# Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Problems Formulation

#### 2.1. Notation

#### 2.2. Preliminaries

- (i)
- The eigenvalues of L satisfy $0={\lambda}_{1}\left(L\right)\le {\lambda}_{2}\left(L\right)\le \cdots \le {\lambda}_{N}\left(L\right)$, if $\mathcal{G}$ is connected, one has$${\lambda}_{2}\left(L\right)={\displaystyle \underset{z\perp {1}_{n}}{min}}\frac{{z}^{T}Lz}{{\u2225z\u2225}^{2}}>0.$$
- (ii)
- L is a positive semi-definite matrix that satisfies the following sum-of-squares (SOS) property:$${z}^{T}Lz=\frac{1}{2}{\displaystyle \sum _{i,j\in \mathcal{E}}}{a}_{ij}{({z}_{j}-{z}_{i})}^{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}z\in {\mathbb{R}}^{n}.$$

**Lemma**

**1.**

#### 2.3. System Model

**Assumption**

**1.**

#### 2.4. Problem Statement

**Definition**

**1.**

## 3. Flocking via Distributed ETC

#### 3.1. Design of Action Function

#### 3.2. Controller Design and Stability Analysis

**Remark**

**1.**

**Theorem**

**1.**

- (i)
- $\mathcal{G}\left(t\right)$ is connected and no collisions occur for $\forall t\ge 0$;
- (ii)
- Flocking motion is achieved asymptotically.

**Proof.**

**Remark**

**2.**

**Corollary**

**1.**

- (i)
- $\mathcal{G}\left(t\right)$ is connected and no collisions occur for $\forall t\ge 0$;
- (ii)
- Flocking motion is achieved asymptotically.

## 4. Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**3-D flocking for 4 agents applying algorithm (6). (

**a**) Initial states; (

**b**) states at $t=50$ s; (

**c**) trajectories of all agents; (

**d**) velocity convergence.

**Figure 4.**The controller update time instants of each agent with algorithm (6).

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**MDPI and ACS Style**

Shen, Y.; Kong, Z.; Ding, L.
Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control. *Appl. Sci.* **2019**, *9*, 1336.
https://doi.org/10.3390/app9071336

**AMA Style**

Shen Y, Kong Z, Ding L.
Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control. *Applied Sciences*. 2019; 9(7):1336.
https://doi.org/10.3390/app9071336

**Chicago/Turabian Style**

Shen, Yanhua, Zhengmin Kong, and Li Ding.
2019. "Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control" *Applied Sciences* 9, no. 7: 1336.
https://doi.org/10.3390/app9071336