﻿ 多智能体系统的二部一致性问题研究

# 多智能体系统的二部一致性问题研究Research on Bipartite Consensus Problems of Multi-Agent Systems

Abstract: For a first-order discrete-time multi-agent system in which cooperation and competition mechanism coexist, we establish its bipartite consensus theories under undirected and directed topology graph based on graph theory, nonnegative matrix theory and stability theory through gauge transformation.

1. 引言

2. 预备知识

3. 无向拓扑图下的一阶离散系统的二部一致性

${x}_{i}\left(k+1\right)={x}_{i}\left(k\right)+{u}_{i}\left(k\right)$ (1)

${u}_{i}\left(k\right)=-\epsilon \underset{j\in {N}_{i}}{\sum }|{a}_{ij}|\left({x}_{i}\left(k\right)-\mathrm{sgn}\left({a}_{ij}\right){x}_{j}\left(k\right)\right)$ (2)

$x\left(k+1\right)={\left[\begin{array}{ccc}{x}_{1}\left(k+1\right)& {x}_{2}\left(k+1\right)& \begin{array}{cc}\cdots & {x}_{n}\left(k+1\right)\end{array}\end{array}\right]}^{\text{T}}$，则

$x\left(k+1\right)=x\left(k\right)-\epsilon Lx\left(k\right)=\left(I-\epsilon L\right)x\left(k\right)=\stackrel{^}{P}x\left(k\right)$ (3)

$z\left(k\right)=Dx\left(k\right),D\in {D}^{*}$ (4)

$\begin{array}{c}z\left(k+1\right)=Dx\left(k+1\right)=D\stackrel{^}{P}x\left(k\right)=D\stackrel{^}{P}{D}^{-1}z\left(k\right)=D\left(I-\epsilon L\right){D}^{-1}z\left(k\right)\\ =\left(I-\epsilon DL{D}^{-1}\right)z\left(k\right)=\left(I-\epsilon {L}_{D}\right)z\left(k\right)={P}_{d}z\left(k\right)\end{array}$ (5)

${l}_{D,ih}=\left\{\begin{array}{l}\underset{j\in {N}_{i}}{\sum }|{a}_{ij}|,h=i\\ {\sigma }_{i}{\sigma }_{h}{a}_{ih},h\ne i\end{array}$ (6)

(1) $G\left(A\right)$ 的所有环是正环；

(2) 存在 $D\in {D}^{*}$，使得 $DAD$ 是非负矩阵，且在规范变换下得到的新的Laplacian矩阵

${L}_{D}=\left[\begin{array}{cccc}\underset{j\in {N}_{1}}{\sum }|{a}_{1j}|& -|{a}_{12}|& \cdots & -|{a}_{1n}|\\ -|{a}_{21}|& \underset{j\in {N}_{2}}{\sum }|{a}_{2j}|& \cdots & -|{a}_{2n}|\\ ⋮& ⋮& \ddots & ⋮\\ -|{a}_{n1}|& -|{a}_{n2}|& \cdots & \underset{j\in {N}_{n}}{\sum }|{a}_{nj}|\end{array}\right]$ 是行和为零的对角占优矩阵。

(3) 0是L的特征值。

(1) $G\left(A\right)$ 有一个或多个环是负的；

(2) 不存在 $D\in {D}^{*}$，使得 $DAD$ 是非负的；

(3) L的所有特征值大于零，即 $\varphi \left(x\right)>0$

${G}_{i}=\left\{s||s-{a}_{ii}|\le {R}_{i},s\in \overline{)C}\right\}$$i=1,2,\cdots ,n$ 为矩阵A的第i个Gershgorin圆，并称 ${R}_{i}$ 为Gershgorin圆的半径。

Perron矩阵的参数，则矩阵具有如下性质：

(1)是一个非负行随机矩阵，并且有一个平凡特征值1；

(2)的所有特征值均在一个单位圆内；

(3) 若是平衡的，则是一个双随机矩阵；

(4) 若，则是一个本原矩阵。

(7)

(2) 令的第j个特征值，则的第j个特征值为。由Gershgorin定理，的所有特征值在圆内，令，则有，即的所有特征值均在一个单位圆内。

(3) 若是一个平衡图，则的左特征向量，即，可得，这说明的列和为1。又由(1)知，是一个非负行随机矩阵，所以是一个双随机矩阵。

(4) 若是强连通的，则是一个不可约矩阵。对任意，令，将圆映射到严格位于单位圆内的圆，其中，这说明在时只有一个模长为1的单特征值，可得是一个本原矩阵。

(8)

(9)

，得：

(10)

4. 有向拓扑图下一阶离散系统的二部一致性

(1)是结构平衡的；

(2)的所有环是正环；

(3) 存在，使得是非负矩阵；

(4) 0是L的特征值。

(1)不是结构平衡的；

(2)至少有一个负有向环；

(3) 不存在，使得是非负矩阵。

。若是加权平衡的，则。若不是结构平衡的，则对.

，显然对应的符号图是一个无向连通图，则由定理1类似可得，系统(1)可获得二部一致性。引理4 (3)保证了D的存在，由定理1知，，由

，可得：

(11)

5. 有向拓扑图下一阶离散系统的二部一致性

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