﻿ 基于事件的具有时变时滞多智能体系统的平均一致性

# 基于事件的具有时变时滞多智能体系统的平均一致性Event-Based Average Consensus of Multi-Agent Systems with Time-Varying Delays

Abstract: In this paper, the method of periodic sampling and event control is used to analyze the delay ro-bustness of the event-triggered average consensus problem for first-order multi-agent systems with time-varying communication delays. By designing consensus protocol and event-triggering conditions, all agents can achieve consensus under lower communication frequency. By using Lyapunov stability theory, it is proved that average consensus of multi-agent systems can be achieved and the obtained results improve the existing ones. Finally, the validity of the theoretical results is illustrated by numerical simulations.

1. 引言

2. 问题陈述

2.1. 图论知识

$\stackrel{^}{A}=\left[{\stackrel{^}{a}}_{ij}\right]\in {R}^{n×n}$${\stackrel{^}{a}}_{ij}={\stackrel{^}{a}}_{ji}=\left({a}_{ij}+{a}_{ji}\right)/2$$\stackrel{^}{G}$ 的拉普拉斯矩阵记为 $\stackrel{^}{L}$

2.2. 模型建立

(1)

${u}_{i}\left(t\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,\tau \left(0\right)\right)，\hfill \\ -\underset{j\in {N}_{i}}{\sum }{a}_{ij}\left[{\stackrel{^}{x}}_{i}\left(t-\stackrel{^}{\tau }\left(t\right)\right)-{\stackrel{^}{x}}_{j}\left(t-\stackrel{^}{\tau }\left(t\right)\right)\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[\tau \left(0\right),+\infty \right).\hfill \end{array}$ (2)

$\stackrel{˙}{x}\left(t\right)=-L\stackrel{^}{x}\left(t-\stackrel{^}{\tau }\left(t\right)\right),$ (3)

${e}_{i}^{2}\left({t}_{k}^{i}+lh\right)>\left({\sigma }^{2}/\underset{j\in {N}_{i}}{\sum }{a}_{ij}\right){\stackrel{^}{y}}_{i}^{2}\left({t}_{k}^{i}+\left(l-1\right)h\right),l\in {Z}^{+},$ (4)

${\stackrel{^}{x}}_{i}\left({t}_{k+1}^{i}\right)={x}_{i}\left({t}_{k+1}^{i}\right)$

3. 主要结果

${x}^{\text{T}}Lx\ge {\alpha }_{2}{‖x-\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{x}_{i}{1}_{n}‖}^{2},$

${x}^{\text{T}}\left(kh\right){L}^{\text{T}}Lx\left(kh\right)\le \frac{{\beta }_{n}}{{\alpha }_{2}}{x}^{\text{T}}\left(kh\right)Lx\left(kh\right),$

$\begin{array}{l}-\left[\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(k+1\right)h\right)+\sigma \right)\left(h+\tau \left(\left(k+2\right)h\right)-\tau \left(\left(k+1\right)h\right)\right)\\ +\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(k+1\right)h\right)\right)-1+\sigma \right)\right]\ge {\epsilon }_{0},\end{array}$ (5)

$V\left(t\right)=V\left(x\left(t\right)\right)=\frac{1}{2}{x}^{\text{T}}\left(t\right)x\left(t\right),t\in \left[\tau \left(0\right),+\infty \right).$

$\begin{array}{c}x\left(t\right)=x\left(kh+\tau \left(kh\right)\right)-\left(t-\left(kh+\tau \left(kh\right)\right)\right)L\stackrel{^}{x}\left(kh\right)\\ =x\left(kh\right)-\tau \left(kh\right)L\stackrel{^}{x}\left(\left(k-1\right)h\right)-\left(t-\left(kh+\tau \left(kh\right)\right)\right)L\stackrel{^}{x}\left(kh\right),k\in {Z}^{+}.\end{array}$ (6)

$t\in \left[kh+\tau \left(kh\right),\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)$，对 $V\left(t\right)$ 进行求导，结合(6)可以得到

(7)

$\begin{array}{c}|{\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right){L}^{\text{T}}L\stackrel{^}{x}\left(kh\right)|\le \frac{1}{2}\left({\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right){L}^{\text{T}}L\stackrel{^}{x}\left(\left(k-1\right)h\right)+{\stackrel{^}{x}}^{\text{T}}\left(kh\right){L}^{\text{T}}L\stackrel{^}{x}\left(kh\right)\right)\\ \le \frac{{\beta }_{n}}{2{\alpha }_{2}}\left({\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right)L\stackrel{^}{x}\left(\left(k-1\right)h\right)+{\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\right),k\in {Z}^{+}.\end{array}$ (8)

$\begin{array}{c}|{e}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)|=|\underset{i=1}{\overset{n}{\sum }}\left[{e}_{i}\left(kh\right)\underset{j\in {N}_{i}}{\sum }{a}_{ij}\left({\stackrel{^}{x}}_{i}\left(kh\right)-{\stackrel{^}{x}}_{j}\left(kh\right)\right)\right]|\\ \le \frac{1}{2}\underset{i=1}{\overset{n}{\sum }}\underset{j\in {N}_{i}}{\sum }\left[\frac{{e}_{i}^{2}\left(kh\right){a}_{ij}}{\sigma }+\sigma {a}_{ij}{\left({\stackrel{^}{x}}_{i}\left(kh\right)-{\stackrel{^}{x}}_{j}\left(kh\right)\right)}^{2}\right]\text{ }\text{ }.\end{array}$ (9)

$\begin{array}{l}V\left(\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)\\ =V\left(kh+\tau \left(kh\right)\right)-\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){e}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\tau \left(kh\right)\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right){L}^{\text{T}}L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}{\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)}^{2}{\stackrel{^}{x}}^{\text{T}}\left(kh\right){L}^{\text{T}}L\stackrel{^}{x}\left( k h \right)\end{array}$

$\begin{array}{l}\le V\left(kh+\tau \left(kh\right)\right)-\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)\underset{i=1}{\overset{n}{\sum }}\underset{j\in {N}_{i}}{\sum }\left[\frac{{e}_{i}^{2}\left(kh\right){a}_{ij}}{\sigma }+\sigma {a}_{ij}{\left({\stackrel{^}{x}}_{i}\left(kh\right)-{\stackrel{^}{x}}_{j}\left(kh\right)\right)}^{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(kh\right)\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right)L\stackrel{^}{x}\left(\left(k-1\right)h\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(kh\right)\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\beta }_{n}}{2{\alpha }_{2}}{\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)}^{2}{\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left( k h \right)\end{array}$

$\begin{array}{l}=V\left(kh+\tau \left(kh\right)\right)+\frac{1}{2}\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)\underset{i=1}{\overset{n}{\sum }}\underset{j\in {N}_{i}}{\sum }\left[\frac{{e}_{i}^{2}\left(kh\right){a}_{ij}}{\sigma }+\sigma {a}_{ij}{\left({\stackrel{^}{x}}_{i}\left(kh\right)-{\stackrel{^}{x}}_{j}\left(kh\right)\right)}^{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{+}\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(kh\right)\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right)L\stackrel{^}{x}\left(\left(k-1\right)h\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(k+1\right)h\right)\right)-1\right){\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right),k\in {Z}^{+}.\end{array}$

$\begin{array}{l}V\left(\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)\\ =V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+{\int }_{h+\tau \left(h\right)}^{2h+\tau \left(2h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\cdots +{\int }_{kh+\tau \left(kh\right)}^{\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t,\end{array}$

$\begin{array}{l}V\left(\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)\\ \le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{q=1}{\overset{k}{\sum }}\frac{1}{2}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\underset{i=1}{\overset{n}{\sum }}\underset{j\in {N}_{i}}{\sum }\left[\frac{{e}_{i}^{2}\left(qh\right){a}_{ij}}{\sigma }+\sigma {a}_{ij}{\left({\stackrel{^}{x}}_{i}\left(qh\right)-{\stackrel{^}{x}}_{j}\left(qh\right)\right)}^{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\beta }_{n}}{2{\alpha }_{2}}\underset{q=1}{\overset{k}{\sum }}\tau \left(qh\right)\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(\left(q-1\right)h\right)L\stackrel{^}{x}\left(\left(q-1\right)h\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{q=1}{\overset{k}{\sum }}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left[\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(k+1\right)h\right)\right)-1\right]{\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left( k h \right)\end{array}$

$\begin{array}{l}=V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(h\right)\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{q=1}{\overset{k}{\sum }}\frac{1}{2}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\underset{i=1}{\overset{n}{\sum }}\underset{j\in {N}_{i}}{\sum }\left[\frac{{e}_{i}^{2}\left(qh\right){a}_{ij}}{\sigma }+\sigma {a}_{ij}{\left({\stackrel{^}{x}}_{i}\left(qh\right)-{\stackrel{^}{x}}_{j}\left(qh\right)\right)}^{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)\left[\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(k+1\right)h\right)\right)-1\right]{\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{q=1}{\overset{k-1}{\sum }}\left[\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(q+1\right)h\right)\left(h+\tau \left(\left(q+2\right)h\right)-\tau \left(\left(q+1\right)h\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1\right)\right]{\stackrel{^}{x}}^{\text{T}}\left(qh\right)L\stackrel{^}{x}\left( q h \right)\end{array}$

$\begin{array}{l}\le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(h\right)\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{q=1}{\overset{k}{\sum }}\frac{1}{2}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\underset{i=1}{\overset{n}{\sum }}\underset{j\in {N}_{i}}{\sum }\left[\frac{{e}_{i}^{2}\left(qh\right){a}_{ij}}{\sigma }+\sigma {a}_{ij}{\left({\stackrel{^}{x}}_{i}\left(qh\right)-{\stackrel{^}{x}}_{j}\left(qh\right)\right)}^{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{q=1}{\overset{k}{\sum }}\left[\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(q+1\right)h\right)\left(h+\tau \left(\left(q+2\right)h\right)-\tau \left(\left(q+1\right)h\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1\right)\right]{\stackrel{^}{x}}^{\text{T}}\left(qh\right)L\stackrel{^}{x}\left(qh\right),k\in {Z}^{+}.\end{array}$

$\begin{array}{c}\frac{1}{2}\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(kh\right)=-\frac{1}{2}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{l}_{ij}{\left({\stackrel{^}{x}}_{i}\left(kh\right)-{\stackrel{^}{x}}_{j}\left(kh\right)\right)}^{2}\\ =-\frac{1}{2}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{l}_{ij}\left({\stackrel{^}{x}}_{i}^{2}\left(kh\right)+{\stackrel{^}{x}}_{j}^{2}\left(kh\right)-2{\stackrel{^}{x}}_{i}\left(kh\right){\stackrel{^}{x}}_{j}\left(kh\right)\right)\\ ={\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right).\end{array}$ (10)

${e}_{i}^{2}\left(kh\right)\le \left({\sigma }^{2}/\underset{j\in {N}_{i}}{\sum }{a}_{ij}\right){\stackrel{^}{y}}_{i}^{2}\left(\left(k-1\right)h\right),k\in {Z}^{+}.$ (11)

$\begin{array}{l}V\left(\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)\\ \le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(h\right)\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\sigma }{2}\underset{q=1}{\overset{k}{\sum }}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(\left(q-1\right)h\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\underset{q=1}{\overset{k}{\sum }}\left[\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(q+1\right)h\right)\left(h+\tau \left(\left(q+2\right)h\right)-\tau \left(\left(q+1\right)h\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1+\sigma \right)\right]\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left( q h \right)\end{array}$

$\begin{array}{l}\le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(h\right)\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\sigma }{2}\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{y}}_{i}^{2}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\underset{q=1}{\overset{k}{\sum }}\left[\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(q+1\right)h\right)+\sigma \right)\left(h+\tau \left(\left(q+2\right)h\right)-\tau \left(\left(q+1\right)h\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1+\sigma \right)\right]\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(qh\right).\end{array}$ (12)

$\begin{array}{c}0\le \underset{k\to \infty }{\mathrm{lim}}V\left(\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)\\ \le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\sigma }{2}\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{y}}_{i}^{2}\left(0\right)-{\epsilon }_{0}\underset{k=1}{\overset{+\infty }{\sum }}\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(kh\right),\end{array}$

$\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(kh\right)$ 非负，所以可以得到 $\underset{k\to \infty }{\mathrm{lim}}\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(kh\right)=0$。注意到 ，所以 $\underset{k\to \infty }{\mathrm{lim}}{\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)=0$。由(11)可以进一步得到 $\underset{k\to \infty }{\mathrm{lim}}e\left(kh\right)={0}_{n}$。由于 $\underset{k\to \infty }{\mathrm{lim}}{\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)=0$，由引理1可得 $\underset{k\to \infty }{\mathrm{lim}}L\stackrel{^}{x}\left(kh\right)={0}_{n}$。所以 $\underset{k\to \infty }{\mathrm{lim}}Lx\left(kh\right)=\underset{k\to \infty }{\mathrm{lim}}Le\left(kh\right)+\underset{k\to \infty }{\mathrm{lim}}L\stackrel{^}{x}\left(kh\right)={0}_{n}$，进而由引理1和(6)得到 ，因此所有智能体状态达到平均一致。

$\begin{array}{l}|{x}_{i}\left({t}_{k}^{i}+lh\right)-{x}_{i}\left({t}_{k}^{i}\right)|\\ >\sigma |\underset{j\in {N}_{i}}{\sum }{a}_{ij}\left({\stackrel{^}{x}}_{i}\left({t}_{k}^{i}+\left(l-1\right)h\right)-{\stackrel{^}{x}}_{j}\left({t}_{k}^{i}+\left(l-1\right)h\right)\right)+{b}_{i}\left({\stackrel{^}{x}}_{i}\left({t}_{k}^{i}+\left(l-1\right)h\right)-{x}_{0}\left(0\right)\right)|,l\in {Z}^{+},\end{array}$ (13)

${e}_{i}^{2}\left({t}_{k}^{i}+lh\right)>\mathrm{max}\left\{\left({\sigma }^{2}/\underset{j\in {N}_{i}}{\sum }{a}_{ij}\right){\stackrel{^}{y}}_{i}^{2}\left({t}_{k}^{i}+\left(l-1\right)h\right),{\sigma }^{2}{\left(\underset{j\in {N}_{i}}{\sum }{a}_{ij}{\stackrel{^}{z}}_{ij}\left({t}_{k}^{i}+\left(l-1\right)h\right)\right)}^{2}\right\},l\in {Z}^{+},$ (14)

$\begin{array}{l}-\left[\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(k+1\right)h\right)+\mu \sigma \right)\left(h+\tau \left(\left(k+2\right)h\right)-\tau \left(\left(k+1\right)h\right)\right)\\ +\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(k+1\right)h\right)\right)-1+\mu \sigma \right)\right]\ge {\epsilon }_{0},\end{array}$

${e}_{i}^{2}\left({t}_{k}^{i}+lh\right)\le {\sigma }^{2}{\left(\underset{j\in {N}_{i}}{\sum }{a}_{ij}{\stackrel{^}{z}}_{ij}\left({t}_{k}^{i}+\left(l-1\right)h\right)\right)}^{2},l\in {Z}^{+}.$ (15)

${e}^{2}\left(kh\right)\le {\sigma }^{2}{\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right){L}^{\text{T}}L\stackrel{^}{x}\left(\left(k-1\right)h\right),k\in {Z}^{+},$

$\begin{array}{c}|{e}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(\left(kh\right)|\le \frac{1}{2}\left(\frac{{e}^{\tau }\left(kh\right)e\left(kh\right)}{\sigma }+\sigma {\stackrel{^}{x}}^{\text{T}}\left(kh\right){L}^{\text{T}}L\stackrel{^}{x}\left(kh\right)\right)\\ \le \frac{\sigma }{2}\left({\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right){L}^{\text{T}}L\stackrel{^}{x}\left(\left(k-1\right)h\right)+{\stackrel{^}{x}}^{\text{T}}\left(kh\right){L}^{\text{T}}L\stackrel{^}{x}\left(kh\right)\right)\\ \le \frac{\sigma {\beta }_{n}}{2{\alpha }_{2}}\left({\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right)L\stackrel{^}{x}\left(\left(k-1\right)h\right)+{\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\right),k\in {Z}^{+}.\end{array}$

$\begin{array}{l}V\left(\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)\\ =V\left(kh+\tau \left(kh\right)\right)-\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){e}^{\text{T}}\left(kh\right)L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\tau \left(kh\right)\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(\left(k-1\right)h\right){L}^{\text{T}}L\stackrel{^}{x}\left(kh\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}{\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)}^{2}{\stackrel{^}{x}}^{\text{T}}\left(kh\right){L}^{\text{T}}L\stackrel{^}{x}\left( k h \right)\end{array}$

$\begin{array}{l}\le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\sigma {\beta }_{n}}{2{\alpha }_{2}}\underset{q=1}{\overset{k}{\sum }}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left({\stackrel{^}{x}}^{\text{T}}\left(\left(q-1\right)h\right)L\stackrel{^}{x}\left(\left(q-1\right)h\right)+{\stackrel{^}{x}}^{\text{T}}\left(qh\right)L\stackrel{^}{x}\left(qh\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\beta }_{n}}{2{\alpha }_{2}}\underset{q=1}{\overset{k}{\sum }}\tau \left(qh\right)\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right){\stackrel{^}{x}}^{\text{T}}\left(\left(q-1\right)h\right)L\stackrel{^}{x}\left(\left(q-1\right)h\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{q=1}{\overset{k}{\sum }}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left[\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1\right]{\stackrel{^}{x}}^{\text{T}}\left(qh\right)L\stackrel{^}{x}\left( q h \right)\end{array}$

$\begin{array}{l}\le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(h\right)\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\sigma {\beta }_{n}}{4{\alpha }_{2}}\underset{q=1}{\overset{k}{\sum }}\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(\left(q-1\right)h\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\underset{q=1}{\overset{k}{\sum }}\left[\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(q+1\right)h\right)\left(h+\tau \left(\left(q+2\right)h\right)-\tau \left(\left(q+1\right)h\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1+\frac{\sigma {\beta }_{n}}{2{\alpha }_{2}}\right)\right]\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left( q h \right)\end{array}$

$\begin{array}{l}\le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(h\right)\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\sigma {\beta }_{n}}{4{\alpha }_{2}}\left(h+\tau \left(2h\right)-\tau \left(h\right)\right)\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\underset{q=1}{\overset{k}{\sum }}\left[\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(q+1\right)h\right)+\frac{\sigma {\beta }_{n}}{2{\alpha }_{2}}\right)\left(h+\tau \left(\left(q+2\right)h\right)-\tau \left(\left(q+1\right)h\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1+\frac{\sigma {\beta }_{n}}{2{\alpha }_{2}}\right)\right]\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(qh\right),k\in {Z}^{+}.\end{array}$ (16)

$\begin{array}{l}V\left(\left(k+1\right)h+\tau \left(\left(k+1\right)h\right)\right)\\ \le V\left(\tau \left(0\right)\right)+{\int }_{\tau \left(0\right)}^{h+\tau \left(h\right)}\stackrel{˙}{V}\left(t\right)\text{d}t+\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(h\right)\left(h+\tau \left(2h\right)-\tau \left(h\right)\right){\stackrel{^}{x}}^{\text{T}}\left(0\right)L\stackrel{^}{x}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\mu \sigma \left(h+\tau \left(2h\right)-\tau \left(h\right)\right)\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\underset{q=1}{\overset{k}{\sum }}\left[\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\tau \left(\left(q+1\right)h\right)+\mu \sigma \right)\left(h+\tau \left(\left(q+2\right)h\right)-\tau \left(\left(q+1\right)h\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(h+\tau \left(\left(q+1\right)h\right)-\tau \left(qh\right)\right)\left(\frac{{\beta }_{n}}{2{\alpha }_{2}}\left(h+\tau \left(\left(q+1\right)h\right)\right)-1+\mu \sigma \right)\right]\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{y}}_{i}^{2}\left(qh\right).\end{array}$

$\begin{array}{l}-\left[\left(\frac{{\lambda }_{n}}{2}\tau \left(\left(k+1\right)h\right)+\mu \sigma \right)\left(h+\tau \left(\left(k+2\right)h\right)-\tau \left(\left(k+1\right)h\right)\right)\\ +\left(h+\tau \left(\left(k+1\right)h\right)-\tau \left(kh\right)\right)\left(\frac{{\lambda }_{n}}{2}\left(h+\tau \left(\left(k+1\right)h\right)\right)-1+\mu \sigma \right)\right]\ge {\epsilon }_{0},\end{array}$

4. 数值模拟

Figure 1. Communication topology

$L=\left[\begin{array}{ccccc}3& -1& -1& -1& 0\\ -1& 2& 0& 0& -1\\ -1& 0& 2& -1& 0\\ -1& 0& -1& 2& 0\\ 0& -1& 0& 0& 1\end{array}\right],$

Figure 2. States of the agents

Figure 3. Event-triggering time instants under (4)

Figure 4. Event-triggering time instants under (13)

Figure 5. Event-triggering time instants under (14)

5. 结论

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