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Equivalent Conditions for the Synchronization of Identical Linear Systems over Arbitrary Interconnections

Nicola Zaupa, Giulia Giordano, Isabelle Queinnec, Sophie Tarbouriech, Luca Zaccarian

Abstract

We propose necessary and sufficient conditions for the synchronization of $N$ identical single-input-single-output (SISO) systems, connected through a directed graph {without imposing any assumption on the graph interconnection}. We consider both the continuous-time and the discrete-time case, and we provide conditions that {are equivalent to} the uniform global exponential stability, {with guaranteed convergence rate,} of the closed {and unbounded} attractor that corresponds to the synchronization set.

Equivalent Conditions for the Synchronization of Identical Linear Systems over Arbitrary Interconnections

Abstract

We propose necessary and sufficient conditions for the synchronization of identical single-input-single-output (SISO) systems, connected through a directed graph {without imposing any assumption on the graph interconnection}. We consider both the continuous-time and the discrete-time case, and we provide conditions that {are equivalent to} the uniform global exponential stability, {with guaranteed convergence rate,} of the closed {and unbounded} attractor that corresponds to the synchronization set.
Paper Structure (6 sections, 6 theorems, 27 equations, 2 figures)

This paper contains 6 sections, 6 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Equivalent conditions to $\alpha$--synchronization
  3. Proof of the main theorem
  4. A few technical lemmas
  5. Proof of Theorem th:consensus_alpha
  6. Conclusions

Key Result

Theorem 1

Consider the continuous-time (resp. discrete-time) system in eq:sys, eq:laplacian, the attractor $\mathcal{A}$ in eq:setA and the parameter ${\alpha^\star}\geq0$ (resp. ${\alpha^\star}\in(0,1]$). The following statements are equivalent:

Figures (2)

  • Figure 1: Block diagram of the closed-loop system.
  • Figure 2: Structure of the proof of Theorem th:consensus_alpha.

Theorems & Definitions (16)

  • Definition 1: $\alpha$--UGES
  • Remark 1
  • Definition 2: $\alpha$--synchronization
  • Theorem 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • Remark 4
  • Lemma 1
  • proof
  • ...and 6 more