﻿ 异质多智能体系统二分一致性的充要条件
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 智能系统学报  2020, Vol. 15 Issue (4): 679-686  DOI: 10.11992/tis.201901008 0

### 引用本文

WANG Xiaoyu, LIU Kaien, JI Zhijian, et al. Necessary and sufficient conditions for bipartite consensus of heterogeneous multi-agent systems[J]. CAAI Transactions on Intelligent Systems, 2020, 15(4): 679-686. DOI: 10.11992/tis.201901008.

### 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 自动化工程学院，山东 青岛 266071

Necessary and sufficient conditions for bipartite consensus of heterogeneous multi-agent systems
WANG Xiaoyu 1, LIU Kaien 1, JI Zhijian 2, LIANG Jingxian 1
1. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China;
2. School of Automation Engineering, Qingdao University, Qingdao 266071, China
Abstract: To investigate the bipartite consensus problem of heterogeneous multi-agent systems composed of first- and second-order agents, in this study, we designed bipartite consensus protocols for continuous and discrete systems. Based on a structurally balanced topology, we employ gauge transformation to transform a system with antagonistic interactions into one with non-negative connection weights. Accordingly, the bipartite consensus problem is transformed into a general consensus problem. We use algebraic graph theory and matrix theory to analyze the dynamic characteristics of the closed-loop control system and obtain the necessary and sufficient conditions to guarantee that heterogeneous multi-agent systems reach bipartite consensus asymptotically. Finally, we present numerical simulations to illustrate the effectiveness of the obtained theoretical results.
Key words: heterogeneous multi-agent systems    bipartite consensus    gauge transformation    structural balance    continuous systems    discrete systems    algebraic graph theory    matrix theory

1 预备知识

1) ${V_1} \cup {V_2} = V,\;{V_1} \cap {V_2} = \text{Ø}$

2) ${a_{ij}} \geqslant 0,\forall {v_i},{v_j} \in {V_q}(q \in \{ 1,2\} );$

 ${a_{ij}} \leqslant 0,\forall {v_i} \in {V_q},{v_j} \in {V_r},q \ne r(q,r \in \{ 1,2\} ) 。$

${{\rm{R}}^n}$ 中所有规范变换矩阵的集合为 ${{\cal D}} = \{ {{D}} =$ ${\rm{diag}}\{{{\rm \sigma} _1},{{\rm \sigma} _2}, \cdots ,{{\rm \sigma} _n}\} ,{{\rm \sigma} _i} \in \{ \pm 1\} \}$ 。当多智能体系统的拓扑结构为结构平衡时，通过选取合适的 ${{D}} \in { {\cal D}}$ 可以使得 ${{\rm{DAD}}}$ 中的元素非负，同时保证 ${{\rm{DLD}}}$ 满足非对角线元素非正且行和为零。

2 问题陈述

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot x}_i}(t) = {v_i}(t),\;\;{{\dot v}_i}(t) = {u_i}(t),\;\;i \in {I_1}} \\ {{{\dot x}_i}(t) = {u_i}(t),\;\;i \in {I_2}} \end{array}} \right.$ (1)

 $\left\{ \begin{array}{l} {u_i}(t) = - \alpha {v_i}(t) + {\alpha ^2}\left[\displaystyle\sum\limits_{j \in {N_{i1}}} {{a_{ij}}} ({x_j}(t) - {y_i}(t)) - \right. \\ \;\;\;\;\;\;\;\;\; \left. \displaystyle\sum\limits_{j \in {N_{i2}}} \mid {a_{ij}}\mid ({y_i}(t) + {x_j}(t))\right],\;\;i \in {I_1} \\ {u_i}(t) = \alpha \left[\displaystyle\sum\limits_{j \in {N_{i2}}} {{a_{ij}}} ({x_j}(t) - {x_i}(t)) - \right. \\ \;\;\;\;\;\;\;\;\; \left. \displaystyle\sum\limits_{j \in {N_{i1}}} \mid {a_{ij}}\mid ({x_i}(t) + {x_j}(t))\right],\;\;i \in {I_2} \\ \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} {x_i}(k + 1) = {x_i}(k) + T{v_i}(k),i \in {I_1} \\ {v_i}(k + 1) = {v_i}(k) + T{u_i}(k),i \in {I_1} \\ {x_i}(k + 1) = {x_i}(k) + T{u_i}(k),i \in {I_2} \\ \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} {u_i}(k) = - \alpha {v_i}(k) + {\alpha ^2}\left[\displaystyle\sum\limits_{j \in {N_{i1}}} {{a_{ij}}} ({x_j}(k) - {x_i}(k)) + \right. \\ \;\;\;\;\;\;\;\;\; \left. \displaystyle\sum\limits_{j \in {N_{i2}}} {{a_{ij}}} {x_j}(k)\right],\;\;\;i \in {I_1} \\ {u_i}(k) = \alpha \left[\displaystyle\sum\limits_{j \in {N_{i2}}} {{a_{ij}}} ({x_j}(k) - {x_i}(k)) + \right. \\ \;\;\;\;\;\;\;\;\; \left. \displaystyle\sum\limits_{j \in {N_{i1}}} {{a_{ij}}} {x_j}(k)\right],\;\;\;\;i \in {I_2} \\ \end{array} \right.$ (4)

 $\left\{ \begin{array}{l} {u_i}(k) = - \alpha {v_i}(k) + {\alpha ^2}\left[\displaystyle\sum\limits_{j \in {N_{i1}}} {{a_{ij}}} ({x_j}(k) - {y_i}(k)) - \right. \\ \;\;\;\;\;\;\;\; \left. \displaystyle\sum\limits_{j \in {N_{i2}}} \mid {a_{ij}}\mid ({y_i}(k) + {x_j}(k))\right],\;\;i \in {I_1} \\ {u_i}(k) = \alpha \left[\displaystyle\sum\limits_{j \in {N_{i2}}} {{a_{ij}}} ({x_j}(k) - {x_i}(k)) - \right. \\ \;\;\;\;\;\;\;\; \left. \displaystyle\sum\limits_{j \in {N_{i1}}} \mid {a_{ij}}\mid ({x_i}(k) + {x_j}(k))\right],\;i \in {I_2} \\ \end{array} \right.$ (5)

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {\lim }\limits_{t \to \infty } \Bigg| {x_i}(t) - \dfrac{{| {a_{ij}}| }}{{{a_{ij}}}}{x_j}(t)\Bigg| = 0,}&{\forall i \in {I_1},j \in I}\\ {\mathop {\lim }\limits_{t \to \infty } | {v_i}(t)| = 0,}\quad{\forall i \in {I_1}}\\ {\mathop {\lim }\limits_{t \to \infty } \Bigg| {x_i}(t) - \dfrac{{| {a_{ij}}| }}{{{a_{ij}}}}{x_j}(t)\Bigg| = 0,}&{\forall i \in {I_2},j \in I} \end{array}} \right.$ (6)

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {\lim }\limits_{k \to \infty } \Bigg| {x_i}(k) - \dfrac{{| {a_{ij}}| }}{{{a_{ij}}}}{x_j}(k)\Bigg| = 0,}&{\forall i \in {I_1},j \in I}\\ {\mathop {\lim }\limits_{k \to \infty } | {v_i}(k)| = 0,\;}\quad{\forall i \in {I_1}}\\ {\mathop {\lim }\limits_{k \to \infty } \Bigg| {x_i}(k) - \dfrac{{| {a_{ij}}| }}{{{a_{ij}}}}{x_j}(k)\Bigg| = 0,}&{\forall i \in {I_2},j \in I} \end{array}} \right.$ (7)

3 一致性分析

 ${{A}} = \left[ {\begin{array}{*{20}{c}} {{{{A}}_1}}&{{{{A}}_2}} \\ {{{{A}}_3}}&{{{{A}}_4}} \end{array}} \right]$ (8)

3.1 连续系统的一致性分析

 $\dot \xi = - {\bar{ L}}\xi$ (9)

 ${\bar{ L}} = \left[ {\begin{array}{*{20}{c}} {\alpha {{{I}_m}}}&{ - \alpha {{{I}}_m}}&{{{0}}} \\ { - \alpha {{\tilde{{A}}}_1}}&{\alpha ({{{\varLambda}} _1} + {{{\varLambda}} _2})}&{\alpha {{\tilde{{A}}}_{\rm{2}}}} \\ {\alpha {{\tilde{{A}}}_{\rm{3}}}}&{{{0}}}&{ - \alpha {{\tilde{{A}}}_{\rm{4}}} + \alpha {{{\varLambda}} _3} + \alpha {{{\varLambda}} _4}} \end{array}} \right]$ (10)

 ${\dot{\hat{\xi}}} = - {\hat{ L}}{\hat{\xi}} ,$ (11)

 ${\hat{ L}} = \left[ {\begin{array}{*{20}{c}} {\alpha {{{I}}_m}}&{ - \alpha {{{I}}_m}}&{\bf 0} \\ { - \alpha {{{\tilde{ A}}}_1}}&{\alpha ({{{\varLambda}} _1} + {{{\varLambda}} _2})}&{ - \alpha {{{\tilde{ A}}}_{\rm{2}}}} \\ { - \alpha {{{\tilde{ A}}}_3}}&{\bf 0}&{ - \alpha {{{\tilde{ A}}}_4} + \alpha {{{\varLambda}} _3} + \alpha {{{\varLambda}} _4}} \end{array}} \right]$

 $\begin{gathered} {\hat{ L}} \to \left[ {\begin{array}{*{20}{c}} {\alpha {{{I}}_m}}&{\bf 0} \\ {\bf 0}&{\alpha ({{{\varLambda}} _1} + {{{\varLambda}} _2}) - \alpha {{{\tilde{ A}}}_1}} \\ {\bf 0}&{ - \alpha {{{\tilde{ A}}}_3}} \end{array}} \right.\left. {\begin{array}{*{20}{c}} {\bf 0} \\ { - \alpha {{{\tilde{ A}}}_2}} \\ { - \alpha {{{\tilde{ A}}}_4} + \alpha {{{\varLambda}} _3} + \alpha {{{\varLambda}} _4}} \end{array}} \right] \;\;\;\; \buildrel \Delta \over = \alpha \left[ {\begin{array}{*{20}{c}} {{{{I}}_m}}&{\bf{0}} \\ {\bf{0}}&{{L}} \end{array}} \right] \\ \end{gathered}$

3.2 离散系统的一致性分析

 $\left\{ \begin{gathered} {{{x}}_m}(k + 1) = (1 - \alpha T){{{x}}_m}(k) + \alpha T{{{y}}_m}(k) \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \\ {{{y}}_m}(k + 1) = {{{I}}_m}{{{y}}_m}(k) + \alpha T[{{{A}}_1}{{{x}}_m}(k) - {{{\varLambda}} _1}{{{y}}_m}(k) +\;\; \\ \;\;\;\;\;\;\;\;\; {{{A}}_2}{{{x}}_f}(k) - {{{\varLambda}} _2}{{{y}}_m}(k)] \\ {{{x}}_f}(k + 1) = {{{I}}_{n - m}}{{{x}}_f}(k) + \alpha T[{{{A}}_4}{{{x}}_f}(k) - {{{\varLambda}} _4}{{{x}}_f}(k) +\\ \;\;\;\;\;\;\;\;\; {{{A}}_3}{{{x}}_m}(k) - {{{\varLambda}} _3}{{{x}}_f}(k)] \\ \end{gathered} \right.$ (12)

${ z}(k)$ = $[{ x}_m^{\rm{T}}(k)$ ${ y}_m^{\rm{T}}(k)$ ${ x}_f^{\rm{T}}(k){]^{\rm{T}}}$ ，将式(12)写成紧凑形式得：

 ${ z}(k + {\rm{1}}) = { U}{ z}(k)$ (13)

 $\begin{gathered} {{U}} = \left[ {\begin{array}{*{20}{c}} {(1 - \alpha T){{{I}}_m}}&{\alpha T{{{I}}_m}} \\ {\alpha T{{{\tilde{ A}}}_1}}&{{{{I}}_m} - \alpha T({{{\varLambda}} _1} + {{{\varLambda}} _2})} \\ { - \alpha T{{{\tilde{ A}}}_3}}&{\bf 0} \end{array} \begin{array}{*{20}{c}} {\bf 0} \\ { - \alpha T{{{\tilde{ A}}}_2}} \\ {{{{I}}_{n - m}} + \alpha T({{{\tilde{ A}}}_4} - {{{\varLambda}} _3} - {{{\varLambda}} _4})} \end{array}} \right] \\ \end{gathered}$

 ${\tilde{ z}}(k + 1) = {\tilde{ U}} {\tilde{ z}}(k)$ (14)

 $\begin{gathered} {\tilde{ U}} = \left[ {\begin{array}{*{20}{c}} {(1 - \alpha T){{{I}}_m}}&{\alpha T{{{I}}_m}} \\ {\alpha T{{{\tilde{ A}}}_1}}&{{{{I}}_m} - \alpha T({{{\varLambda}} _1} + {{{\varLambda}} _2})} \\ {\alpha T{{{\tilde{ A}}}_3}}&{\bf 0} \end{array}\begin{array}{*{20}{c}} {\bf 0} \\ {\alpha T{{{\tilde{ A}}}_2}} \\ {{{{I}}_{n - m}} + \alpha T({{{\tilde{ A}}}_4} - {{{\varLambda}} _3} - {{{\varLambda}} _4})} \end{array}} \right] \\ \end{gathered}$

 $\begin{array}{l} {\tilde{ U}} \to \left[ {\begin{array}{*{20}{c}} {(1 - \alpha T){{{I}}_m}}&{\bf 0} \\ {\alpha T{{{\tilde{ A}}}_3}}&{{{{I}}_{n - m}} + \alpha T({{{\tilde{ A}}}_4} - {{{\varLambda}} _3} - {{{\varLambda}} _4})} \\ {\alpha T{{{\tilde{ A}}}_1}}&{\alpha T{{{\tilde{ A}}}_2}} \end{array}\begin{array}{*{20}{c}} {\alpha T{{{I}}_m}} \\ {\bf 0} \\ {{{{I}}_m} - \alpha T({{{\varLambda}} _1} + {{{\varLambda}} _2})} \end{array}} \right] {\rm{ }} \triangleq {\hat{ U}} \\ \end{array}$

 ${\hat{ z}}(k + 1) = {\hat{ U}}{\hat{ z}}(k)$ (15)

 ${{V}} = \left[ {\begin{array}{*{20}{c}} {{{{I}}_m}}&{\bf 0}&{ - {{{I}}_m}} \\ { - {{{\tilde{ A}}}_3}}&{ - {{{\tilde{ A}}}_4} + {{{\varLambda}} _3} + {{{\varLambda}} _4}}&{\bf{0}} \\ { - {{{\tilde{ A}}}_1}}&{ - {{{\tilde{ A}}}_2}}&{{{{\varLambda}} _1} + {{{\varLambda}} _2}} \end{array}} \right]$

 $\begin{gathered} { P} = [{{{\varphi}} _{r1}},{{{\varphi}} _{r2}}, \cdots ,{{{\varphi}} _{r(n + m)}}] \in {{ R}^{(n + m) \times (n + m)}}\\ {{ P}^{ - 1}} = {[{{\varphi}} _{l1}^{\rm{T}},{{\varphi}} _{l2}^{\rm{T}}, \cdots ,{{\varphi}} _{l(n + m)}^{\rm{T}}]^{\rm{T}}} \in {{ R}^{(n + m) \times (n + m)}} \end{gathered}$

 ${\hat{{\textit{z}}}}(k)={\hat{ U}} {\hat{{\textit{z}}}}(k - 1) = {{\hat{ U}}^2} {\hat{{\textit{z}}}}(k - 2)= \cdots ={{\hat{ U}}^k} {\hat{{\textit{z}}}}(0)$

 $\mathop {\lim }\limits_{k \to \infty } {\hat{ z}}(k) = \mathop {\lim }\limits_{k \to \infty } \left[ {\begin{array}{*{20}{c}} {{{{\hat{ x}}}_n}(k)} \\ {{{{\hat{ y}}}_m}(k)} \end{array}} \right] = \mathop {\lim }\limits_{k \to \infty } {{\hat{ U}}^k}{\hat{ z}}(0) = \left[ {\begin{array}{*{20}{c}} {{{{\textit{1}}}_n}{{\varphi}} _{l1}^{\rm{T}}{\hat{ z}}(0)} \\ {{{{\textit{1}}}_m}{{\varphi}} _{l1}^{\rm{T}}{\hat{ z}}(0)} \end{array}} \right]$

4 数值模拟

 Download: 图 1 有向拓扑图−包含一棵有向生成树 Fig. 1 Directed topology which contains a directed spanning tree
 Download: 图 2 系统 (1) 中每个智能体的位置 Fig. 2 Position of each agent of system (1)
 Download: 图 3 系统 (1) 分组 ${ V_1}$ 中每个智能体的速度 Fig. 3 Velocity of each agent in ${ V_1}\$ of system (1)

 Download: 图 4 系统 (3) 中每个智能体的位置 ( $v\alpha T = 0.2$ ) Fig. 4 Position of each agent of system (3) ( $\alpha T = 0.2$ )
 Download: 图 5 系统 (3) 分组 ${V_1}$ 中每个智能体的速度 ( $\alpha T = 0.2$ ) Fig. 5 Velocities of each agent in ${V_1}$ of system (3) ( $\alpha T = 0.2$ )
 Download: 图 6 系统 (3) 中每个智能体的位置 ( $\alpha T = 0.756$ ) Fig. 6 Position of each agent of system (3) ( $\alpha T = 0.756$ )
 Download: 图 7 系统(3)分组 ${V_1}$ 中每个智能体的速度 ( $\alpha T = 0.756$ ) Fig. 7 Velocity of each agent in ${V_1}$ of system (3) ( $\alpha T = 0.756$ )
5 结束语

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