«上一篇
 文章快速检索 高级检索

 智能系统学报  2019, Vol. 14 Issue (2): 355-361  DOI: 10.11992/tis.201710002 0

引用本文

CHEN Shiming, QIU Yun, HUANG Yue, et al. Research on group consensus of heterogeneous multi-agent systems via pinning control[J]. CAAI Transactions on Intelligent Systems, 2019, 14(2): 355-361. DOI: 10.11992/tis.201710002.

文章历史

Research on group consensus of heterogeneous multi-agent systems via pinning control
CHEN Shiming , QIU Yun , HUANG Yue , JIANG Jihai
School of Electrical and Automation Engineering, East China JiaoTong University, Nanchang 330013, China
Abstract: In order to research the group consensus problem of heterogeneous multi-agent systems composed of the linear first- and second-order integrator agents together with the nonlinear Euler-Lagrange(EL) agents, and the states of agents in the same subgroup can converge to the desired state, for undirected fixed topology, a distributed control protocol via pinning scheme is proposed; Based on graph theory, Lyapunov theory and Barbalat’s Lemma, the feasibility of the controllers is proved. Finally, Simulation results show that the group consensus of the heterogeneous multi-agent systems with the nonlinear EL structure can be realized under the control protocol proposed, Compared with group consensus algorithm without pinning control, each subgroups can be tend to a desired state.
Key words: Euler-Lagrange agent    heterogeneous multi-agent systems    group consensus    desired state    pinning control    fixed topology    Lyapunov theory    Barbalat’s Lemma

1 预备知识和问题描述 1.1 代数图论

 ${l_{ij}} = \left\{ {\begin{array}{*{20}{l}} - {a_{ij}} \;\; , \; \;i \ne j \\ \displaystyle \sum\limits_{j = 1,j \ne i}^n {{a_{ij}}}\; \; , \;\; i = j \\ \end{array}} \right.$

1.2 问题描述

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

 $\left\{ \begin{array}{c} {{{{\dot x}}}_{{i}}}\left( {{t}} \right) = {{{v}}_{{i}}}\left( {{t}} \right) \\ \mathop {{{{{\dot v}}}_{{i}}}\left( {{t}} \right) = {{{u}}_{{i}}}\left( {{t}} \right)}\limits^. \\ \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} {{{{\dot x}}}_{{i}}}\left( {{t}} \right) = {{{v}}_{{i}}}\left( {{t}} \right)\\ {{{M}}_{{i}}}\left( {{{{x}}_{{i}}}} \right){{{{\dot v}}}_{{i}}} + {{{C}}_{{i}}}\left( {{{{x}}_{{i}}}{{,}}{{{v}}_{{i}}}} \right){{{v}}_{{i}}} = {{{u}}_{{i}}} \\ \end{array} \right.$ (3)

 $0 < {\lambda _m}\left\{ {{{{M}}_{{i}}}\left( {{{{x}}_{{i}}}} \right)} \right\}{{I}} \leqslant {{{{ M}}}_{{\iota }}}\left( {{{{\xi }}_{{\iota }}}} \right) \leqslant {\lambda _M}\left\{ {{{{M}}_{{i}}}\left( {{{{x}}_{{i}}}} \right)} \right\}{{I}} < \infty$ (4)

 ${{{r}}^{\rm{T}}}\left( {\mathop {{{{M}}_{{i}}}}\limits^{{.}} \left( {{{{x}}_{{i}}}} \right) - {{2}}{{{C}}_{{i}}}\left( {{{{x}}_{{i}}}{{,}}{{{v}}_{{i}}}} \right)} \right){{r}} = 0$ (5)

 $\left\{ \begin{array}{c} \mathop {\lim }\limits_{t \to \infty } \left\| {{{{x}}_{{i}}}\left( {{t}} \right) - {{{x}}_{{j}}}\left( {{t}} \right)} \right\| = 0,{\rm if} {\sigma _i} \in {\sigma _j}, \\ \forall i,j \in \left\{ {1,2,\cdots ,n} \right\},\forall {\sigma _i},{\sigma _j} \in \left\{ {1,2,\cdots ,k} \right\} \\ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{x}}_{{i}}}\left( {{t}} \right) - {{{x}}_{{j}}}\left( {{t}} \right)} \right\| \ne 0,{\rm if}{\sigma _i} \ne {\sigma _j} , \\ \forall i,j \in \left\{ {1,2,\cdots ,n} \right\},\forall {\sigma _i},{\sigma _j} \in \left\{ {1,2,\cdots ,k} \right\} \\ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{v}}_{{i}}}\left( {{t}} \right)} \right\| = 0,\forall i \in \left\{ {l + 1,l + 2,\cdots ,n} \right\} \\ \end{array} \right.$ (6)

1)当 $\left\| x \right\| \to \infty$ 时， $V\left( x \right) \to \infty$

2)对于任意的 $x \in {{\rm R}^n},\mathop V\limits^. \left( x \right) \leqslant 0$

2 异质多智能体系统的群一致性控制

 ${{{u}}_{{i}}} = \left\{ \begin{array}{c} \displaystyle \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right) + \sum\limits_{j = 1}^n {{l_{ij}}{{{x}}_{{{{\sigma }}_{{j}}}}}} - {b_i}\left( {{{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}} \right)},\\ \displaystyle i = 1,2,\cdots ,l; \\ \displaystyle\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right) + \sum\limits_{j = 1}^n {{l_{ij}}{{{x}}_{{{{\sigma }}_{{j}}}}} - {{\kappa }}} } {{{v}}_{{i}}} - {b_i}\left( {{{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}} \right) ,\\ \displaystyle i = l + 1,2,\cdots ,m ; \\ \displaystyle\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right) + \sum\limits_{j = 1}^n {{l_{ij}}{{{x}}_{{{{\sigma }}_{{j}}}}} - {{\lambda }}} } {{{v}}_{{i}}} - {b_i}\left( {{{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}} \right),\\ \displaystyle i = m + 1,2,\cdots ,n \\ \end{array} \right.$ (7)

 $\left\{ \begin{array}{c} \displaystyle{{{{\dot x}}}_{{i}}}\left( {{t}} \right) = \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right) + \sum\limits_{j = 1}^n {{l_{ij}}{{{x}}_{{{{\sigma }}_{{j}}}}}} } {\rm{ - }}{b_i}\left( {{{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}} \right),\qquad\quad \\ \displaystyle i = 1,2,\cdots ,l \\ {{{{\dot x}}}_{{i}}}\left( {{t}} \right) = {{{v}}_{{i}}}\left( {{t}} \right) \qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\qquad\,\\ \displaystyle{{{{\dot v}}}_{{i}}}\left( {{t}} \right) = \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right) + \sum\limits_{j = 1}^n {{l_{ij}}{{{x}}_{{{{\sigma }}_{{j}}}}} - } } {{\kappa }}{{{v}}_{{i}}} {\rm{ - }}{b_i}\left( {{{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}} \right),\; \;\;\\ \displaystyle i = l + 1,l + 2,\cdots ,m \\ \displaystyle{{{{\dot x}}}_{{i}}}\left( {{t}} \right) = {{{v}}_{{i}}}\left( {{t}} \right) \qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\quad\;\\ \displaystyle {{{M}}_{{i}}}\left( {{{{x}}_{{i}}}} \right){{{{\dot v}}}_{{i}}} + {{{C}}_{{i}}}\left( {{{{x}}_{{i}}}{{,}}{{{v}}_{{i}}}} \right){{{v}}_{{i}}} = \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right) + \sum\limits_{j = 1}^n {{l_{ij}}{{{x}}_{{{{\sigma }}_{{j}}}}} - }} \\ \displaystyle{{{\lambda }}} {{{v}}_{{i}}} {\rm{ - }}{b_i}( {{{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}} ),\qquad\qquad\qquad\qquad\quad\qquad\\ \displaystyle i = m + 1,m+2,\cdots ,n \\ \end{array} \right.$ (8)

${{{e}}_{{i}}} = {{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}},i = 1,2,\cdots ,n$ ，则有：

 $\begin{gathered} \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right)} + \sum\limits_{j = 1}^n {{l_{ij}} ={{{x}}_{{{{\sigma }}_{{j}}}}}} \\ \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{x}}_{{j}}} - {{{x}}_{{i}}}} \right) + \sum\limits_{j = 1,j \ne i}^n {{l_{ij}}{{{x}}_{{{{\sigma }}_{{j}}}}}} + {l_{ii}}{{{x}}_{{{{\sigma }}_{{i}}}}}} =\\ \sum\limits_{j = 1}^n {{a_{ij}}\left( {( {{{{x}}_{{j}}} - {{{x}}_{{{{\sigma }}_{{j}}}}}} ) - \left( {{{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}} \right)} \right)} = \\ \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} \\ \end{gathered}$ (9)

 $\left\{ \begin{gathered} {{{{\dot e}}}_{{i}}} = \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} {\rm{ - }}{b_i}{{{e}}_{{i}}},i = 1,2,\cdots ,l \qquad\qquad\\ {{{{\dot e}}}_{{i}}} = {{{v}}_{{i}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\ \;{{{{\dot v}}}_{{i}}} = \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} - {{\kappa }}{{{v}}_{{i}}}{\rm{ - }}{b_i}{{{e}}_{{i}}},i = l + 1,l + 2,\cdots m \\ {{{{\dot e}}}_{{i}}} = {{{v}}_{{i}}} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ {{{M}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}} \right){{{{\dot v}}}_{{i}}} + {{{C}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}{{,}}{{{v}}_{{i}}}} \right){{{v}}_{{i}}}= \qquad\qquad\qquad\\ \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} - {{\lambda }}{{{v}}_{{i}}}{\rm{ - }}{b_i}{{{e}}_{{i}}},i = m + 1,m + 2,\cdots ,n \\ \end{gathered} \right.$ (10)

 $\begin{gathered} V = \sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{{M}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}} \right){{{v}}_{{i}}} } + \sum\limits_{i = l + 1}^m {{{v}}_{{i}}^{\rm{T}}{{{v}}_{{i}}}} +\\ \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}{{\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)}^2}} } + \sum\limits_{i = 1}^n {{b_i}{{e}}_{{i}}^{\rm{T}}{{{e}}_{{i}}}} \\ \\ \end{gathered}$ (11)

 $\begin{gathered} \dot V = \sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{{{\dot M}}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}} \right){{{v}}_{{i}}}} + 2\sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{{M}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}} \right){{{{\dot v}}}_{{i}}}} + \\ \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)\left( {{{{{\dot e}}}_{{i}}} - {{{{\dot e}}}_{{j}}}} \right)} } + 2\sum\limits_{i = l + 1}^m {{{v}}_{{i}}^{\rm{T}}{{{{\dot v}}}_{{i}}}} {\rm{ + }}2\sum\limits_{i = 1}^n {{b_i}{{e}}_{{i}}^{\rm{T}}{{{{\dot e}}}_{{i}}}} \\ \end{gathered}$ (12)

 $\begin{gathered} \sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{{{\dot M}}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}} \right){{{v}}_{{i}}}} + 2\sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{{M}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}} \right){{{{\dot v}}}_{{i}}}} =\\ 2\sum\limits_{i = m + 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}{{v}}_{{i}}^{\rm{T}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} } - 2\sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{\lambda }}{{{v}}_{{i}}}} - 2\sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{b_i}{{{e}}_{{i}}}} \\ \end{gathered}$ (13)

 $\begin{gathered} \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)\left( {{{{{\dot e}}}_{{j}}} - {{{{\dot e}}}_{{i}}}} \right)} } = \\ \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{{\dot e}}}_{{j}}} - } } \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{{\dot e}}}_{{i}}}} } = \\ \sum\limits_{i = 1}^{\rm{n}} {\sum\limits_{j = 1}^l {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{{\dot e}}}_{{j}}}} } + \sum\limits_{i = 1}^n {\sum\limits_{j = l + 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{v}}_{{j}}}} } - \\ \sum\limits_{i = 1}^l {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{{\dot e}}}_{{i}}}} } - \sum\limits_{i = l + 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{v}}_{{i}}}} } \\ \end{gathered}$ (14)

 $\begin{gathered} \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^l {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{{\dot e}}}_{{j}}}} } = - \sum\limits_{i = 1}^l {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{{\dot e}}}_{{i}}}} } \\ \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^l {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{v}}_{{i}}}} = - \sum\limits_{i = 1}^l {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{v}}_{{j}}}} } } \\ \end{gathered}$ (15)

 $\begin{array}{ccccc} \displaystyle\sum\limits_{i = 1}^n {\displaystyle\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{\bf{i}}}} \right)\left( {{{{{\dot e}}}_{{j}}} - {{{\bf{\dot e}}}_{{i}}}} \right)} } = \\ - 2\displaystyle\sum\limits_{i = 1}^l {\displaystyle\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{\bf{i}}}} \right){{{{\dot e}}}_{\bf{i}}}} } {\rm{ }} - 2 \displaystyle\sum\limits_{i = l + 1}^n {\displaystyle\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right){{{v}}_{{i}}}} } \end{array}$ (16)

 $\begin{gathered} \mathop V\limits^. \left( t \right) = {\rm{ - }}2\sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{\lambda }}{{{v}}_{{i}}}} {\rm{ - }}2\sum\limits_{i = 1}^l {\sum\limits_{j = 1}^n {{{{a}}_{{{ij}}}}{{\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)}^{\rm{T}}}{{{{\dot e}}}_{{i}}}} } + \\ 2\sum\limits_{i = 1}^l {{{{b}}_{{i}}}{{e}}_{{i}}^{\rm{T}}{{{{\dot e}}}_{{i}}}} - 2\sum\limits_{i = l + 1}^m {{{v}}_{{i}}^{\rm{T}}{{\kappa }}{{{v}}_{{i}}}} {\rm{ = }}\\ -2\sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{\lambda }}{{{v}}_{{i}}}} - 2{\sum\limits_{i = 1}^l {\left( {{{{{\dot e}}}_{{i}}}} \right)} ^2} - 2\sum\limits_{i = l + 1}^m {{{v}}_{{i}}^{\rm{T}}{{\kappa }}{{{v}}_{{i}}}} \leqslant 0 \\ \end{gathered}$ (17)

 $\left\{ \begin{gathered} {{{{\dot e}}}_{{i}}} = 0,\quad i = 1,2,\cdots ,l \qquad\;\\ \!\! \! {{{v}}_{{i}}} = 0,\quad i = l + 1,l + 2,\cdots ,n \\ \end{gathered} \right.$ (18)

 $\left\{ \begin{gathered} \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} {\rm{ - }}{b_i}{{{e}}_{{i}}} = 0,\quad i = 1,2,\cdots ,l \\ {{{v}}_{{i}}} = 0,\quad i = l + 1,l + 2,\cdots ,n \qquad\qquad\\ \end{gathered} \right.$ (19)

 $\sum\limits_{i = 1}^n {{{{e}}_{{i}}}\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} {\rm{ - }}} {b_i}{{{e}}_{{i}}} = 0$ (20)
 $\sum\limits_{j = 1}^n {{{{e}}_{{j}}}\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} {\rm{ - }}} {b_i}{{{e}}_{{i}}} = 0$ (21)

 $- \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}{{\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)}^2}} - 2\sum\limits_{i = 1}^n {{d_i}{{e}}_{{i}}^{\rm{2}}} } = 0$ (22)

 $\left\{ {\begin{array}{*{20}{l}} {{{e}}_{{i}}}\left( {{t}} \right) = 0, &i = 1,2,\cdots ,n \\ {{{v}}_{{i}}}\left( {{t}} \right) = 0, &i = l + 1,l + 2,\cdots ,n \\ \end{array}} \right.$ (23)

${{{e}}_{{i}}} = {{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}$ 代入到式(23)，可以知道对于 ${{x}}\left( {{0}} \right)$ ${{v}}\left( {{0}} \right)$ ，当 $t \to \infty$ 有:

 $\left\{ \begin{array}{c} \mathop {\lim }\limits_{t \to \infty } \left\| {{{{x}}_{{i}}}\left( {{t}} \right) - {{{x}}_{{j}}}\left( {{t}} \right)} \right\| = 0,\;\; \;{\sigma _i} = {\sigma _j} \qquad\quad\quad\\ \forall i,j \in \left\{ {1,2,\cdots ,n} \right\},\forall {\sigma _i},{\sigma _j} \in \left\{ {1,2,\cdots ,k} \right\} \\ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{x}}_{{i}}}\left( {{t}} \right) - {{{x}}_{{j}}}\left( {{t}} \right)} \right\| \ne 0,\;\; \;{\sigma _i} \ne {\sigma _j} \qquad\quad\quad \\ \forall i,j \in \left\{ {1,2,\cdots ,n} \right\},\forall {\sigma _i},{\sigma _j} \in \left\{ {1,2,\cdots ,k} \right\} \\ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{x}}_{{i}}}\left( {{t}} \right) - {{{x}}_{{{{\sigma }}_{{i}}}}}} \right\| = 0,\;\;\; {\sigma _i} = {\sigma _j} \qquad\quad \quad\;\\ \forall i,j \in \left\{ {1,2,\cdots ,n} \right\},\forall {\sigma _i},{\sigma _j} \in \left\{ {1,2,\cdots ,k} \right\} \\ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{v}}_{{i}}}\left( {{t}} \right)} \right\| = 0,\forall i \in \left\{ {l + 1,l + 2,\cdots ,n} \right\}\quad\quad \\ \end{array} \right.$ (24)

 $\begin{gathered} V = \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}{{\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)}^2}} } + \sum\limits_{i = l + 1}^m {{{v}}_{{i}}^{\rm{T}}{{{v}}_{{i}}}}+ \\ \sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{{M}}_{{i}}}\left( {{{{e}}_{{i}}}{{ + }}{{{x}}_{{{{\sigma }}_{{i}}}}}} \right){{{v}}_{{i}}}} \\ \end{gathered}$ (25)

 $\begin{gathered} \dot V{\rm{ = }} - 2\sum\limits_{i = 1}^l {{{\left( {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} } \right)}^{\rm{T}}}\left( {\sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} } \right)}- \\ 2\sum\limits_{i = m + 1}^n {{{v}}_{{i}}^{\rm{T}}{{\lambda }}{{{v}}_{{i}}}} - 2\sum\limits_{i = l + 1}^m {{{v}}_{{i}}^{\rm{T}}{{\kappa }}{{{v}}_{{i}}}} \leqslant 0 \\ \end{gathered}$ (26)

 $\left\{ \begin{gathered} {{{v}}_{{i}}} = 0,\quad i = l + 1,l + 2,\cdots ,n \qquad\\ \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} = 0, \quad i = 1,2,\cdots ,l \\ \end{gathered} \right.$ (27)

 $\sum\limits_{i = 1}^n {{{{e}}_{{i}}}} \sum\limits_{j = 1}^n {{a_{ij}}\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)} = 0$ (28)
 $\sum\limits_{j = 1}^n {{{{e}}_{{j}}}} \sum\limits_{i = 1}^n {{a_{ij}}\left( {{{{e}}_{{i}}} - {{{e}}_{{j}}}} \right)} = 0$ (29)

 $- \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{a_{ij}}{{\left( {{{{e}}_{{j}}} - {{{e}}_{{i}}}} \right)}^2} = 0} }$ (30)

 $\left\{ {\begin{array}{*{20}{l}} {{{e}}_{{i}}} = {{{e}}_{{j}}},&i,j = 1,2,\cdots ,n \\ {{{v}}_{{i}}} = 0,&i = l + 1,l + 2,\cdots ,n \\ \end{array}} \right.$ (31)

${{{e}}_{{i}}} = {{{x}}_{{i}}} - {{{x}}_{{{{\sigma }}_{{i}}}}}$ 代入到式(31)中，可以得到：

 $\left\{ \begin{array}{c} \mathop {\lim }\limits_{t \to \infty } \left\| {{{{x}}_{{i}}}\left( {{t}} \right) - {{{x}}_{{j}}}\left( {{t}} \right)} \right\| = 0,\;\;\;{\sigma _i} = {\sigma _j} \qquad\quad\quad\\ \forall i,j \in \left\{ {1,2,\cdots ,n} \right\},\forall {\sigma _i},{\sigma _j} \in \left\{ {1,2,\cdots ,k} \right\} \\ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{x}}_{{i}}}\left( {{t}} \right) - {{{x}}_{{j}}}\left( {{t}} \right)} \right\| \ne 0,\;\;\;{\sigma _i} \ne {\sigma _j} \qquad\quad\quad \\ \forall i,j \in \left\{ {1,2,\cdots ,n} \right\},\forall {\sigma _i},{\sigma _j} \in \left\{ {1,2,\cdots ,k} \right\} \\ \mathop {\lim }\limits_{t \to \infty } \left\| {{{{v}}_{{i}}}\left( {{t}} \right)} \right\| = 0,\forall i \in \left\{ {l + 1,l + 2,\cdots ,n} \right\} \quad\quad\\ \end{array} \right.$ (32)

3 数值仿真

 Download: 图 1 异质多智能体系统的通信拓扑图 Fig. 1 Conmmunication topology of heterogeneous multi-agent systems

${{{x}}_{{1}}}\left( {{0}} \right) = {\left[ { - 1.5\;\;1.9} \right]^{\rm{T}}},{{{x}}_{{2}}}\left( {{0}} \right) = {\left[ {1.9\;\;1.5} \right]^{\rm{T}}}$ ${{{x}}_{{3}}}\left( {{0}} \right) = {\left[ { - 1.8\;\; - 1.5} \right]^{\rm{T}}},{{{x}}_{{4}}}\left( {{0}} \right) = {\left[ {0.1\;\;0.6} \right]^{\rm{T}}},{{{x}}_{{5}}}\left( {{0}} \right) = {\left[ {1.4\;\;1.9} \right]^{\rm{T}}}$ ${{{x}}_{{6}}} \left( {{0}} \right) ={\left[ { - 0.5\;\; - 1.8} \right]^{\rm{T}}},{{{v}}_{{3}}}\left( {{0}} \right) = {\left[ { - 0.2\;\;0.1} \right]^{\rm{T}}},{{{v}}_{{4}}}\left( {{0}} \right) = {\left[ { - 0.8\;\;1.9} \right]^{\rm{T}}}$ ${{{v}}_{{5}}}\left( {{0}} \right) = {\left[ {0.1\;\; - 0.3} \right]^{\rm{T}}},{{{v}}_{{6}}}\left( {{0}} \right) = {\left[ {0.6\;\;0.8} \right]^{\rm{T}}}$

${{\kappa }} = {\rm{diag}}\left[ {1,1} \right],{{\lambda }} = {\rm diag}\left[ {1,1} \right]$ ${{{M}}_{{i}}} = \left[ {\begin{array}{*{20}{c}} {{m_{11}}}&{{m_{12}}} \\ {{m_{21}}}&{{m_{22}}} \end{array}} \right]$ ${m_{11}} = {a_1} + 2{a_3}\cos {x_{i\left( 2 \right)}} + 2{a_4}\sin {x_{i\left( 2 \right)}}$ ${m_{12}} = {a_2} + {a_3}\cos {x_{i\left( 2 \right)}} + {a_4}\sin {x_{i\left( 2 \right)}}$ ${m_{21}} = {a_2} + {a_3}\cos {x_{i\left( 2 \right)}} + {a_4}\sin {x_{i\left( 2 \right)}}$ ${m_{22}} = {a_2};$ Ci = $\left[ {\begin{array}{*{20}{c}} {{c_{11}}}&{{c_{12}}} \\ {{c_{21}}}&{{c_{22}}} \end{array}} \right]$

${c_{11}} = {a_4}\cos {x_{i\left( 2 \right)}} - {a_3}\sin {x_{i\left( 2 \right)}}$ , ${c_{12}} = \left( {{a_4}\cos {x_{i\left( 2 \right)}} - {a_3}\sin {x_{i\left( 2 \right)}}} \right)\left( {{{\mathop x\limits^. }_{i\left( 1 \right)}} + {{\mathop x\limits^. }_{i\left( 2 \right)}}} \right)$ , ${c_{21}} = {a_3}\sin {x_{i\left( 2 \right)}} - {a_4}\cos {x_{i\left( 2 \right)}},$ ${c_{22}} = 0$

 Download: 图 2 牵制控制作用下的仿真结果 Fig. 2 Simulation restults under the effect of pinning control

 Download: 图 3 未加牵制控制的仿真结果 Fig. 3 Simulation results of without pinning control

4 结束语

 [1] LIU Yuan, MIN Haibo, WANG Shicheng, et al. Distributed consensus of a class of networked heterogeneous multi-agent systems[J]. Journal of the franklin institute, 2014, 351(3): 1700-1716. DOI:10.1016/j.jfranklin.2013.12.020 (0) [2] 王玉振, 杜英雪, 王强. 多智能体时滞和无时滞网络的加权分组一致性分析[J]. 控制与决策, 2015, 30(11): 1993-1998. WANG Yuzhen, DU Yingxue, WANG Qiang. Weighted group-consensus analysis of multi-agent systems with and without time-delay network[J]. Control and decision, 2015, 30(11): 1993-1998. (0) [3] WANG Yuzhen, ZHANG Chenghui, LIU Zhenbin. A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems[J]. Automatica, 2012, 48(7): 1227-1236. DOI:10.1016/j.automatica.2012.03.024 (0) [4] XIA Weiguo, CAO Ming. Clustering in diffusively coupled networks[J]. Automatica, 2011, 47(11): 2395-2405. DOI:10.1016/j.automatica.2011.08.043 (0) [5] YU Junyan, WANG Long. Group consensus of multi-agent systems with directed information exchange[J]. International journal of systems science, 2012, 43(2): 334-348. DOI:10.1080/00207721.2010.496056 (0) [6] YU Junyan, WANG Long. Group consensus in multi-agent systems with switching topologies and communication delays[J]. Systems & control letters, 2010, 59(6): 340-348. (0) [7] LIU Cong, ZHOU Qiang, HU Xiaoguang. Group consensus of heterogeneous multi-agent systems with fixed topologies[J]. International journal of intelligent computing and cybernetics, 2015, 8(4): 294-311. DOI:10.1108/IJICC-03-2015-0009 (0) [8] WEN Guoguang, HUANG Jun, WANG Chunyan, et al. Group consensus control for heterogeneous multi-agent systems with fixed and switching topologies[J]. International journal of control, 2016, 89(2): 259-269. DOI:10.1080/00207179.2015.1072876 (0) [9] WEN Guoguang, HUANG Jun, PENG Zhaoxia, et al. On pinning group consensus for heterogeneous multi-agent system with input saturation[J]. Neurocomputing, 2016, 207: 623-629. DOI:10.1016/j.neucom.2016.05.046 (0) [10] WEN Guoguang, YU Yongguang, PENG Zhaoxia, et al. Dynamical group consensus of heterogenous multi-agent systems with input time delays[J]. Neurocomputing, 2016, 175: 278-286. DOI:10.1016/j.neucom.2015.10.060 (0) [11] CHENG Yujuan, YU Hui. Adaptive group consensus of multi-agent networks via pinning control[J]. International journal of pattern recongnition and artificial intelligence, 2016, 30(5): 1659041. (0) [12] 张先迪, 李正良. 图论及其应用[M]. 北京: 高等教育出版社, 2005. (0) [13] WANG Hanlei. Flocking of networked uncertain Euler-Lagrange systems on directed graphs[J]. Automatica, 2013, 49(9): 2774-2779. DOI:10.1016/j.automatica.2013.05.029 (0) [14] AEYELS D. Asymptotic stability of nonautonomous systems by Liapunov’s direct method[J]. Systems & control letters, 1995, 25(4): 273-280. (0)