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Structured stability analysis of networked systems with uncertain links

Simone Mariano, Michael Cantoni

Abstract

An input-output approach to stability analysis is explored for networked systems with uncertain link dynamics. The main result consists of a collection of integral quadratic constraints, which together imply robust stability of the uncertain networked system, under the assumption that stability is achieved with ideal links. The conditions are decentralized inasmuch as each involves only agent and uncertainty model parameters that are local to a corresponding link. This makes the main result, which imposes no restriction on network structure, suitable for the study of large-scale systems.

Structured stability analysis of networked systems with uncertain links

Abstract

An input-output approach to stability analysis is explored for networked systems with uncertain link dynamics. The main result consists of a collection of integral quadratic constraints, which together imply robust stability of the uncertain networked system, under the assumption that stability is achieved with ideal links. The conditions are decentralized inasmuch as each involves only agent and uncertainty model parameters that are local to a corresponding link. This makes the main result, which imposes no restriction on network structure, suitable for the study of large-scale systems.
Paper Structure (9 sections, 7 theorems, 30 equations, 3 figures)

This paper contains 9 sections, 7 theorems, 30 equations, 3 figures.

Table of Contents

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

Key Result

Lemma 1

If $G:\mathbf{L}_{2e}^q\rightarrow \mathbf{L}_{2e}^p$ and $\Delta:\mathbf{L}_{2e}^p\rightarrow \mathbf{L}_{2e}^q$ are both stable, and $G$ is linear, then the following are equivalent:

Figures (3)

  • Figure 2: Standard feedback interconnection.
  • Figure 3: Example network graph $\mathcal{G}$ (left) and corresponding sub-system graph $\mathcal{G}_s$ (right). The sub-system graph has $2m=8$ vertices and $m=4$ edges, while the network graph has $n=4$ nodes and $m=4$ edges.
  • Figure 4: Networked system model $[\![P,R\circ T\circ H]\!]$, and loop transformations for robust stability analysis.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Theorem 1
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • ...and 8 more