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Mode Consensus Algorithms With Finite Convergence Time

Chao Huang, Hyungbo Shim, Siliang Yu, Brian D. O. Anderson

TL;DR

This paper studies the distributed mode consensus problem in a multi-agent system, in which the agents each possess a certain attribute and they aim to agree upon the mode via distributed computation via distributed computation.

Abstract

This paper studies the distributed mode consensus problem in a multi-agent system, in which the agents each possess a certain attribute and they aim to agree upon the mode (the most frequent attribute owned by the agents) via distributed computation. Three algorithms are proposed. The first one directly calculates the frequency of all attributes at every agent, with protocols based on blended dynamics, and then returns the most frequent attribute as the mode. Assuming knowledge at each agent of a lower bound of the mode frequency as a priori information, the second algorithm is able to reduce the number of frequencies to be computed at every agent if the lower bound is large. The third algorithm further eliminates the need for this information by introducing an adaptive updating mechanism. The algorithms find the mode in finite time, and estimates of convergence time are provided. The proposed first and second algorithms enjoy the plug-and-play property with a dwell time.

Mode Consensus Algorithms With Finite Convergence Time

TL;DR

This paper studies the distributed mode consensus problem in a multi-agent system, in which the agents each possess a certain attribute and they aim to agree upon the mode via distributed computation via distributed computation.

Abstract

This paper studies the distributed mode consensus problem in a multi-agent system, in which the agents each possess a certain attribute and they aim to agree upon the mode (the most frequent attribute owned by the agents) via distributed computation. Three algorithms are proposed. The first one directly calculates the frequency of all attributes at every agent, with protocols based on blended dynamics, and then returns the most frequent attribute as the mode. Assuming knowledge at each agent of a lower bound of the mode frequency as a priori information, the second algorithm is able to reduce the number of frequencies to be computed at every agent if the lower bound is large. The third algorithm further eliminates the need for this information by introducing an adaptive updating mechanism. The algorithms find the mode in finite time, and estimates of convergence time are provided. The proposed first and second algorithms enjoy the plug-and-play property with a dwell time.
Paper Structure (15 sections, 4 theorems, 34 equations, 10 figures, 3 algorithms)

This paper contains 15 sections, 4 theorems, 34 equations, 10 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Underlying network
  4. $f$-consensus and broad problem setting
  5. Two relevant examples of $f$-consensus
  6. Distributed computation of network size
  7. $k$-th smallest element consensus
  8. The problem of interest: Mode consensus
  9. Plug-and-play
  10. Direct mode consensus algorithm
  11. Mode consensus algorithm considering the frequency of mode
  12. Algorithm with a priori knowledge of the least frequency of the mode
  13. Algorithm with learned knowledge of the least frequency of the mode
  14. Simulations
  15. Conclusion

Key Result

Theorem 1

Suppose Assumptions Asu:connectivity and Asu:finiteattributes hold. If $\gamma_y \ge \bar{N}^3$, then for any initial condition $y_i(0) \in {\mathcal{K}}_y := [-0.5,\bar{N}+0.5]$, the solution of the consensus protocol Mode-consensus--ell satisfies where, with $M_{{\mathcal{K}}_y} = \bar{N} + 1$ (the size of ${\mathcal{K}}_y$),

Figures (10)

  • Figure 1: The mode $a^*$ estimated at each agent with Algorithm \ref{['Alg1']}, converging to $4$.
  • Figure 2: The estimated frequency of the mode, i.e., $\mathcal{F}\left(a^*\right)$, at each agent with Algorithm \ref{['Alg1']}, converging to $16$.
  • Figure 3: The network size ,i.e. $N$, estimated at each agent with protocol (\ref{['Nconsensus']}).
  • Figure 4: Top figure: the estimated $14$-th smallest element at each agent with protocol (\ref{['k-th1']}), converging to $3$; bottom figure: the estimated $28$-th smallest element at each agent with protocol (\ref{['k-th1']}), converging to $4$.
  • Figure 5: Top figure: The estimated frequency of the $14$-th smallest element at each agent with protocol (\ref{['Mode-consensus']})--(\ref{['ell']}), converging to $7$; bottom figure: The estimated frequency of the $28$-th smallest element at each agent with protocol (\ref{['Mode-consensus']})--(\ref{['ell']}), converging to $16$;
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 1: $f$-consensus
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Lemma 1
  • ...and 3 more