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 智能系统学报  2020, Vol. 15 Issue (4): 780-786  DOI: 10.11992/tis.201908027 0

### 引用本文

JI Xiukun, XIE Guangming, WEN Jiayan, et al. Bipartite consensus for multi-agent systems subject to time delays[J]. CAAI Transactions on Intelligent Systems, 2020, 15(4): 780-786. DOI: 10.11992/tis.201908027.

### 文章历史

1. 广西科技大学 电气与信息工程学院，广西 柳州 545006;
2. 北京大学 工学院，北京 100871

Bipartite consensus for multi-agent systems subject to time delays
JI Xiukun 1, XIE Guangming 1,2, WEN Jiayan 1, LUO Wenguang 1
1. School of Electrical and Information Engineering, Guangxi University of Science and Technology, Liuzhou 545006, China;
2. College of Engineering, Peking University, Beijing 100871, China
Abstract: The aim of this paper is to consider the practical constraint of interaction network associated with multi-agent systems (MASs) subject to communication delay, the bipartite consensus seeking is investigated for the first-order MASs with fixed time-delays and time-varying delays over an undirected signed graph, respectively. To solve the concerned problems, the corresponding algorithms oriented for fixed time-delays or time-varying counterpart are proposed. In other words, the consensus is achieved within each subgroup during evolution, in which the state of each subgroup will converge to the same modulus but different symbol. Sufficient condition of bipartite consensus for the interaction network of MASs suffered from fixed time-delays can be obtained by the use of the generalized Nyquist criterion; Furthermore, based on the tools of integral inequality and linear matrix inequality, the sufficient condition for the case of bipartite consensus for the interaction network of MASs with time-varying delays also can be derived through constructing a reasonable Lyapunov function with triple integral term. Numerical simulations are provided that demonstrate the effectiveness of our theoretical results.
Key words: multi-agent systems    undirected graph    bipartite consensus    fixed-time delays    time-varying delays    Nyquist criterion    linear matrix inequality    free matrix

1 预备知识 1.1 图论与数学基础

 ${a_{ij}} = \left\{ {\begin{array}{*{20}{l}} { {\text{非}} 0,}\quad{\left( {{v_j},{v_i}} \right) \in E}\\ {0,}\quad{\text{其他}} \end{array}} \right.$

 ${l_{ij}} = \left\{ {\begin{array}{*{20}{l}} {\displaystyle\sum \limits_{j \in {N_i}} \left| {{a_{ij}}} \right|{\rm{,}}}\quad{j = i}\\ { - {a_{ij}}{\rm{,}}}\quad{j \ne i} \end{array}} \right.$

1)若 $\forall {v}_{i},{v}_{j}\in {{ V}}_{l}\left(l\in \left\{\mathrm{1,2}\right\}\right)$ ，则所有权重 ${a}_{ij} \geqslant 0$

2)若 $\forall {v}_{i}\in {{ V}}_{l},{v}_{j}\in {{ V}}_{q},l\ne q\left(l,q\in \left\{\mathrm{1,2}\right\}\right)$ ，则所有权重 ${a}_{ij}\leqslant 0$

1.2 问题描述

 ${\dot{x}}_{i}\left(t\right)={u}_{i}\left(t\right),i=\mathrm{1,2},\cdots ,n$ (1)

 $\begin{array}{l} {u_i}\left( t \right) = - \displaystyle\sum\limits_{j \in {N_i}} {\left| {{a_{ij}}} \right|} ({x_i}\left( {t - {\tau _0}} \right) - {\rm{sgn}}\left( {{a_{ij}}} \right){x_j}(t - {\tau _0})) \end{array}$ (2)

 $\begin{array}{l} {u}_{i}\left(t\right)=-\displaystyle\sum \limits_{j\in {N}_{i}}\left|{a}_{ij}\right|({x}_{i}\left(t-\tau \left(t\right)\right)- \mathrm{sgn}\left({a}_{ij}\right){x}_{j}(t-\tau (t\left)\right) \end{array}$ (3)

 $\dot{{ x}}\left(t\right)=-{ L}{ x}\left(t-{\tau }_{0}\right)$ (4)

 $\dot{{ x}}\left(t\right)=-{ L}{ x}(t-\tau (t\left)\right)$ (5)

2 二分一致性分析 2.1 含固定时延的二分一致性

 $\begin{array}{c} \dot{{ Y}}\left(t\right)={ D}\dot{{ x}}\left(t\right)={ D}\left(-{ L}{ x}\left(t-{\tau }_{0}\right)\right)= \\ { D}\left(-{ L}{{ D}}^{-1}{ Y}\left(t-{\tau }_{0}\right)\right)= -{{ L}}_{{ D}}{ Y}\left(t-{\tau }_{0}\right) \end{array}$ (6)

 $\begin{array}{l} {{ L}}_{{ D}}={ D}{ L}{ D}={ C}-{ D}{ A}{ D}= \\ \left\{ {\begin{array}{*{20}{l}} { - |{a_{ij}}|,}\quad{{v_j}\in {N_i}}\\ {\displaystyle \sum \limits_{{v_j}\in {N_i}} \left| {{a_{ij}}} \right|,}\quad{j = i}\\ {0,}\quad{\text{其他}} \end{array}} \right. \end{array}$

 $\mathrm{det}\left(s{ I}+{{ L}}_{{ D}}\left(s\right)\right)=0$

 $\begin{array}{l} {{ L}}_{{ D}}\left(s\right)={\mathrm{e}}^{-{\tau }_{0}s}\left({ C}-{ D}{ A}{ D}\right)= \\ \left\{ {\begin{array}{*{20}{l}} { - |{a_{ij}}|{{\rm{e}}^{ - {\tau _0}s}},}\quad{{v_j} \in {N_i}}\\ {\displaystyle \sum \limits_{{v_j}\in {N_i}} \left| {{a_{ij}}} \right|{{\rm{e}}^{ - {\tau _0}s}},}\quad{j = i}\\ {0,}\quad{\text{其他}} \end{array}} \right. \end{array}$

 $F\left(s\right)=\mathrm{det}\left({s}{ I}+{{ L}}_{{ D}}\left(s\right)\right)$ (7)

1)当 $s=0$ 时， $F\left(0\right)=\mathrm{d}\mathrm{e}\mathrm{t}(0{ I}+{{ L}}_{{ D}}\left(0\right))$ ，根据引理1可得 ${{ L}}_{{ D}}$ 存在单一特征根0，因此当 $s=\;0$ 时， $F\left(s\right)$ 只有一个零点。

2)当 $s\ne 0$ 时，令

 ${F}^{*}\left(s\right)=\mathrm{d}\mathrm{e}\mathrm{t}({ I}+{ G}(s\left)\right)$ (8)

 ${ G}\left(s\right)={\mathrm{e}}^{-{\tau }_{0}s}\frac{{ C}-{ D}{ A}{ D}}{s}$ (9)

 ${ G}\left(\mathrm{j}\omega \right)={\mathrm{e}}^{-{\tau }_{0}\mathrm{j}\omega }\frac{{ C}-{ D}{ A}{ D}}{s}$ (10)

 $\lambda \left({{G}}\left(\mathrm{j}\omega \right)\right)\in {\cup }_{i\in { N}}{G}_{i}$ (11)
 $\begin{array}{c} {G}_{i}=\left\{\zeta \epsilon C,|\zeta -\displaystyle\sum \limits_{{v}_{j}\epsilon {{ N}}_{i}}\left|{a}_{ij}\right|\dfrac{{\mathrm{e}}^{-{\tau }_{0}\mathrm{j}\omega }}{\mathrm{j}\omega } \leqslant \left|\displaystyle\sum \limits_{{v}_{j}\epsilon {{ N}}_{i}}\left|{a}_{ij}\right|\dfrac{{\mathrm{e}}^{-{\tau }_{0}\mathrm{j}\omega }}{\mathrm{j}\omega }\right|\right\} \end{array}$ (12)

 ${G}_{i0}\left(\mathrm{j}\omega \right)={\varepsilon }_{i}\frac{{\mathrm{e}}^{-{\tau }_{0}\mathrm{j}\omega }}{\mathrm{j}\omega }$

 ${W}_{i}\left(\mathrm{j}\omega \right)=2{\varepsilon }_{i}\frac{{\mathrm{e}}^{-{\tau }_{0}\mathrm{j}\omega }}{\mathrm{j}\omega }$

 $\begin{array}{c} \gamma {\rm{C}}\mathrm{o}({0\cup \{E}_{i}\left(\mathrm{j}\omega \right),i\in { N}\})\ni \\ {\gamma }_{i}{\rm{C}}\mathrm{o}({0\cup \{E}_{i}\left(\mathrm{j}\omega \right),i\in { N}\})={\rm{C}}\mathrm{o}({W}_{i}\left(\mathrm{j}\omega \right),i\in { N}\left\}\right) \end{array}$

2.2 含时变时延的二分一致性

 $\dot{{ Y}}\left(t\right)=-{{ L}}_{{ D}}{ Y}\left(t-\tau \left(t\right)\right)$ (13)

 ${{\psi = }}\left[ {\begin{array}{*{20}{c}} {{{{\Lambda }}_{11}}}&{{{{\Lambda }}_{12}}}&{{{{\Lambda }}_{13}}}&{{{{P}}_{12}}}&{ - \dfrac{1}{h}{{W}}_{11}^{\rm{T}} + \dfrac{1}{h}{{Z}}}\\ * &{{{{\Lambda }}_{22}}}&{ - {{L}}_{{D}}^{\rm{T}}}&0&{ - {{L}}_{{D}}^{\rm{T}}{{{P}}_{13}}}\\ * & * &{{{{\Lambda }}_{33}}}&{{{{P}}_{22}}}&{ - {{{P}}_{33}} + \dfrac{1}{h}{{W}}_{12}^{\rm{T}}}\\ * & * & * &{ - {{{Q}}_{22}}}&{{{{P}}_{23}}}\\ * & * & * & * &{ - \dfrac{1}{h}{{{W}}_{11}} - \dfrac{1}{{{h^2}}}{{Z}}} \end{array}} \right] < 0$ (14)
 $\begin{array}{c} { P}=\left[\begin{array}{c}{{ P}}_{11}\; {{ P}}_{12}\; {{ P}}_{13}\\ {{{ P}}^{\mathrm{T}}_{12}}\; {{ P}}_{22} \;{{ P}}_{23}\\ {{{ P}}^{\mathrm{T}}_{13}}\; {{{ P}}^{\mathrm{T}}_{23}}\; {{ P}}_{33}\end{array}\right]>0;{ Q}=\left[\begin{array}{c}{{ Q}}_{11} \;{{ Q}}_{12}\\ {{{ Q}}^{\mathrm{T}}_{12}}\; {{ Q}}_{22}\end{array}\right]>0 \; \\ { W}=\left[\begin{array}{c}{{ W}}_{11} \;{{ W}}_{12}\\ {{{ W}}^{\mathrm{T}}_{12}} \;{{ W}}_{22}\end{array}\right]>0 ;\; { Z}>0 \end{array}$ (15)

 $\begin{array}{c} {{ \varLambda }}_{11}={{ P}}_{13}+{{{ P}}_{13}}+{{ Q}}^{\mathrm{T}}_{11}+h{{ W}}_{11} \\ {{ \varLambda }}_{12}=-{{ P}}_{11}{{ L}}_{{ D}}-{{ Q}}_{12}{{ L}}_{{ D}}-h{{ W}}_{11}{{ L}}_{{ D}} \\ {{ \varLambda }}_{13}=-{{ P}}_{13}+{{{ P}}^{\mathrm{T}}_{23}}+\dfrac{1}{h}{{ W}}_{22} \\ {{ \varLambda }}_{22}={{ Q}}_{22}-{h}{{{ L}}^{\mathrm{T}}_{{ D}}}{{ W}}_{22}{{ L}}_{{ D}}+\dfrac{{{h}}^{2}}{2}{{{ L}}^{\mathrm{T}}_{{ D}}}{ Z}{{ L}}_{{ D}} \\ {{ \varLambda }}_{33}=-{{ P}}_{23}-{{{ P}}^{\mathrm{T}}_{23}}-{{ Q}}_{11}-\dfrac{1}{h}{{ W}}_{22} \end{array}$

 ${ V}\left(t\right)={{ V}}_{1}\left(t\right)+{{ V}}_{2}\left(t\right)+{{ V}}_{3}\left(t\right)+{{ V}}_{4}\left(t\right)$ (16)

 $\begin{array}{c} {{ V}}_{1}\left(t\right)={{ \varsigma }\left(t\right)}^{\mathrm{T}}{ P}{ \varsigma }\left(t\right) \\ {{ V}}_{2}\left(t\right)=\displaystyle\int_{t-\tau \left(t\right)}^{t}{{ \rho }\left(s\right)}^{\mathrm{T}}{ Q}{ \rho }\left(s\right)\mathrm{d}s \\ {{ V}}_{3}\left(t\right)=\displaystyle\int_{-h}^{0}\int_{t+\theta }^{t}{{ \rho }\left(s\right)}^{\mathrm{T}}{ W}{ \rho }\left(s\right)\mathrm{d}s\mathrm{d}\theta \\ {{ V}}_{4}\left(t\right)=\displaystyle\int_{-h}^{0}\int_{\theta }^{0}\int_{t+\lambda }^{t}{\dot{{ Y}}\left(s\right)}^{\mathrm{T}}{ R}\dot{{ Y}}\left(s\right)\mathrm{d}s\mathrm{d}\lambda \mathrm{d}\theta \\ { \varsigma }\left(t\right)={\left[{{ Y}\left(t\right)}^{\mathrm{T}}\quad{{ Y}\left(t-h\right)}^{\mathrm{T}}\quad\displaystyle\int_{t-h}^{t}{{ Y}\left(s\right)}^{\mathrm{T}}{\rm{d}}s\right]}^{\mathrm{T}} \\ { \rho }\left(s\right)={\left[{{ Y}\left(s\right)}^{\mathrm{T}}\quad{\dot{{ Y}}\left(s\right)}^{\mathrm{T}}\right]}^{\mathrm{T}} \end{array}$

 ${ Y}\left(t-h\right)={ Y}\left(t\right)-\int_{t - h}^t \dot{{ Y}}\left(s\right){\rm{d}}s$ (21)
 $\begin{array}{c} \displaystyle\int_{-h}^{0}\displaystyle\int_{t+\theta }^{t}\dot{{ Y}}\left(\mathrm{s}\right)\mathrm{d}s\mathrm{d}\theta =\displaystyle\int_{-h}^{0}\left[{ Y}\left(t\right)-{ Y}\left(t-\theta \right)\right]\mathrm{d}\theta = \\ h{ Y}\left(t\right)-\displaystyle\int_{t-h}^{t}{ Y}\left(s\right){\rm{d}}s \end{array}$ (22)

 $\begin{array}{c} -\displaystyle\int_{t-h}^{t}{{ \rho }\left(s\right)}^{\mathrm{T}}{ W}{ \rho }\left(s\right)\mathrm{d}s\leqslant \\ -\dfrac{1}{h}\displaystyle\int_{t-h}^{t}{{ \rho }\left(s\right)}^{\mathrm{T}}\mathrm{d}s{ W}\displaystyle\int_{t-h}^{t}{ \rho }\left(s\right)\mathrm{d}s \end{array}$ (23)
 $\begin{array}{c} \displaystyle\int_{-h}^{0}\displaystyle\int_{t+\theta }^{t}{\dot{{ Y}}\left(s\right)}^{\mathrm{T}}{ R}\dot{{ Y}}\left(s\right)\mathrm{d}s\mathrm{d}\theta \leqslant \\ -\dfrac{2}{{h}^{2}}\displaystyle\int_{-h}^{0}\displaystyle\int_{t+\theta }^{t}{\dot{{ Y}}\left(s\right)}^{\mathrm{T}}\mathrm{d}s\mathrm{d}\theta { R}\displaystyle\int_{-h}^{0}\displaystyle\int_{t+\theta }^{t}\dot{{ Y}}\left(s\right)\mathrm{d}s\mathrm{d}\theta \end{array}$ (24)

 $\begin{array}{c} {\dot{{ V}}}_{1}\left(t\right)={\dot{{ \varsigma }}\left(t\right)}^{\mathrm{T}}{ P}{ \varsigma }\left(t\right)+{{ \varsigma }\left(t\right)}^{\mathrm{T}}{ P}\dot{{ \varsigma }}\left(t\right)= \\ \left[{{ Y}\left(t\right)}^{\mathrm{T}}\;\;\;{{ Y}\left(t-h\right)\;}^{\mathrm{T}}\;\;\;\displaystyle\int_{t-h}^{t}{{ Y}\left(s\right)}^{\mathrm{T}}{\rm{d}}s\right].\\ \left[\begin{array}{ccc}{{ P}}_{11}& {{ P}}_{12}& {{ P}}_{13}\\ *& {{ P}}_{22}& {{ P}}_{23}\\ *& *& {{ P}}_{33}\end{array}\right]\left[\begin{array}{c}{ Y}\left(t\right)\\ { Y}\left(t-h\right)\\ \displaystyle\int_{t-h}^{t}{ Y}\left(s\right){\rm{d}}s\end{array}\right] \end{array}$ (25)
 $\begin{array}{c} {\dot{{ V}}}_{2}\left(t\right)={{ \rho }\left(t\right)}^{\mathrm{T}}{ Q}{ \rho }\left(t\right)-{{ \rho }\left(t-h\right)}^{\mathrm{T}}{ Q}{ \rho }\left(t-h\right)= \\ \left[{{ Y}\left(t\right)}^{\mathrm{T}}\quad{\dot{{ Y}}\left(t\right)}^{\mathrm{T}}\right]\left[\begin{array}{cc}{{ Q}}_{11}& {{ Q}}_{12}\\ *& {{ Q}}_{22}\end{array}\right]\left[\begin{array}{c}{ Y}\left(t\right)\\ \dot{{ Y}}\left(t\right)\end{array}\right]- \\ {\left[\begin{array}{c}{ Y}\left(t-h\right)\\ \dot{{ Y}}\left(t-h\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{{ Q}}_{11}& {{ Q}}_{12}\\ *& {{ Q}}_{22}\end{array}\right]\left[\begin{array}{c}{ Y}\left(t-h\right)\\ \dot{{ Y}}\left(t-h\right)\end{array}\right] \end{array}$ (26)
 $\begin{array}{c} {\dot{{ V}}}_{3}\left(t\right)=h{{ \rho }\left(t\right)}^{\mathrm{T}}{ W}{ \rho }\left(t\right)-\displaystyle\int_{t-h}^{t}{{ \rho }\left(s\right)}^{\mathrm{T}}{ W}{ \rho }\left(s\right)\mathrm{d}s \leqslant \\ h\left[{{ Y}\left(t\right)}^{\mathrm{T}}{\dot{{ Y}}\left(t\right)}^{\mathrm{T}}\right]\left[\begin{array}{cc}{{ W}}_{11}& {{ W}}_{12}\\ *& {{ W}}_{22}\end{array}\right]\left[\begin{array}{c}{ Y}\left(t\right)\\ \dot{{ Y}}\left(t\right)\end{array}\right]- \\ \dfrac{1}{h}\displaystyle\int_{t-h}^{t}{\left[\begin{array}{c}{ Y}\left(s\right)\\ \dot{{ Y}}\left(s\right)\end{array}\right]}^{\mathrm{T}}\mathrm{d}s\left[\begin{array}{cc}{{ W}}_{11}& {{ W}}_{12}\\ *& {{ W}}_{22}\end{array}\right]\displaystyle\int_{t-h}^{t}\left[\begin{array}{c}{ Y}\left(s\right)\\ \dot{{ Y}}\left(s\right)\end{array}\right]\mathrm{d}s \end{array}$ (27)
 $\begin{array}{c} {\dot{{ V}}}_{4}\left(t\right)=\dfrac{{h}^{2}}{2}{\dot{{ Y}}\left(t\right)}^{\mathrm{T}}{ Z}\dot{{ Y}}\left(t\right)-\displaystyle\int_{-h}^{0}\displaystyle\int_{t+\theta }^{t}{\dot{{ Y}}\left(s\right)}^{\mathrm{T}}{ Z}\dot{{ Y}}\left(s\right)\mathrm{d}s\mathrm{d}\theta - \\ \displaystyle\int_{-h}^{0}\displaystyle\int_{t+\theta }^{t}{\dot{{ Y}}\left(s\right)}^{\mathrm{T}}{ Z}\dot{{ Y}}\left(s\right)\mathrm{d}s\mathrm{d}\theta \leqslant \\ \dfrac{{h}^{2}}{2}{{ Y}\left(t-\tau \left(t\right)\right)}^{\mathrm{T}}{{{ L}}_{{ D}}}^{\mathrm{T}}{ Z}{{ L}}_{{ D}}{ Y}\left(t-\tau \left(t\right)\right)- \\ \dfrac{2}{{h}^{2}}\left\{\left[h{{ Y}\left(t\right)}^{\mathrm{T}}-\displaystyle\int_{t-h}^{t}{{ Y}\left(s\right)}^{\mathrm{T}}\mathrm{d}s\right]{ Z}\left[h{ Y}\left(t\right)-\displaystyle\int_{t-h}^{t}{ Y}\left(s\right)\mathrm{d}s\right]\right\} \end{array}$ (28)

 $\dot{{ V}}\left(t\right)\leqslant {{ \xi }\left(t\right)}^{\mathrm{T}}{ \psi }{ \xi }\left(t\right)$

 ${ \xi }\left(t\right)=[{{ Y}\left(t\right)}^{\mathrm{T}}\quad{{ Y}\left(t-\tau \left(t\right)\right)}^{\mathrm{T}}\quad{{ Y}\left(t-h\right)}^{\rm{T}}{\begin{array}{c}\\ \end{array}}$
 ${\dot{{ Y}}\left(t-h\right)}^{\mathrm{T}}\quad\int_{t-h}^{t}{{ Y}\left(s\right)}^{\mathrm{T}}\mathrm{d}s{]}^{\mathrm{T}}$

3 数值仿真分析

 ${{L}} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 2\\ { - 1}\\ { - 1} \end{array}}&{\begin{array}{*{20}{c}} { - 1}\\ 2\\ { - 1} \end{array}}&{\begin{array}{*{20}{c}} { - 1}\\ { - 1}\\ 3 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0\\ 1 \end{array}}\\ 0&0&1&3\\ 0&0&0&{ - 1}\\ 0&0&0&{ - 1} \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}\\ { - 1}\\ 2\\ { - 1} \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}}\\ { - 1}\\ { - 1}\\ 2 \end{array}} \right]$
3.1 含固定时延的系统

 Download: 图 2 固定时延条件下智能体状态 Fig. 2 States of multi-agents with fixed-time delays

3.2 含时变时延的系统

 Download: 图 3 时变时延条件下智能体状态 Fig. 3 States of multi-agents under time-varying delays

4 结束语

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