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Randomized Matrix Weighted Consensus

Nhat-Minh Le-Phan, Minh Hoang Trinh, Phuoc Doan Nguyen

TL;DR

First, the notion of expected matrix weighted network is introduced, which captures the multi-dimensional interactions between any two agents in a probabilistic sense, and under some mild assumptions on the distribution of the expected matrix weights, the proposed asynchronous pairwise update algorithms drive the network to achieve a consensus in expectation.

Abstract

In this paper, randomized gossip-type matrix-weighted consensus algorithms are proposed for both leaderless and leader-follower topologies. First, we introduce the notion of expected matrix-weighted network, which captures the multi-dimensional interactions between any two agents in a probabilistic sense. Under some mild assumptions on the distribution of the expected matrix weights and the upper bound of the updating step size, the proposed asynchronous pairwise update algorithms drive the network to achieve a consensus in expectation. An upper bound of the $ε$-convergence time of the algorithm is then derived. Furthermore, the proposed algorithms are applied to the bearing-based network localization and formation control problems. The theoretical results are supported by several numerical examples.

Randomized Matrix Weighted Consensus

TL;DR

First, the notion of expected matrix weighted network is introduced, which captures the multi-dimensional interactions between any two agents in a probabilistic sense, and under some mild assumptions on the distribution of the expected matrix weights, the proposed asynchronous pairwise update algorithms drive the network to achieve a consensus in expectation.

Abstract

In this paper, randomized gossip-type matrix-weighted consensus algorithms are proposed for both leaderless and leader-follower topologies. First, we introduce the notion of expected matrix-weighted network, which captures the multi-dimensional interactions between any two agents in a probabilistic sense. Under some mild assumptions on the distribution of the expected matrix weights and the upper bound of the updating step size, the proposed asynchronous pairwise update algorithms drive the network to achieve a consensus in expectation. An upper bound of the -convergence time of the algorithm is then derived. Furthermore, the proposed algorithms are applied to the bearing-based network localization and formation control problems. The theoretical results are supported by several numerical examples.
Paper Structure (31 sections, 15 theorems, 61 equations, 6 figures)

This paper contains 31 sections, 15 theorems, 61 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Matrix weighted graphs and the expected graphs
  4. Randomized distributed protocol
  5. Randomized matrix weighted consensus with Leaderless topology
  6. Convergence in expectation
  7. Randomized matrix weighted average consensus
  8. Convergence of the second moment
  9. Upper bound of the $\epsilon$-consensus time
  10. Randomized matrix weighted consensus with leader-following topology
  11. Discrete-time matrix weighted consensus with Leader-Following topology
  12. Randomized matrix weighted consensus with Leader-Following topology
  13. Upper bound of the $\epsilon$-consensus time
  14. Application in bearing-based network localization problem
  15. Problem Formulation
  16. ...and 16 more sections

Key Result

Lemma 1

THM2018 The expected Laplacian matrix $\mathbf{L}^{\rm M}$ is symmetric and positive semi-definite. Moreover, ${\rm null}(\mathbf{L}^{\rm M}) = {\rm span}\{ {\rm range}(\mathbf{1}_n \otimes \mathbf{I}_d), \{ \mathbf{v}=[v_1^\top,\dots, v_n^\top]^\top \in \mathbb{R}^{nd}|~(v_j-v_j)\in {\rm null}(\mat

Figures (6)

  • Figure 1: A directed matrix weighted graph $\mathcal{G}$ and its corresponding expected graph $\mathcal{G}^{\rm M}$.
  • Figure 2: Examples of infinitesimally/non-infinitesimally bearing rigid frameworks in three-dimensional space.
  • Figure 3: Randomized Randomized matrix weighted of 4 agents with Leaderless topology.
  • Figure 4: Randomized Randomized matrix weighted of 4 agents with Leader-Following topology
  • Figure 5: Simulation of a sensor network consisting of 62 nodes under the gossip-based network localization protocol \ref{['algorithmNL0']}, \ref{['algorithmNL1']}, \ref{['algorithmNL2']}: (a) - the network $\mathcal{G}(\bar{\mathbf{x})}$ (beacon nodes are denoted with '${black}{\boldsymbol{\Delta}}$', normal nodes are denoted by 'o', respectively); From (b) to (f) - the estimate configurations at different time instances; (g) the bearing error vs time.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Definition 1
  • Lemma 4
  • Lemma 5
  • ...and 12 more