposed scalable approach for analysing multi-agent systems with stochastic packet loss by allowing for heterogeneous transmission probabilities and temporal correlation in the communication model.

Abstract

An important aspect in jointly analysing networked control systems and their communication is to model the networking in a sufficiently rich but at the same time mathematically tractable way. As such, this paper improves on a recently proposed scalable approach for analysing multi-agent systems with stochastic packet loss by allowing for heterogeneous transmission probabilities and temporal correlation in the communication model. The key idea is to consider the transmission probabilities as uncertain, which facilitates the use of tools from robust control. Due to being formulated in terms of linear matrix inequalities that grow linearly with the number of agents, the result is applicable to very large multi-agent systems, which is demonstrated by numerical simulations with up to 10000 agents.

A Scalable Approach for Analysing Multi-Agent Systems with Heterogeneous Stochastic Packet Loss

TL;DR

This paper improves on a recently proposed scalable approach for analysing multi-agent systems with stochastic packet loss by allowing for heterogeneous transmission probabilities and temporal correlation in the communication model.

Abstract

An important aspect in jointly analysing networked control systems and their communication is to model the networking in a sufficiently rich but at the same time mathematically tractable way. As such, this paper improves on a recently proposed scalable approach for analysing multi-agent systems with stochastic packet loss by allowing for heterogeneous transmission probabilities and temporal correlation in the communication model. The key idea is to consider the transmission probabilities as uncertain, which facilitates the use of tools from robust control. Due to being formulated in terms of linear matrix inequalities that grow linearly with the number of agents, the result is applicable to very large multi-agent systems, which is demonstrated by numerical simulations with up to 10000 agents.
Paper Structure (13 sections, 6 theorems, 35 equations, 3 figures)

This paper contains 13 sections, 6 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Notation and Definitions
  4. Multi-Agent Systems with Packet Loss
  5. Stochastic Jump Linear Systems
  6. Modelling Packet Loss using Stochastic Laplacians
  7. Decomposable Markov Jump Linear Systems
  8. Expected Laplacians with Heterogeneous Probabilities
  9. Analysis with Uncertain Probabilities
  10. Robust Stability
  11. Robust Performance
  12. Application Example
  13. Conclusions

Key Result

Theorem 1

The MJLS eq:mjls is mean-square stable if and only if there exist $X_i \succ 0$ such that

Figures (3)

  • Figure 1: Graph structures that are used in the example section.
  • Figure 2: Best upper bound on the $\mathcal{H}_2$-Performance that can be obtained from Theorem \ref{['thm:robust_h2']} for transition probabilities in the interval $[\rho_l, 1]$ with $N = 4$ (\ref{['fig:uncertainty_sweep_line_N4']}) and $N = 6$ (\ref{['fig:uncertainty_sweep_line_N6']}). For comparison, an estimate obtained from Theorem \ref{['thm:mjls_h2']} by evaluating all combinations of $p^{ij} \in \set{\rho_l, 1}$ with the same $X$ is included.
  • Figure 3: $\mathcal{H}_2$-Performance as obtained from Theorem \ref{['thm:robust_h2']} for different agent counts with transition probabilities in $[0.4, 0.6]$. The second axis shows the computation time required to solve the optimization problem, averaged over ten runs. $\mathcal{H}_2$-Performance and computation time as obtained from Theorem 7 in Hespe2022 for $p = 0.5$ are shown for reference.

Theorems & Definitions (15)

  • Definition 1: Mean-Square Stability Costa2005
  • Theorem 1: Stability Test Costa1993
  • Definition 2: MJLS $\mathcal{H}_2$-Norm Costa2005
  • Theorem 2: $\mathcal{H}_2$-Norm Calculation Fioravanti2012
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark
  • Theorem 5: Robust Stability Test
  • ...and 5 more