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Matrix-Scaled Consensus over Undirected Networks

Minh Hoang Trinh, Hoang Huy Vu, Nhat-Minh Le-Phan, Quyen Ngoc Nguyen

TL;DR

This paper studies matrix-scaled consensus algorithms for linear dynamical agents interacting over an undirected network using observer-based matrix-scaled consensus algorithms designed for homogeneous and then extended for heterogeneous linear-time invariant agents.

Abstract

In this paper, we propose matrix-scaled consensus algorithms for linear dynamical agents interacting over an undirected network. Under the proposed algorithms, the state vectors of all agents to asymptotically agree up to some matrix scaling weights. First, the algebraic properties of the matrix-scaled Laplacian and the geometry of the matrix-scaled consensus space are studied. Second, we examine matrix-scaled consensus algorithms for networks of single-integrators with or without constant parametric uncertainties. Nonlinear and finite-time matrix-scaled consensus algorithms are also proposed. Third, observer-based matrix-scaled consensus algorithms for homogeneous or heterogeneous linear-time invariant agents are designed. The convergence of the proposed algorithms is asserted by rigorous mathematical analysis and supported by numerical simulations.

Matrix-Scaled Consensus over Undirected Networks

TL;DR

This paper studies matrix-scaled consensus algorithms for linear dynamical agents interacting over an undirected network using observer-based matrix-scaled consensus algorithms designed for homogeneous and then extended for heterogeneous linear-time invariant agents.

Abstract

In this paper, we propose matrix-scaled consensus algorithms for linear dynamical agents interacting over an undirected network. Under the proposed algorithms, the state vectors of all agents to asymptotically agree up to some matrix scaling weights. First, the algebraic properties of the matrix-scaled Laplacian and the geometry of the matrix-scaled consensus space are studied. Second, we examine matrix-scaled consensus algorithms for networks of single-integrators with or without constant parametric uncertainties. Nonlinear and finite-time matrix-scaled consensus algorithms are also proposed. Third, observer-based matrix-scaled consensus algorithms for homogeneous or heterogeneous linear-time invariant agents are designed. The convergence of the proposed algorithms is asserted by rigorous mathematical analysis and supported by numerical simulations.
Paper Structure (27 sections, 13 theorems, 52 equations, 8 figures)

This paper contains 27 sections, 13 theorems, 52 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Graph theory
  5. The multi-agent system and the objective
  6. Matrix-scaled consensus space and matrix-scaled Laplacian
  7. Geometry of the matrix-scaled consensus space in $\mathbb{R}^2$
  8. Algebratic properties of the matrix-scaled Laplacian
  9. Matrix-scaled consensus algorithms for single-integrator agents
  10. Matrix-scaled consensus of single-integrators
  11. Adaptive matrix-scaled consensus of single-integrators with parametric uncertainties
  12. Matrix-scaled consensus algorithms for time-invariant linear dynamical agents
  13. Simplified linear agents
  14. The agents' dynamics are identical
  15. Simplified heterogeneous linear agents with unknown system matrix and full control input
  16. ...and 12 more sections

Key Result

Lemma 3.1

NicholsonLAA The matrix $\mathbf{S}_i = \in \mathbb{R}^{2\times 2}$ is positive definite (resp., negative definite) if and only if $a+d>0$ and $4ad > (b+c)^2$ (resp., $a + d < 0$ and $4ad > (b+c)^2$). Particularly,

Figures (8)

  • Figure 1: Geometry of the MSC space in $\mathbb{R}^2$ for several types of the scaling matrices $\mathbf{S}_i$.
  • Figure 2: A snowflake-like formation in $\mathcal{C}_S$ corresponding to $\mathbf{S}_i$ in Example \ref{['eg:snowflake']}.
  • Figure 3: Simulation of the 18-agent system in Example \ref{['eg:snowflake']} under the MSC algorithm \ref{['eq:msc_single_integrator']}.
  • Figure 4: The agents converge to three clusters in 2D under the matrix-scaled consensus algorithms \ref{['eq:msc_single_integrator']} (Figs. (a)--(d)) and \ref{['eq:msc_nonl']} (Figs. (e)--(h)).
  • Figure 5: The six-agent system under the adaptive MSC algorithm \ref{['eq:AMWC-1']}--\ref{['eq:AMWC-2']}.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Lemma 3.1
  • Example 3.1: A snowflake formation
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.1
  • Theorem 4.1
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.2
  • Remark 4.3
  • ...and 9 more