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Non-trivial consensus on directed signed matrix-weighted networks with compound measurement noises and time-varying topologies

Tianmu Niu, Xiaoqun Wu

Abstract

This paper studies non-trivial consensus--a relatively novel and unexplored convergence behavior--on directed signed matrix-weighted networks subject to both additive and multiplicative measurement noises under time-varying topologies. Building upon grounded matrix-weighted Laplacian properties, a stochastic dynamic model is established that simultaneously captures inter-dimensional cooperative and antagonistic interactions, compound measurement noises and time-varying network structures. Based on stochastic differential equations theory, protocols that guarantee mean square and almost sure non-trivial consensus are proposed. Specifically, for any predetermined non-trivial consensus state, all agents are proven to converge toward this non-zero value in the mean-square and almost-sure senses. The design of control gain function in our protocols highlights a balanced consideration of the cumulative effect over time, the asymptotic decay property and the finite energy corresponding to measurement noises. Notably, the conditions on time-varying topologies in our protocols only require boundedness of elements in edge weight matrices, which facilitate the practicality of concept "time-varying topology" in matrix-weighted network consensus algorithms. Furthermore, the proposed protocols operate under milder connectivity conditions and no requirements on structural (un)balance properties. The work in this paper demonstrates that groups with both cooperative and antagonistic inter-dimensional interactions can achieve consensus even in the presence of compound measurement noises and time-varying topologies, challenging the conventional belief that consensus is attainable only in fully cooperative settings.

Non-trivial consensus on directed signed matrix-weighted networks with compound measurement noises and time-varying topologies

Abstract

This paper studies non-trivial consensus--a relatively novel and unexplored convergence behavior--on directed signed matrix-weighted networks subject to both additive and multiplicative measurement noises under time-varying topologies. Building upon grounded matrix-weighted Laplacian properties, a stochastic dynamic model is established that simultaneously captures inter-dimensional cooperative and antagonistic interactions, compound measurement noises and time-varying network structures. Based on stochastic differential equations theory, protocols that guarantee mean square and almost sure non-trivial consensus are proposed. Specifically, for any predetermined non-trivial consensus state, all agents are proven to converge toward this non-zero value in the mean-square and almost-sure senses. The design of control gain function in our protocols highlights a balanced consideration of the cumulative effect over time, the asymptotic decay property and the finite energy corresponding to measurement noises. Notably, the conditions on time-varying topologies in our protocols only require boundedness of elements in edge weight matrices, which facilitate the practicality of concept "time-varying topology" in matrix-weighted network consensus algorithms. Furthermore, the proposed protocols operate under milder connectivity conditions and no requirements on structural (un)balance properties. The work in this paper demonstrates that groups with both cooperative and antagonistic inter-dimensional interactions can achieve consensus even in the presence of compound measurement noises and time-varying topologies, challenging the conventional belief that consensus is attainable only in fully cooperative settings.
Paper Structure (13 sections, 6 theorems, 63 equations, 8 figures)

This paper contains 13 sections, 6 theorems, 63 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Graph Theory
  5. Model Description
  6. Problem formulation
  7. Main results
  8. Fixed topology
  9. Time-varying topology
  10. Numerical examples
  11. fixed topology
  12. time-varying topology
  13. Conclusions

Key Result

Lemma 1

Suppose that Assumption directed Assumption holds, and $\sum_{j\in\varOmega_{i}(t)}|A_{ij}(t)|\succ\boldsymbol{0}$. For any $t\geq0$, denote choose the informed agents set as $\mathcal{V}_{\mathcal{I}}(t)=\mathcal{U}(t)$, take $\delta_{i}(t)=\delta(t),\ \forall i\in\mathcal{V}_{\mathcal{I}}(t)$. If the non-zero coupling coefficient $\delta(t)>C(t)$, and the coupling matrix weights $B_{i}(t)$ and

Figures (8)

  • Figure 1: Directed topology $\mathcal{G}_{1}$ for original FAN \ref{['FAN_noise']} and $\widehat{\mathcal{G}}_{1}$ for the corresponding SAN \ref{['SAN_noise']} under Lemma \ref{['Basic Lemma']}. The blue and red solid (dashed) lines represent positive and negative (semi-) definite edges, respectively.
  • Figure 2: Mean square non-trivial consensus errors of directed signed matrix-weighted network with fixed topology $\mathcal{G}_{1},\ \widehat{\mathcal{G}}_{1}$ in Figure \ref{['G1']}.
  • Figure 3: States evolution of directed matrix-weighted network with fixed topology $\widehat{\mathcal{G}}_{1}$ in Figure \ref{['G1']}.
  • Figure 4: Topology $\mathcal{G}_{2},\ \widehat{\mathcal{G}}_{2}$ and $\mathcal{G}_{3},\ \widehat{\mathcal{G}}_{3}$ for switching network \ref{['FAN_noise']}\ref{['SAN_noise']} under Lemma \ref{['Basic Lemma']}. The blue and red solid (dashed) lines represent positive and negative (semi-) definite edges, respectively.
  • Figure 5: Mean square non-trivial consensus error of directed matrix-weighted network with time-varying topology $\widehat{\mathcal{G}}(t)$ switching among $\widehat{\mathcal{G}}_{1}$ in Figure \ref{['G1']} and $\widehat{\mathcal{G}}_{2}$, $\widehat{\mathcal{G}}_{3}$ in Figure \ref{['switching G2 and G3']}.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Remark 1
  • Definition 1: Mean square non-trivial consensus
  • Definition 2: Almost sure non-trivial consensus
  • Definition 3: positive-negative path
  • Definition 4: in-degree-dominated
  • Lemma 1: my_matrix-weighted_NTC
  • Remark 2
  • Theorem 1
  • Lemma 2: semimartingale, semimartingale convergence theorem
  • Theorem 2
  • ...and 5 more