﻿ 时滞高阶多智能体系统的一致性 Higher-Order Consensus in Multi-Agent System with Delay

Vol.06 No.09(2017), Article ID:23073,7 pages
10.12677/AAM.2017.69133

Higher-Order Consensus in Multi-Agent System with Delay

Hao Wen, Xiao Wang

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha Hunan

Received: Nov. 24th, 2017; accepted: Dec. 12th, 2017; published: Dec. 19th, 2017

ABSTRACT

In this paper, the consensus problem of high-order multi-agent system with delay is considered. Via constructing Lyapunov function, the result of high-order multi-agent system consensus is obtained when the delay is smaller than a certain value. Finally, a numerical simulation is used to verify our main result.

Keywords:Multi-Agent System, Delay, Consensus, Lyapunov Function

1. 引言

$\left\{\begin{array}{l}{\stackrel{˙}{\xi }}_{i}^{1}\left(t\right)={\xi }_{i}^{2}\left(t\right)\hfill \\ {\stackrel{˙}{\xi }}_{i}^{2}\left(t\right)={\xi }_{i}^{3}\left(t\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}⋮\hfill \\ {\stackrel{˙}{\xi }}_{i}^{m}\left(t\right)={u}_{i}\left(t\right),\text{ }i=1,2,\cdots ,N.\hfill \end{array}$ (1)

${u}_{i}\left(t\right)=c\sum _{j=1,j\ne i}^{N}{G}_{ij}\sum _{k=1}^{m}{\alpha }_{k}\left({\xi }_{j}^{k}\left(t\right)-{\xi }_{i}^{k}\left(t\right)\right),\text{ }i=1,2,\cdots ,N.$ (2)

$\left\{\begin{array}{l}{\stackrel{˙}{\xi }}_{i}^{1}\left(t\right)={\xi }_{i}^{2}\left(t\right)\hfill \\ {\stackrel{˙}{\xi }}_{i}^{2}\left(t\right)={\xi }_{i}^{3}\left(t\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }⋮\hfill \\ {\stackrel{˙}{\xi }}_{i}^{m}\left(t\right)={u}_{i}\left(t-\tau \right),\text{ }i=1,2,\cdots ,N.\hfill \end{array}$ (3)

2. 预备知识及引理

${u}_{i}\left(t-\tau \right)=-c\sum _{j=1}^{N}{L}_{ij}\sum _{k=1}^{m}{\alpha }_{k}{\xi }_{j}^{k}\left(t-\tau \right),\text{ }i=1,2,\cdots ,N.$ (4)

$C=\left[\begin{array}{cccc}0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\\ 0& 0& \cdots & 0\end{array}\right],\text{\hspace{0.17em}}D=\left[\begin{array}{cccc}0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0\\ {\alpha }_{1}& {\alpha }_{2}& \cdots & {\alpha }_{m}\end{array}\right],$

${\stackrel{˙}{\eta }}_{i}\left(t\right)=C{\eta }_{i}\left(t\right)-c\sum _{j=1}^{N}{L}_{ij}D{\eta }_{j}\left(t-\tau \right),\text{ }i=1,2,\cdots ,N,$ (5)

$\stackrel{˙}{\eta }\left(t\right)=\left({I}_{N}\otimes C\right)\eta \left(t\right)-c\left(L\otimes D\right)\eta \left(t-\tau \right).$ (6)

$\left({P}^{\text{T}}\otimes {I}_{m}\right)\stackrel{˙}{\eta }\left(t\right)=\left({P}^{\text{T}}\otimes C\right)\eta \left(t\right)-c\left({P}^{\text{T}}\otimes D\right)\eta \left(t-\tau \right)=\left({I}_{N}\otimes C\right)\left({P}^{\text{T}}\otimes {I}_{m}\right)\eta \left(t\right)-c\left(\Lambda \otimes D\right)\left({P}^{\text{T}}\otimes D\right)\eta \left(t-\tau \right).$

$\zeta \left(t\right)=\left({P}^{\text{T}}\otimes {I}_{m}\right)\eta \left(t\right)={\left({\zeta }_{1}^{\text{T}},{\zeta }_{2}^{\text{T}},\cdots ,{\zeta }_{N}^{\text{T}}\right)}^{\text{T}}$ ，则 $\zeta \left(t\right)$ 满足

$\stackrel{˙}{\zeta }\left(t\right)=\left({I}_{N}\otimes C\right)\zeta \left(t\right)-c\left(\Lambda \otimes D\right)\zeta \left(t-\tau \right).$ (7)

${\stackrel{˙}{\zeta }}_{i}\left(t\right)=C{\zeta }_{i}\left(t\right)-c{\lambda }_{i}D{\zeta }_{i}\left(t-\tau \right),\text{ }i=1,2,\cdots ,N.$ (8)

$\underset{t\to \infty }{\mathrm{lim}}‖{\zeta }_{i}\left(t\right)‖=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=2,\cdots ,N.$ (9)

3. 主要结果

${v}_{\alpha }=\left\{X|v\left(X\right)\le \alpha \right\}.$

${\zeta }_{i}\left(t\right)\in {v}_{\alpha },\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=2,3,\cdots ,N.$ (10)

${\zeta }_{i}\left(t\right)=\zeta \left(0\right)+{\int }_{0}^{t}C{\zeta }_{i}\left(s\right)-c{\lambda }_{i}D{\zeta }_{i}\left(s-\tau \right)\text{d}s.$

$‖{\zeta }_{i}\left(t\right)‖\le \delta +{\int }_{0}^{t}\left(‖C‖‖{\zeta }_{i}\left(s\right)‖+‖c{\lambda }_{i}D‖\delta \right)\text{d}s\le \delta \left(1+‖c{\lambda }_{i}D‖\tau \right){\text{e}}^{‖C‖\tau }\le {\left(\alpha \right)}^{1/2}.$

$‖{\zeta }_{i}\left(t\right)-{\zeta }_{i}\left(t-\tau \right)‖<\epsilon ‖{\zeta }_{i}\left(t\right)‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=2,3,\cdots ,N.$ (11)

$\underset{t-2\tau \le s\le t}{\mathrm{sup}}‖{\zeta }_{i}\left(s\right)‖\le \frac{\sqrt{\alpha }}{\sqrt{{\lambda }_{\mathrm{min}}\left(I\right)}}=‖{\zeta }_{i}\left(t\right)‖$

$‖{\zeta }_{i}\left(t\right)-{\zeta }_{i}\left(t-\tau \right)‖=$ $‖{\int }_{t-\tau }^{t}C{\zeta }_{i}\left(s\right)-c{\lambda }_{i}D{\zeta }_{i}\left(s-\tau \right)\text{d}s‖\le \tau \left(‖C‖+‖c{\lambda }_{i}D‖\right)\underset{t-2\tau \le s\le t}{\mathrm{sup}}‖{\zeta }_{i}\left(s\right)‖\le \tau \left(‖C‖+‖c{\lambda }_{i}D‖\right)‖{\zeta }_{i}\left(t\right)‖.$

$\begin{array}{c}\frac{\text{d}v\left({\zeta }_{i}\left(t\right)\right)}{\text{d}t}=〈{\left(C{\zeta }_{i}\left(t\right)-c{\lambda }_{i}D{\zeta }_{i}\left(t-\tau \right)\right)}^{\text{T}},{\zeta }_{i}\left(t\right)〉+〈{\zeta }_{i}{\left(t\right)}^{\text{T}},C{\zeta }_{i}\left(t\right)-c{\lambda }_{i}D{\zeta }_{i}\left(t-\tau \right)〉\\ =〈{\zeta }_{i}{\left(t\right)}^{\text{T}}{C}^{\text{T}},{\zeta }_{i}\left(t\right)〉-〈c{\lambda }_{i}{\zeta }_{i}{\left(t\right)}^{\text{T}}{D}^{\text{T}},{\zeta }_{i}\left(t\right)〉+〈{\zeta }_{i}{\left(t\right)}^{\text{T}},C{\zeta }_{i}\left(t\right)〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-〈{\zeta }_{i}{\left(t\right)}^{\text{T}},c{\lambda }_{i}D{\zeta }_{i}\left(t\right)〉+〈c{\lambda }_{i}{\zeta }_{i}{\left(t\right)}^{\text{T}}{D}^{\text{T}},{\zeta }_{i}\left(t\right)〉-〈c{\lambda }_{i}{\zeta }_{i}^{\text{T}}\left(t-\tau \right){D}^{\text{T}},{\zeta }_{i}\left(t\right)〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+〈{\zeta }_{i}{\left(t\right)}^{\text{T}},c{\lambda }_{i}D{\zeta }_{i}\left(t\right)〉-〈{\zeta }_{i}{\left(t\right)}^{\text{T}},c{\lambda }_{i}D{\zeta }_{i}\left(t-\tau \right)〉\\ \le -{p}_{i}{‖{\zeta }_{i}\left(t\right)‖}^{2}+2\epsilon ‖c{\lambda }_{i}D‖{‖{\zeta }_{i}\left(t\right)‖}^{2}=\left(2\epsilon ‖c{\lambda }_{i}D‖-{p}_{i}\right){‖{\zeta }_{i}\left(t\right)‖}^{2}.\end{array}$

$\frac{\text{d}v\left({\zeta }_{i}\left(t\right)\right)}{\text{d}t}\le -\gamma {‖{\zeta }_{i}\left(t\right)‖}^{2}$

4. 数值模拟

(a) 特征值 ${\lambda }_{2}=1$ ，初值为 $\left({\zeta }_{21},{\zeta }_{22},{\zeta }_{23}\right)=\left(2,1,3\right)$ 时系统(8)的解的图像 (b) 特征值 ${\lambda }_{3}=2$ ，初值为 $\left({\zeta }_{31},{\zeta }_{32},{\zeta }_{33}\right)=\left(1,2,3\right)$ 时系统(8)的解的图像

Figure 1. The trajectories of system (8) when $\tau =\frac{1}{4}$

(c) 特征值 ${\lambda }_{2}=1$ ，初值为 $\left({\zeta }_{21},{\zeta }_{22},{\zeta }_{23}\right)=\left(2,1,3\right)$ 时系统(8)的解的图像 (d) 特征值 ${\lambda }_{3}=2$ ，初值为 $\left({\zeta }_{31},{\zeta }_{32},{\zeta }_{33}\right)=\left(1,2,3\right)$ 时系统(8)的解的图像

Figure 2. The trajectories of system (8) when $\tau =\frac{\text{2}}{\text{3}}$

Higher-Order Consensus in Multi-Agent System with Delay[J]. 应用数学进展, 2017, 06(09): 1098-1104. http://dx.doi.org/10.12677/AAM.2017.69133

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