# Cluster-Delay Mean Square Consensus of Stochastic Multi-Agent Systems with Impulse Time Windows

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## Abstract

**:**

## 1. Introduction

- In this paper, the cluster-delay consensus problem of MASs is studied based on the concept of the impulse time window for the first time. From this perspective, our contribution is mainly reflected in solving the problem of how to reasonably preset the impulsive time sequence under the new application background. In other words, setting the corresponding impulse time window layout according to our research results can ensure that MASs achieve cluster-delay consensus under the action of non-fixed position impulsive control signals.
- This paper studies the cluster-delay mean square consensus problem of MASs based on the uncertainty model for the first time, and gives a sufficient mean square consensus criterion through the It$\hat{o}$ formula, which deepens and expands the current research jobs to a certain extent.

## 2. Notation and Preliminaries

## 3. Problem Description and Model Construction

**Assumption**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Lemma**

**1**

**([38]).**For vectors $x,\hat{y}\in {\mathbb{R}}^{n}$ and constant $\sigma >0$, we can get ${x}^{T}\hat{y}+{\hat{y}}^{T}x\le \sigma {x}^{T}x+{\sigma}^{-1}{\hat{y}}^{T}\hat{y}$.

**Assumption**

**2.**

**Definition**

**1**

**([23]).**The SMASs with (3) and (5) are said to reach cluster mean square consensus, if there exist the solutions of (3) and (5) such that ${lim}_{t\to +\infty}\mathrm{E}(\parallel {\hat{x}}_{i}\left(t\right){\parallel}^{2})=0$, where ${\hat{x}}_{i}\left(t\right)={x}_{i}\left(t\right)-{S}_{\hat{i}}\left(t\right)$.

**Definition**

**2**

**([23]).**The SMASs with (3) are said to reach delay mean square consensus, if there exist the solutions of (3) such that ${lim}_{t\to +\infty}\mathrm{E}(\parallel {e}_{y}\left(t\right){\parallel}^{2})=0$, where ${e}_{y}\left(t\right)={S}_{y}\left(t\right)-{S}_{1}(t-{\tau}_{y})$.

**Definition**

**3.**

**Remark**

**4.**

## 4. Consensus Analysis

**Theorem**

**1.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 5. Numerical Simulation

**Example**

**1.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MASs | Multi-agent systems |

SMASs | Stochastic multi-agent systems |

ITM | Impulse time window |

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**Figure 6.**The error trajectories between ${S}_{2}\left(t\right)$, ${S}_{3}\left(t\right)$ and their virtual leaders’ states when ${\rho}_{2}={\rho}_{3}=20$.

**Figure 8.**The error trajectories between ${S}_{2}\left(t\right)$, ${S}_{3}\left(t\right)$ and their virtual leaders’ states when ${\rho}_{2}={\rho}_{3}=30$.

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**MDPI and ACS Style**

Luo, H.; Wang, Y.; Zhan, R.; Zhang, X.; Wen, H.; Yang, S.
Cluster-Delay Mean Square Consensus of Stochastic Multi-Agent Systems with Impulse Time Windows. *Entropy* **2021**, *23*, 1033.
https://doi.org/10.3390/e23081033

**AMA Style**

Luo H, Wang Y, Zhan R, Zhang X, Wen H, Yang S.
Cluster-Delay Mean Square Consensus of Stochastic Multi-Agent Systems with Impulse Time Windows. *Entropy*. 2021; 23(8):1033.
https://doi.org/10.3390/e23081033

**Chicago/Turabian Style**

Luo, Huan, Yinhe Wang, Ruidian Zhan, Xuexi Zhang, Haoxiang Wen, and Senquan Yang.
2021. "Cluster-Delay Mean Square Consensus of Stochastic Multi-Agent Systems with Impulse Time Windows" *Entropy* 23, no. 8: 1033.
https://doi.org/10.3390/e23081033