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Neighbourhood conditions for network stability with link uncertainty

Simone Mariano, Michael Cantoni

TL;DR

This work addresses robust stability of large-scale networked systems with uncertain links using an input-output IQC framework. It introduces a neighborhood-based decomposition that yields decentralized, agent-local stability conditions and connects them to a global IQC certificate through $G=H\circ (P-T\circ H)^{-1}$ and $\Delta=(R-I_{2m})\circ T$. The main contribution is a set of per-agent matrices $W_i,X_i,Y_i,Z_i$ (Lemma \ref{lem:PVRNodes}) producing local inequalities that collectively guarantee stability, potentially reducing conservativeness relative to prior link-wise approaches (e.g., MCLinks). The results enable scalable robust stability certification for sparsely interconnected networks and are demonstrated to offer tighter conditions in a numerical example, with avenues for extension to asynchronous delays and dynamic quantization.

Abstract

The main result relates to structured robust stability analysis of an input-output model for networks with link uncertainty. It constitutes a collection of integral quadratic constraints, which together imply robust stability of the uncertain networked dynamics. Each condition is decentralized in the sense that it depends on model data pertaining to the neighbourhood of a specific agent. By contrast, pre-existing conditions for the network model are link-wise decentralized, with each involving conservatively more localized problem data. A numerical example is presented to illustrate the advantage of the new broader neighbourhood conditions.

Neighbourhood conditions for network stability with link uncertainty

TL;DR

This work addresses robust stability of large-scale networked systems with uncertain links using an input-output IQC framework. It introduces a neighborhood-based decomposition that yields decentralized, agent-local stability conditions and connects them to a global IQC certificate through and . The main contribution is a set of per-agent matrices (Lemma \ref{lem:PVRNodes}) producing local inequalities that collectively guarantee stability, potentially reducing conservativeness relative to prior link-wise approaches (e.g., MCLinks). The results enable scalable robust stability certification for sparsely interconnected networks and are demonstrated to offer tighter conditions in a numerical example, with avenues for extension to asynchronous delays and dynamic quantization.

Abstract

The main result relates to structured robust stability analysis of an input-output model for networks with link uncertainty. It constitutes a collection of integral quadratic constraints, which together imply robust stability of the uncertain networked dynamics. Each condition is decentralized in the sense that it depends on model data pertaining to the neighbourhood of a specific agent. By contrast, pre-existing conditions for the network model are link-wise decentralized, with each involving conservatively more localized problem data. A numerical example is presented to illustrate the advantage of the new broader neighbourhood conditions.
Paper Structure (8 sections, 8 theorems, 41 equations, 2 figures)

This paper contains 8 sections, 8 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation
  4. Signals and systems
  5. Graphs
  6. Networked system model and robust stability analysis
  7. Main result: Neighbourhood conditions
  8. Conclusions

Key Result

Theorem 1

Given stable system $\Delta:\mathbf{L}_{2e}^p \rightarrow \mathbf{L}_{2e}^q$, and bounded linear map $\Pi:\mathbf{L}_{2\,}^p\times\mathbf{L}_{2\,}^q \rightarrow \mathbf{L}_{2\,}^p\times \mathbf{L}_{2\,}^q$ that is self-adjoint in the sense $(\forall g_1,g_2\in\mathbf{L}_{2\,}^p\times \mathbf{L}_{2\, for all $(y,\alpha)\in\mathbf{L}_{2\,}^p \times [0,1]$. Further, given stable system $G:\mathbf{L}_

Figures (2)

  • Figure 1: Standard feedback interconnection.
  • Figure 2: Networked system model $[\![P,R\circ T\circ H]\!]$, and loop transformations for robust stability analysis.

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 1
  • ...and 2 more