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Real Eventual Exponential Positivity of Complex-valued Laplacians: Applications to Consensus in Multi-agent Systems

Aditi Saxena, Twinkle Tripathy, Rajasekhar Anguluri

TL;DR

It is shown that the property of eventual exponential positivity (EEP) in complex matrices holds for the real part of the matrix exponential for a certain class of complex matrices.

Abstract

In this paper, we explore the property of eventual exponential positivity (EEP) in complex matrices. We show that this property holds for the real part of the matrix exponential for a certain class of complex matrices. Next, we present the relation between the spectral properties of the Laplacian matrix of an unsigned digraph with complex edge-weights and the property of real EEP. Finally, we show that the Laplacian flow system of a network is stable when the negated Laplacian admits real EEP. Numerical examples are presented to demonstrate the results.

Real Eventual Exponential Positivity of Complex-valued Laplacians: Applications to Consensus in Multi-agent Systems

TL;DR

It is shown that the property of eventual exponential positivity (EEP) in complex matrices holds for the real part of the matrix exponential for a certain class of complex matrices.

Abstract

In this paper, we explore the property of eventual exponential positivity (EEP) in complex matrices. We show that this property holds for the real part of the matrix exponential for a certain class of complex matrices. Next, we present the relation between the spectral properties of the Laplacian matrix of an unsigned digraph with complex edge-weights and the property of real EEP. Finally, we show that the Laplacian flow system of a network is stable when the negated Laplacian admits real EEP. Numerical examples are presented to demonstrate the results.

Paper Structure

This paper contains 14 sections, 5 theorems, 22 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Matrix Algebra
  4. Graph theory
  5. MAIN RESULTS
  6. Irreduciblity in directed networks
  7. Real Eventually Exponentially Positive Laplacians
  8. Undirected unsigned networks
  9. Directed unsigned networks
  10. SIMULATION RESULTS
  11. Complex-valued Laplacian: Synthetic Networks
  12. Complex-valued Laplacian: Power Network
  13. Counterexample
  14. CONCLUSIONS & FUTURE WORK

Key Result

Lemma 1

Consider a digraph for $n\geq 2$, $L \in \mathbb{C}^{n,n}$, then $L$ is irreducible if and only if $\mathcal{G}(A)$ is strongly connected.

Figures (2)

  • Figure 1:
  • Figure 3: Convergence properties of complex Laplacian flow systems

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 7 more