﻿ 基于二阶邻居事件触发多智能体系统的一致性
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 智能系统学报  2017, Vol. 12 Issue (6): 833-840  DOI: 10.11992/tis.201702008 0

### 引用本文

XIA Qianqian, LIU Kaien, JI Zhijian. Event-triggered consensus of multi-agent systems based on second-order neighbors[J]. CAAI Transactions on Intelligent Systems, 2017, 12(6): 833-840. DOI: 10.11992/tis.201702008.

### 文章历史

1. 青岛大学 数学与统计学院，山东 青岛 266071;
2. 青岛大学 自动化与电气工程学院，山东 青岛 266071

Event-triggered consensus of multi-agent systems based on second-order neighbors
XIA Qianqian1, LIU Kaien1, JI Zhijian2
1. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China;
2. School of Automation and Electrical Engineering, Qingdao University, Qingdao 266071, China
Abstract: In this study, to prove the average consensus of multi-agent systems and guarantee that the state of each agent can be quickly converged to the average value of the initial states of all agents, a Lyapunov Function method was adopted. By combining with specific event-triggered conditions, an event-based consensus protocol was designed, and the information on second-order neighbors was used to speed up the consensus convergence of the multi-agent systems. Firstly, under a fixed topology network, we looked at speeding up the consensus convergence by getting the multi-agent systems to apply information to their second-order neighbors; then, similar problems were analyzed under a switching topology network; finally, the protocol was applied in a numerical simulation and was compared with a protocol that applied information to only the first-order neighbors. The simulation results show that the proposed protocol can accelerate the convergence speed.
Key words: first-order dynamics equation    multi-agent systems    consensus    event-triggering    second-order neighbors    Lyapunov function    convergence rate    simulation

1 准备知识 1.1 代数图论

$G = \{ {\cal V},{\cal E},{\mathit{\boldsymbol{A}}}\}$ 表示含有n个节点的无向图，其中 ${\cal V} = \{ {v_1},{v_2}, \cdots ,{v_n}\}$ 表示节点的集合， ${\cal E} \subseteq {\cal V} \times {\cal V}$ 表示边的集合，若 $({v_i},{v_j}) \in {\cal E}$ ，那么vivj称为是相邻的。 ${N_i} = \{ j|({v_i},{v_j}) \in {\cal E},j \ne i\}$ 表示节点vi的一阶邻居， $N_i^2 = \{ k|k \in {N_j},j \in {N_i},k \ne i,k \ne j\}$ 表示节点vi的二阶邻居。邻接矩阵 ${\mathit{\boldsymbol{A}}} = {[{a_{ij}}]_{n \times n}}$ 定义为：若 $({v_i},{v_j}) \in {\cal E}$ ，那么 ${a_{ij}} = 1$ ；否则 ${a_{ij}} = 0$ 。由于在无向图G中， ${a_{ij}} = {a_{ji}}$ $\forall i \ne j$ ，所以A为对称阵。在无向图G中，度矩阵 ${\mathit{\boldsymbol{D}}} = {\rm{diag}}( {d_1},{d_2}, \cdots ,{d_n})$ 是一个对角阵，其中di表示节点vi的邻居集Ni的势。矩阵 ${\mathit{\boldsymbol{L}}} = {\mathit{\boldsymbol{D}}} - {\mathit{\boldsymbol{A}}}$ 称为与图G中一阶邻居信息对应的拉普拉斯矩阵。类似地可以定义二阶邻居信息对应的拉普拉斯矩阵 ${\tilde{L}}$ 。在无向图G中，L是对称的半正定矩阵，即 ${\mathit{\boldsymbol{L}}} = {{\mathit{\boldsymbol{L}}}^{\rm{T}}} \geqslant 0$ ，因此它的特征值都是非负实数[3]，记为 ${\lambda _1} \leqslant {\lambda _2} \leqslant \cdots \leqslant$ ${\lambda _n}$ 。对于无向图G，如果两个节点 ${v_{{i_0}}}$ ${v_{{i_r}}}$ 之间存在一组边 $({v_{{i_0}}},{v_{{i_1}}}),({v_{{i_1}}},{v_{{i_2}}}), \cdots ,({v_{{i_{r - 1}}}},{v_{{i_r}}})$ ，则称从节点 ${v_{{i_0}}}$ ${v_{{i_r}}}$ 存在长度为r的一条路。如果对于G中的任意两个顶点都有一条路，则称G为连通图。在连通图中， ${\lambda _1} = 0$ ${\lambda _2}$ 是最小的非零特征值，且 ${{\bf{1}}_n} = {[1 \,\, 1 \,\, \cdots \,\, 1]^{\rm{T}}}$ 为零特征值所对应的特征向量。在无向连通图中， ${\bf{1}}_n^{\rm{T}}{\mathit{\boldsymbol{L}}} = {\bf{0}}_n^{\rm{T}}$ ，其中 ${{\bf{0}}_n} = {[0 \,\, 0 \,\, \cdots \,\, 0]^{\rm{T}}}$

1.2 模型描述

 ${\dot x_i}(t) = {u_i}(t),\,\, i = 1,2, \cdots ,n$ (1)

 $||{e_i}(t_r^i + lh)||_2^2 \leqslant {\sigma _i}||{z_i}(t_r^i + lh)||_2^2,\,\, l = 1,2, \cdots$ (2)

 $t_{r + 1}^i = t_r^i + h\inf \{ l:||{e_i}(t_r^i + lh)||_2^2 > {\sigma _i}||{z_i}(t_r^i + lh)||_2^2\}$

 ${\hat x_i}(t) \buildrel \Delta \over = {x_i}(t_r^i),\;t_r^i \leqslant t < t_{r + 1}^i$

 ${u_i}(t) = - (\sum\limits_{j \in {{\cal N}_i}} {({{\hat x}_i}(} t) - {\hat x_j}(t)) + \sum\limits_{k \in {\cal N}_i^2} {({{\hat x}_i}(} t) - {\hat x_k}(t)))$ (3)
2 固定拓扑网络下的一致性研究

 ${\dot x_i}(t) = - (\sum\limits_{j \in {{\cal N}_i}} {({{\hat x}_i}(} t) - {\hat x_j}(t)) + \sum\limits_{k \in {\cal N}_i^2} {({{\hat x}_i}(} t) - {\hat x_k}(t)))$

 $\begin{array}{c}{{\dot x}_i}(t) = - (\displaystyle\sum\limits_{j \in {{\cal N}_i}} {({x_i}(} t_r^i) - {x_j}(t_{r'}^j)) + \sum\limits_{k \in {\cal N}_i^2} {({x_i}(} t_r^i) - {x_k}(t_{r''}^k))) = \\ - \displaystyle\sum\limits_{j \in {{\cal N}_i}} {({x_i}(} t_r^i + lh) - {x_j}(t_r^i + lh)) - \\\sum\limits_{j \in {{\cal N}_i}} {({x_i}(} t_r^i) - {x_i}(t_r^i + lh)) + \displaystyle\sum\limits_{j \in {{\cal N}_i}} {({x_j}(} t_{r'}^j) - {x_j}(t_r^i + lh)) - \\\displaystyle\sum\limits_{k \in {\cal N}_i^2} {({x_i}(} t_r^i + lh) - {x_k}(t_r^i + lh)) - \\\displaystyle\sum\limits_{k \in {\cal N}_i^2} {({x_i}(} t_r^i) - {x_i}(t_r^i + lh)) + \sum\limits_{k \in {\cal N}_i^2} {({x_k}(} t_{r''}^k) - {x_k}(t_r^i + lh)) = \\ - \displaystyle\sum\limits_{j \in {{\cal N}_i}} {({x_i}(} t_r^i + lh) - {x_j}(t_r^i + lh)) - \sum\limits_{j \in {{\cal N}_i}} {({e_i}(} t_r^i + lh) - \\{e_j}(t_r^i + lh)) - \displaystyle\sum\limits_{k \in {\cal N}_i^2} {({x_i}(} t_r^i + lh) - {x_k}(t_r^i + lh)) - \\\displaystyle\sum\limits_{k \in {\cal N}_i^2} {({e_i}(} t_r^i + lh) - {e_k}(t_r^i + lh))\end{array}$ (4)

 ${\dot{x}}(t) = - ({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) - ({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh)$ (5)

 $|2{{\mathit{\boldsymbol{a}}}^{\rm{T}}}{\mathit{\boldsymbol{Ab}}}| \leqslant {{\mathit{\boldsymbol{a}}}^{\rm{T}}}{\mathit{\boldsymbol{Aa}}} + {{\mathit{\boldsymbol{b}}}^{\rm{T}}}{\mathit{\boldsymbol{Ab}}}$

 ${{\mathit{\boldsymbol{a}}}^{\rm{T}}}{\mathit{\boldsymbol{Aa}}} = {{\mathit{\boldsymbol{a}}}^{\rm{T}}}{{\mathit{\boldsymbol{U}}}^{\rm{T}}}{\mathit{\boldsymbol{UA}}}{{\mathit{\boldsymbol{U}}}^{\rm{T}}}{\mathit{\boldsymbol{Ua}}} = {{\mathit{\boldsymbol{\alpha }}}^{\rm{T}}}{\mathit{\boldsymbol{\varLambda \alpha }}} = {\lambda _1}\alpha _1^2 + \cdots + {\lambda _n}\alpha _n^2$

${\mathit{\boldsymbol{\beta }}} = {\mathit{\boldsymbol{Ub}}}$ ，则

 ${{\mathit{\boldsymbol{b}}}^{\rm{T}}}{\mathit{\boldsymbol{Ab}}} = {{\mathit{\boldsymbol{b}}}^{\rm{T}}}{{\mathit{\boldsymbol{U}}}^{\rm{T}}}{\mathit{\boldsymbol{UA}}}{{\mathit{\boldsymbol{U}}}^{\rm{T}}}{\mathit{\boldsymbol{Ub}}} = {{\mathit{\boldsymbol{\beta }}}^{\rm{T}}}{\mathit{\boldsymbol{\varLambda \beta }}} = {\lambda _1}\beta _1^2 + \cdots + {\lambda _n}\beta _n^2$

 $\begin{array}{c}{{\mathit{\boldsymbol{a}}}^{\rm{T}}}{\mathit{\boldsymbol{Aa}}} + {{\mathit{\boldsymbol{b}}}^{\rm{T}}}{\mathit{\boldsymbol{Ab}}} = {{\mathit{\boldsymbol{\alpha }}}^{\rm{T}}}{\mathit{\boldsymbol{\varLambda \alpha }}} + {{\mathit{\boldsymbol{\beta }}}^{\rm{T}}}{\mathit{\boldsymbol{\varLambda \beta }}} = \\[3pt]{\lambda _1}\alpha _1^2 + \cdots + {\lambda _n}\alpha _n^2 + {\lambda _1}\beta _1^2 + \cdots + {\lambda _n}\beta _n^2 = \\[3pt]{\lambda _1}(\alpha _1^2 + \beta _1^2) + \cdots + {\lambda _n}(\alpha _n^2 + \beta _n^2) \geqslant \\[5pt]2({\lambda _1}|{\alpha _1}{\beta _1}| + \cdots + {\lambda _n}|{\alpha _n}{\beta _n}|) = 2|{{\mathit{\boldsymbol{\alpha }}}^{\rm{T}}}{\mathit{\boldsymbol{\varLambda \beta }}}| = \\[3pt]2|{{\mathit{\boldsymbol{a}}}^{\rm{T}}}{{\mathit{\boldsymbol{U}}}^{\rm{T}}}{\mathit{\boldsymbol{\varLambda Ub}}}| = 2|{{\mathit{\boldsymbol{a}}}^{\rm{T}}}{\mathit{\boldsymbol{Ab}}}|\end{array}$

 $\bar x(t) = \frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} (t)$

 $\delta (t) = {\mathit{\boldsymbol{x}}}(t) - \bar x(t){{\bf{1}}_n} = {\mathit{\boldsymbol{x}}}(t) - \bar x{{\bf{1}}_n}$

 $V(t) = \frac{1}{2}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(t){\mathit{\boldsymbol{x}}}(t)$ (6)

 $0 < h \leqslant \frac{1}{{2{\lambda _n}}},0 < {\sigma _{\rm{max}}} < \frac{1}{{\lambda _n^2}}$ (7)

 $\frac{1}{{2{\lambda _n}}} < h < \frac{3}{{4{\lambda _n}}},\;0 < {\sigma _{\rm{max}}} < \frac{{3 - 4h{\lambda _n}}}{{(4h{\lambda _n} - 1)\lambda _n^2}}$ (8)

 $\begin{array}{c}\dot V(t) = {{\mathit{\boldsymbol{x}}}^{\rm{T}}}(t){\dot{x}}(t) = - {{\mathit{\boldsymbol{x}}}^{\rm{T}}}(t)({\mathit{\boldsymbol{L}}} + {\tilde{L}})({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh)) = \\[4pt](t - rh){({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh))^{\rm{T}}}{({\mathit{\boldsymbol{L}}} + {\tilde{L}})^2}({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh)) - \\[4pt]{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}})({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh)) \leqslant - {{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + \\[4pt]{\tilde{L}})({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh)) + h{\lambda _n}{({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh))^{\rm{T}}}({\mathit{\boldsymbol{L}}} + \\[4pt]{\tilde{L}})({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh)) = - (1 - h{\lambda _n}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) - \\[5pt]{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh){\mathit{\boldsymbol{(L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh) + h{\lambda _n}{{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh) + \\[4pt]2h{\lambda _n}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh) = - (1 - h{\lambda _n}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + \\[5pt]{\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + (2h{\lambda _n} - 1){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh) + \\[4pt]h{\lambda _n}{{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh)\end{array}$

 $\begin{array}{c}\dot V(t) \leqslant - (1 - h{\lambda _n}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + (1 - 2h{\lambda _n})\\[5pt](\displaystyle\frac{1}{2}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + \displaystyle\frac{1}{2}{{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh)) + \\[5pt]h{\lambda _n}{{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh) = - \displaystyle\frac{1}{2}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + \\[5pt]\displaystyle\frac{1}{2}{{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh)\end{array}$

 $\begin{array}{c}\dot V(t) \leqslant - \displaystyle\frac{1}{2}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + \displaystyle\frac{1}{2}\lambda _n^2{\sigma _{\rm{max}}}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)\\[5pt]({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) = - \displaystyle\frac{1}{2}(1 - \lambda _n^2{\sigma _{\rm{max}}}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh)\end{array}$

$2h{\lambda _n} - 1 > 0$ 时，有

 $\begin{array}{c}\dot V(t) \leqslant - (1 - h{\lambda _n}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + (2h{\lambda _n} - 1)\\[5pt](\displaystyle\frac{1}{2}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + \displaystyle\frac{1}{2}{{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh)) + \end{array}$
 $\begin{array}{c}h{\lambda _n}{{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh) = (2h{\lambda _n} - \displaystyle\frac{3}{2}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + \\[5pt]{\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + (2h{\lambda _n} - \displaystyle\frac{1}{2}){{\mathit{\boldsymbol{e}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh)\end{array}$

 $\begin{array}{c}\dot V(t) \leqslant (2h{\lambda _n} - \displaystyle\frac{3}{2}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh) + \\[5pt](2h{\lambda _n} - \displaystyle\frac{1}{2})\lambda _n^2{\sigma _{max}}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{e}}}(rh) = \\[5pt](2h{\lambda _n} - \displaystyle\frac{3}{2} + (2h{\lambda _n} - \displaystyle\frac{1}{2})\lambda _n^2{\sigma _{max}}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh)({\mathit{\boldsymbol{L}}} + {\tilde{L}}){\mathit{\boldsymbol{x}}}(rh)\end{array}$

 $0 < {\sigma _{\rm{max}}} < \frac{1}{{4{{(n - 1)}^2}}},\;0 < h < \frac{1}{{4(n - 1)}}。$

3 切换拓扑网络下的一致性研究

$V(t) = \displaystyle\frac{1}{2}{{\mathit{\boldsymbol{x}}}^{\rm{T}}}(t)$ ${\mathit{\boldsymbol{x}}}(t)$

 $0 < h \leqslant \frac{1}{{2{\lambda _{\rm{max}}}}},\;0 < {\sigma _{\rm{max}}} < \frac{1}{{\lambda _{\rm{max}}^2}}$

 $\frac{1}{{2{\lambda _{\rm{max}}}}} < h < \frac{3}{{4{\lambda _{\rm{max}}}}},\;0 < {\sigma _{\rm{max}}} < \frac{{3 - 4h{\lambda _{\rm{max}}}}}{{(4h{\lambda _{\rm{max}}} - 1)\lambda _{\rm{max}}^2}},$

 $\dot V(t) = - {{\mathit{\boldsymbol{x}}}^{\rm{T}}}(t){({\mathit{\boldsymbol{L}}} + {\tilde{L}})_{s(rh)}}({\mathit{\boldsymbol{x}}}(rh) + {\mathit{\boldsymbol{e}}}(rh))$

$0 < h \leqslant \displaystyle\frac{1}{{2{\lambda _n}({G_{s(rh)}})}}$ ，且 $0 < {\sigma _{\rm{max}}} < \displaystyle\frac{1}{{\lambda _n^2({G_{s(rh)}})}}$ ，或者 $\displaystyle\frac{1}{{2{\lambda _n}({G_{s(rh)}})}} < h < \displaystyle\frac{3}{{4{\lambda _n}({G_{s(rh)}})}}$ ，且

 $0 < {\sigma _{\rm{max}}} <\displaystyle\frac{{3 - 4h{\lambda _n}({G_{s({{rh}})}})}}{{(4h{\lambda _n}({G_{s({{rh}})}}) - 1)\lambda _n^2({G_{s({{rh}})}})}}$

 $\lambda _n^2({G_{s(rh)}}){\sigma _{\rm{max}}}){{\mathit{\boldsymbol{x}}}^{\rm{T}}}(rh){({\mathit{\boldsymbol{L}}} + {\tilde{L}})_{s(rh)}}{\mathit{\boldsymbol{x}}}(rh)$

4 仿真

 ${\mathit{\boldsymbol{L}}} + {\tilde{L}} = \left[ {\begin{array}{*{20}{c}}3 & { - 1} & { - 1} & { - 1}\\{ - 1} & 3 & { - 1} & { - 1}\\{ - 1} & { - 1} & 3 & { - 1}\\{ - 1} & { - 1} & { - 1} & 3\end{array}} \right]$

 图 1 通信拓扑图 Fig.1 Communication topology

 图 2 智能体的状态 Fig.2 States of the agents

 图 3 非一致向量范数 Fig.3 Norm of the disagreement vector

 图 4 智能体的控制输入 Fig.4 Control inputs for the agents

 图 5 智能体的事件触发时刻 Fig.5 Event times for the agents

 图 6 智能体的状态 Fig.6 States of the agents
 图 7 非一致向量范数 Fig.7 Norm of the disagreement vector

 图 8 切换拓扑图 Fig.8 Switching topology

 图 9 智能体的状态 Fig.9 States of the agents

 图 10 智能体的事件触发时刻 Fig.10 Event times for the agents
5 结束语

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