# A Fast Non-Linear Symmetry Approach for Guaranteed Consensus in Network of Multi-Agent Systems

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Background and Methods

## 3. Theoretical Result

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

- is a non-linear model as well as DeGroot’s model, which means that it has fast convergence;
- is distinguished by a less complicated computation as well as the non-linear stochastic operators of CQSO, DSQO, and EDSQO; and,
- avoids the problem of periodic and non-changing initial values as in the case of EDSQO.

## 4. Discussion and Numerical Solution

#### 4.1. Three Agents

#### 4.2. Four Agents

#### 4.3. Five Agents

## 5. Comparison of the Consensus SSQO Model with Other Consensus Models

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The consensus of three agents by semi symmetry quadratic operator (SSQO) using stochastic matrix (SM) and doubly stochastic matrix (DSM) with initial statuses of (0, 1, 0).

**Figure 2.**The consensus of three agents by SSQO using SM and DSM with initial statuses of (0.162, 0.222, 0.616) (random).

**Figure 4.**The consensus of four agents by SSQO by SM and DSM with initial statuses of (0.12, 0.18, 0.60, 0.10) (random).

**Figure 5.**The consensus of five agents by SSQO using SM and DSM with initial statuses of (0, 1, 0, 0, 0).

**Figure 6.**The consensus of five agents by SSQO by SM and DSM for initial statuses of (0.02, 0.09, 0.15, 0.20, 0.54) (random).

**Figure 9.**The Disagreement Cases of DeGroot, CSQO, DSQO, and EDSQO as compared to SSQO for three agents in one time for three agents in different random initial statuses.

**Figure 10.**The Disagreement Cases of DeGroot, CSQO, DSQO, and EDSQO as compared to SSQO for three agents in 10 times for three agents in different random initial statuses.

Name | Advantages | Disadvantages | Evolutionary Operators | Transition Matrix | |
---|---|---|---|---|---|

1 | DeGroot [63] | - Less computation
| - Linear
- Slow consensus
- Not convergence in some cases
| ${s}_{i}^{\left(\left(t\right)+1\right)}={\sum}_{i=1}^{m}{A}_{ij}{s}_{i}^{\left(t\right)},$ | ${A}_{ij}=\{{a}_{ij}\ge 0,,{\displaystyle {\displaystyle \sum}_{i=1}^{m}}{P}_{ij}=1\}$ |

2 | CSQO [18] | - Non-linear
- Fast consensus
| - More computation (Long polynomial)
- Restricted conditions
| ${s}_{i}^{\left(\left(t\right)+1\right)}={\sum}_{i,j=1}^{m}{s}_{i}^{\left(t\right)}{A}_{ij,k}{s}_{j}^{\left(t\right)},$ | ${A}_{ij,k}=\left\{\begin{array}{c}{a}_{ij,k}\ge 0,{\displaystyle {\displaystyle \sum}_{i=1}^{m}}{a}_{j,k}=1,{\displaystyle {\displaystyle \sum}_{j=1}^{m}}{a}_{i,k}=1,\\ {\displaystyle {\displaystyle \sum}_{k=1}^{m}}{P}_{ij,k}=1,\forall i,j,k=1,\dots ,m\end{array}\right\}$ |

3 | DSQO [2] | - Non-linear
- Fast consensus
| - More computation (Long polynomial)
- Not convergence in some cases
- Restricted conditions
| ${s}_{i}^{\left(\left(t\right)+1\right)}={\sum}_{i,j=1}^{m}{s}_{i}^{\left(t\right)}{A}_{ij,k}{s}_{j}^{\left(t\right)},$ | ${A}_{ij,k}=\left\{\begin{array}{c}{a}_{ij,k}\ge 0,{a}_{ij.k}={a}_{ji,k},\\ {\displaystyle {\displaystyle \sum}_{ij=1}^{m}}{a}_{ij,k}=m,{\displaystyle {\displaystyle \sum}_{k=1}^{m}}{a}_{ij,k}=1,\\ {\displaystyle {\displaystyle {\displaystyle \sum}_{i,j\in \alpha}}}{a}_{ij}\le \left|\alpha \right|,\\ \alpha \subset \left\{1,2,3,\dots ,m\right\},\\ \forall i,j,k=1,\dots ,m\end{array}\right\}$ |

4 | EDSQO [54] | - Non-linear
- Less computation
- Fast consensus
| - Not convergence in some cases
- Restricted conditions
| ${s}_{i}^{\left(\left(t\right)+1\right)}={\sum}_{i,j=1}^{m}{s}_{i}^{\left(t\right)}{A}_{ij,k}{s}_{j}^{\left(t\right)},$ | ${A}_{ij,k}=\left\{\begin{array}{c}{a}_{ii,k}=\left(0|1\right),{a}_{ij,k}=\left(0\left|\frac{1}{2}\right|1\right),\\ {a}_{ji.k}={a}_{ij,k},{\displaystyle {\displaystyle \sum}_{ij=1}^{m}}{a}_{ij,k}=m,\\ {\displaystyle {\displaystyle \sum}_{k=1}^{m}}{a}_{ij,k}=1,{\displaystyle {\displaystyle \sum}_{i,j\in \alpha}}{a}_{ij}\le \left|\alpha \right|,\\ \alpha \subset \left\{1,2,3,\dots ,m\right\},\forall i,j,k=1,\dots ,m\end{array}\right\}$ |

5 | SSQO proposed model | - Non-linear
- Less computation
- Fast consensus
- Convergence in all cases
| - Some restricted conditions
| ${A}_{ij,k}=\left\{\begin{array}{c}{a}_{ii.k}={a}_{ji.k}=(\frac{1}{2}|0),\\ {\sum}_{ij=1}^{m}{a}_{ij,k}=m,{\sum}_{k=1}^{m}{a}_{ij,k}=1\end{array}\right\}$. |

**Table 2.**Description of the cases of the transition matrix for DeGroot, CSQO, DSQO, and EDSQO that cannot reach to consensus with advantaged proposed model SSQO.

Name | Advantages | Disadvantages | |
---|---|---|---|

1 | DeGroot [63] | ${A}_{ij,1}=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$ | This is a transition matrix under the rules of the DeGroot linear consensus model, which is the drawback that cannot reach to consensus. |

2 | CSQO [18] | ${A}_{ij,1}=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\end{array}\right)$, ${A}_{ij,2}=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 0& 0& 1\end{array}\right)$, ${A}_{ij,3}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 1& 0\\ 0& 1& 0\end{array}\right).$ | This is a transition matrix under the rules of the CSQO non-linear consensus model, which is the drawback that cannot reach to consensus. |

3 | DSQO [2] | ${A}_{ij,1}=\left(\begin{array}{ccc}1& 0.7& 0.3\\ 0.7& 0& 0\\ 0.3& 0& 0\end{array}\right)$, ${A}_{ij,2}=\left(\begin{array}{ccc}0& 0.2& 0\\ 0.2& 1& 0.8\\ 0& 0.8& 0\end{array}\right)$, ${A}_{ij,3}=\left(\begin{array}{ccc}0& 0.1& 0.7\\ 0.1& 0& 0.2\\ 0.7& 0.2& 1\end{array}\right)$ | This is a transition matrix under the rules of the DSQO non-linear consensus model, which is the drawback that cannot reach to consensus. |

4 | EDSQO [54] | ${A}_{ij,1}=\left(\begin{array}{ccc}1& 0.5& 0.5\\ 0.5& 0& 0\\ 0.5& 0& 0\end{array}\right)$, ${A}_{ij,2}=\left(\begin{array}{ccc}0& 0.5& 0\\ 0.5& 1& 0.5\\ 0& 0.5& 0\end{array}\right)$, ${A}_{ij,3}=\left(\begin{array}{ccc}0& 0& 0.5\\ 0& 0& 0.5\\ 0.5& 0.5& 1\end{array}\right).$ | This is a transition matrix under the rules of the EDSQO non-linear consensus model, which is the drawback that cannot reach to consensus. |

5 | SSQO proposed model | ${A}_{ij,1}=\left(\begin{array}{ccc}0.5& 0.5& 0.5\\ 0.5& 0.5& 0\\ 0.5& 0& 0\end{array}\right)$, ${A}_{ij,2}=\left(\begin{array}{ccc}0& 0.5& 0\\ 0.5& 0.5& 0.5\\ 0& 0.5& 0.5\end{array}\right)$, ${A}_{ij,3}=\left(\begin{array}{ccc}0.5& 0& 0.5\\ 0& 0& 0.5\\ 0.5& 0.5& 0.5\end{array}\right).$ | This is a transition matrix under the rules of the proposed SSQO non-linear consensus model, which avoids the drawback that cannot reach to consensus. |

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**MDPI and ACS Style**

Abdulghafor, R.; Almotairi, S.
A Fast Non-Linear Symmetry Approach for Guaranteed Consensus in Network of Multi-Agent Systems. *Symmetry* **2020**, *12*, 1692.
https://doi.org/10.3390/sym12101692

**AMA Style**

Abdulghafor R, Almotairi S.
A Fast Non-Linear Symmetry Approach for Guaranteed Consensus in Network of Multi-Agent Systems. *Symmetry*. 2020; 12(10):1692.
https://doi.org/10.3390/sym12101692

**Chicago/Turabian Style**

Abdulghafor, Rawad, and Sultan Almotairi.
2020. "A Fast Non-Linear Symmetry Approach for Guaranteed Consensus in Network of Multi-Agent Systems" *Symmetry* 12, no. 10: 1692.
https://doi.org/10.3390/sym12101692