# On the Role of Matrix-Weights Elements in Consensus Algorithms for Multi-Agent Systems

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^{†}

## Abstract

**:**

## 1. Introduction

- to examine the matrix-weighted graph and show how each element influences the consensus algorithm;
- to study how different forms of consensus can be achieved by the choice of matrix weights;
- to be able to determine the number of clusters and the elements in each of the cluster by looking at the matrix-weight set.

## 2. Problem Formulation

#### 2.1. Agent Dynamics

#### 2.2. Graph Theory

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

#### 2.3. Cluster Consensus

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.4. Objectives

## 3. Results

#### 3.1. Consensus Control Design

**Remark**

**4.**

**Remark**

**5.**

#### 3.2. Non-Diagonal Matrix-Weights

#### 3.2.1. Control Law Depends on Other States in the Same Cluster

**Theorem**

**1.**

- 1.
- If the k-state graph is connected, the kth states of all agents reach k-global consensus (KGC).
- 2.
- If the k-state agents are not connected, the kth states of all agents reach k-cluster consensus (KCC).

**Proof of Theorem 1.**

#### 3.2.2. Control Law Depends on Other States in Different Clusters

**Theorem**

**2.**

- 1.
- There will at least be y clusters for the corresponding agent states in the graph causing a k-cluster consensus (KCC).
- 2.
- In the k-state graph, if ${x}_{j}^{\left[k\right]}$ and ${x}_{i}^{\left[k\right]}$ belong to the same connected component, which has only one incoming link from the other state graphs, then ${x}_{j}^{\left[k\right]},{x}_{i}^{\left[k\right]}\in {\mathcal{C}}_{p}^{k}$.
- 3.
- In the k-state graph, assume that ${x}_{j}^{\left[k\right]}$, ${x}_{i}^{\left[k\right]}$, and ${x}_{p}^{\left[k\right]}$ belong to the same connected components. If ${x}_{i}^{\left[k\right]}$ and ${x}_{p}^{\left[k\right]}$ have incoming links from different connected components in the l-state graph, and ${x}_{j}^{\left[k\right]}$ has no incoming links from other state graphs, then ${x}_{j}^{\left[k\right]}$ will be in a different cluster of ${x}_{i}^{\left[k\right]}$ and ${x}_{p}^{\left[k\right]}$.

**Proof of Theorem 2.**

#### 3.3. Diagonal Matrix Weights

#### 3.3.1. Positive Definite Matrix Weights

**Theorem**

**3.**

- 1.
- If the k-state graph is connected, there will be a global consensus (GC) across the m states of all agents.
- 2.
- If the k-state graph is not connected, there will be a global cluster consensus (GCC) across the m states such that ${\mathcal{C}}^{i}={\mathcal{C}}^{j}$ for all $i\ne j$, $i=1,\dots ,m$, and $j=1,\dots ,m$. Moreover, the number of clusters of the states is determined by the number of connected components of the k-state graph.

**Proof of Theorem 3.**

#### 3.3.2. Positive Semi-Definite Matrix Weights

**Theorem**

**4.**

- 1.
- If the k-state graph is connected, there will be k-global consensus (KGC).
- 2.
- If the k-state graph is not connected, then there is a k-cluster consensus (KCC). Moreover, the number of clusters of the states is determined by the number of connected components of the k-state graph.

**Proof of Theorem 4.**

## 4. Simulations

#### 4.1. Non-Diagonal Matrix-Weights

#### 4.1.1. Control Law Dependent on Other State Values in the Same Cluster

#### 4.1.2. Control Law Dependent on Other State Values in Different Clusters

#### 4.2. Diagonal Matrix-Weights

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MASs | Multi-agent Systems |

PD | Positive definite |

PSD | Positive semi-definite |

GC | Global consensus |

GCC | Global clustered consensus |

KCC | Cluster consensus for state k |

KGC | Global consensus for state k |

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**Figure 1.**Graph topology with non-diagonal matrix weights: $1\mathrm{GC}$, $2\mathrm{GC}$ and $3CC$.

**Figure 2.**State trajectory for non-diagonal matrix weighted graph: $1\mathrm{GC}$, $2\mathrm{GC}$ and $3CC$.

**Figure 5.**State trajectory for diagonal matrix weighted graph: scalar multiple of the identity matrix (GC with similar convergence properties across the states).

**Figure 6.**State trajectory for general diagonal PD matrix weighted graph (GC with different convergence properties across the states).

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**MDPI and ACS Style**

Ogbebor, J.; Meng, X.
On the Role of Matrix-Weights Elements in Consensus Algorithms for Multi-Agent Systems. *Network* **2021**, *1*, 233-246.
https://doi.org/10.3390/network1030014

**AMA Style**

Ogbebor J, Meng X.
On the Role of Matrix-Weights Elements in Consensus Algorithms for Multi-Agent Systems. *Network*. 2021; 1(3):233-246.
https://doi.org/10.3390/network1030014

**Chicago/Turabian Style**

Ogbebor, Joshua, and Xiangyu Meng.
2021. "On the Role of Matrix-Weights Elements in Consensus Algorithms for Multi-Agent Systems" *Network* 1, no. 3: 233-246.
https://doi.org/10.3390/network1030014