﻿ 有向图中网络Euler-Lagrange系统无需相对速度信息的群一致性
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 自动化学报  2018, Vol. 44 Issue (1): 44-51 PDF

Group Consensus for Networked Euler-Lagrangian Systems Under a Directed Graph Without Relative Velocity Information
CAO Ran, MEI Jie
School of Mechanical Engineering and Automation, Harbin Institute of Technology(Shenzhen), Shenzhen 518055
Manuscript received : September 6, 2016, accepted: February 3, 2017.
Foundation Item: Supported by National Natural Science Foundation of China (61403094) and the Foundation Research Project of Shenzhen (JCYJ20160505175231531)
Corresponding author. MEI Jie Associate professor at the School of Mechanical Engineering and Automation, Harbin Institute of Technology (Shenzhen). His research interest covers distributed control of multi-agent systems and its application in formation flying. Corresponding author of this paper
Recommended by Associate Editor XIA Yuan-Qing
Abstract: In this paper, we study group consensus for networked Euler-Lagrangian systems under a general nonsymmetric directed graph. The relationships between interactive agents in same group are cooperative while the relationships between interactive agents among different groups can be either cooperative or competitive. To achieve group consensus, we assume that the interaction topology among groups is acyclic and the associated directed graph satisfies the in-degree balance condition. Considering the fact that the relative velocity information among neighboring agents is difficult to obtain, we propose a distributed adaptive control algorithm for each agent without relative velocity information to achieve group consensus for networked Euler-Lagrangian systems. Finally, simulation results are presented to demonstrate the effectiveness of the proposed control algorithm.
Key words: Multi-agent systems     Euler-Lagrangian systems     group consensus     adaptive control

1 数学背景与问题描述

 \begin{align} M_i(q_i)\ddot q_i+C_i(q_i,\dot q_i)\dot q_i+g_i(q_i)=\tau _i \end{align} (1)

1) 如果有向图 $G$ 包含一个有向生成树, 则其Laplacian矩阵 $\mathcal{L}_{A}$ 有一个单零特征值并且其余特征值均拥有正实部.

2) 如果有向图 $G$ 是强连通的, 那么存在一个向量 $\xi=[\xi_{1},\cdots,\xi_{n}]^{\rm T}\in{\bf R}^n$ , 其中, $\sum_{i=1}^{n}\xi_i=1$ , $\xi_i$ $>$ $0$ , 对于 $\forall i=1,\cdots,n$ , 则有 $\xi^{\rm T} \mathcal{L}_A=0$ 成立.

 \begin{align} a(L_A)=\min\limits_{{\vartheta^{\rm T}\varsigma=0},{\vartheta^{\rm T}\vartheta=1} }\vartheta^{\rm T}B\vartheta>0 \end{align} (2)

 \begin{align} \mathcal {L}_A=\left(\begin{array}{cccc} L_{11} & 0 & \cdots & 0\\ L_{21} & L_{22} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ L_{d1} & L_{d2} & \cdots & L_{dd} \end{array}\right) \end{align} (3)

2 控制律设计

 \begin{align} &\sum\limits_{j\in\mathcal {V}_n}a_{ij}=0,\quad \forall i\in\mathcal {V}_m \end{align} (4)
 \begin{align} &\sum\limits_{j\in\mathcal {V}_m}a_{ij}=0,\quad \forall i\in\mathcal {V}_n \end{align} (5)

 \begin{align} \dot q_{ri}&= -\alpha\sum\limits_{j\in\nu}a_{ij}(q_i-q_j) \end{align} (6)
 \begin{align} s_i&=\dot q_i-\dot q_{ri}=\dot q_i+\alpha\sum\limits_{j=1}^na_{ij}(q_i-q_j) \end{align} (7)

 \begin{align} \tau_i&=-ks_i+Y_i(q_i,\dot q_i,0_p,\dot q_{ri})\widehat{\Theta}_i \end{align} (8)
 \begin{align} \dot{\widehat{\Theta}}_i&=-\Lambda_iY_i^{\rm T}(q_i,\dot q_i,0_p,\dot q_{ri})s_i \end{align} (9)

 \begin{align} M_i(q_i)\dot {s}_i=&-ks_i-C_i(q_i,\dot q_i)s_i-M_i(q_i)\ddot {q}_{ri}-\notag\\\\ &\ Y_i(q_i,\dot q_i,0_p,\dot q_{ri})\widetilde{\Theta}_i \end{align} (10)

 \begin{align} Q=\left(\begin{array}{ccccc}~~{-1+(n-1)v} ~&~ {1-v} ~&~ -v ~&~ \cdots ~&~ -v~\\ ~~{-1+(n-1)v} ~&~ -v ~&~ {1-v} ~&~ \ddots ~&~ \vdots~\\ ~~\vdots & \vdots ~&~ \ddots ~&~ \ddots ~&~ -v ~\\ ~~{-1+(n-1)v} ~&~ -v ~&~ \cdots ~&~ -v ~&~ {1-v}~ \end{array}\right) \end{align} (11)

 \begin{align} &Q{\bf1}_n={\bf0}_{n-1} \end{align} (12)
 \begin{align} & QQ^{\rm T}=I_{n-1} \end{align} (13)
 \begin{align} &Q^{\rm T}Q=I_n-\frac{1}{n}{\bf1}_n{\bf1}_n^{\rm T} \end{align} (14)

 \begin{align} V_1(t)=\sum\limits_{i=1}^{n_1}\left[\frac{1}{2}s_i^{\rm T}M_i(q_i)s_i+\frac{1}{2}\widetilde{\Theta}_i^{\rm T}\Lambda_i^{-1}\widetilde{\Theta}_i+\xi_{1i}\widetilde{q}_i^{\rm T}\widetilde{q}_i\right] \end{align} (15)

 \begin{align}S_1=\dot X_1+\alpha(L_{11}\otimes I_p)\widetilde{X}_1 \end{align} (16)

 \begin{align} \ddot X_{r1}=& -\alpha(L_{11}\otimes I_p)\dot X_{1}=\notag\\\\ & - \alpha(L_{11}\otimes I_p)S_1+\alpha^2(L_{11}\otimes I_p)^2\widetilde{X}_1 \end{align} (17)

 \begin{align} \dot {V}_1(t)=&-kS_1^{\rm T}S_1-S_1^{\rm T}M(X_1)\ddot {X}_{r1}~+\notag\\\\ &\ \dot {\widetilde{X}_1}^{\rm T}(\Xi_1\otimes I_p)\widetilde{X}_1+\widetilde{X}_1^{\rm T}(\Xi_1\otimes I_p)\dot {\widetilde{X}_1} \end{align} (18)

 \begin{align} &S_1^{\rm T}M(X_1)\ddot {X}_{r1}\leq k_{\overline{m}}S_1^{\rm T}\ddot {X}_{r1}\leq\notag\\\\ &\qquad \alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{11})\|S_1\|\|\widetilde{X}_1\|~+\notag\\\\ &\qquad \alpha k_{\overline{m}}\sigma_{\max}(L_{11})\|S_1\|^2 \end{align} (19)

 \begin{align} \dot {\widetilde{X}}_1=&\ \dot {X}_1-{\bf1}_{n_1}\otimes\dot {\overline{q}}_1=\notag\\\\ &\ \dot {X}_1-{\bf1}_{n_1}\otimes[(\zeta_1^{\rm T}\otimes I_p)\dot {X}_1=\notag\\\\ &\ S_1-\alpha(L_{11}\otimes I_p)\widetilde{X}_1~-\notag\\\\ &\ {\bf1}_{n_1}\otimes[(\zeta^{\rm T}_1\otimes I_p){S}_1] \end{align} (20)

 \begin{align} &(\Xi_1\otimes I_p)\{S_1-{\bf1}_{n_1}\otimes[(\zeta_1^{\rm T}\otimes I_p){S}_1]\}=\notag\\\\ &\qquad (\Xi_1\otimes I_p)\{S_1-[({\bf1}_{n_1}\otimes\zeta_1^{\rm T})\otimes I_p]{S}_1\}=\notag\\\\ &\qquad (\Xi_1\otimes I_p)S_1-\{[\Xi_1({\bf1}_{n_1}\otimes\zeta_1^{\rm T})]\otimes I_p\}S_1=\notag\\\\ &\qquad [(\Xi_1-\zeta_1\zeta_1^{\rm T})\otimes I_p]S_1 \end{align} (21)

 \begin{align} \dot {V}_1(t)=&-kS_1^{\rm T}S_1-S_1^{\rm T}M(X_1)\ddot {X}_{r1}~+\notag\\\\ &\ 2\widetilde{X}_1^{\rm T}[(\Xi_1-\zeta_1\zeta_1^{\rm T})\otimes I_p]S_1~-\notag\\\\ &\ \alpha\widetilde{X}_1^{\rm T}(B_1\otimes I_p)\widetilde{X}_1 \end{align} (22)

 \begin{align} \widetilde{X}_1^{\rm T}(B_1\otimes I_p)\widetilde{X}_1\geq a(L_{11})\|\widetilde{X}_1\|^2 \end{align} (23)

${\bf1}_{n_1}^{\rm T}\zeta=1$ , 可知矩阵 $\Xi_1-\zeta_1\zeta_1^{\rm T}$ 是对角占优的, 故矩阵也是对称半正定的.由Gersorin定理[29]可得

 \begin{align} \sigma_{\max}(\Xi_1-\zeta_1\zeta_1^{\rm T})\leq\frac{1}{2} \end{align} (24)

 \begin{align} \dot {V}_1(t)\leq&-k\|S_1^2\|+\alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{11})\|S_1\|\|\widetilde{X}_1\|~+\notag\\\\ &\ \alpha k_{\overline{m}}\sigma_{\max}(L_{11})\|S_1\|^2+\|\widetilde {X}_1\|\|S_1\|~-\notag\\\\ &\ \alpha a(L_{11})\|\widetilde {X}_1\|^2\leq\notag\\\\ &\ -k\|S_1\|^2+\alpha k_{\overline{m}}\sigma_{\max}(L_{11})\|S_1\|^2~+\notag\\\\ &\ \frac{\alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{11})+1}{a(L_{11})}\|S_1\|^2~+\notag\\\\ &\ \frac{a(L_{11})}{4}\|\widetilde {X}_1\|^2-\alpha a(L_{11})\|\widetilde {X}_1\|^2 \end{align} (25)

 \begin{align} k=&\ k_1+k_0=\notag\\\\ &\ \alpha k_{\overline{m}}\sigma_{\max}(L_{11})+\frac{\alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{11})+1}{a(L_{11})}+k_0 \end{align} (26)

 \begin{align} \dot {V}_1(t)\leq-k_0\|S_1\|^2-\frac{3a(L_{11})}{4}\|\widetilde {X}_1\|^2\leq0 \end{align} (27)

 \begin{align} V_2(t)=&\sum\limits_{i=n_1+1}^{n_1+n_2}\left[\frac{1}{2}s_i^{\rm T}M_i(q_i)s_i~+\right.\notag\\\\ &\left.\frac{1}{2}\widetilde{\Theta}_i^{\rm T}\Lambda_i^{-1}\widetilde{\Theta}_i+\xi_{2(i-n_1)}\widetilde{q}_i^{\rm T}\widetilde{q}_i\right] \end{align} (28)

 \begin{align}S_2=\dot X_2+\alpha(L_{21}\otimes I_p)\widetilde{X}_1+\alpha(L_{22}\otimes I_p)\widetilde{X}_2 \end{align} (29)

 \begin{align} \ddot X_{r2}=&-\alpha(L_{21}\otimes I_p)\dot X_{1}-\alpha(L_{22}\otimes I_p)\dot X_{2} \end{align} (30)

 \begin{align} \dot {V}_2(t)=&-kS_2^{\rm T}S_2-S_2^{\rm T}M(X_2)\ddot {X}_{r2}~+\notag\\\\ &\ \dot {\widetilde{X}_2}^{\rm T}(\Xi_2\otimes I_p)\widetilde{X}_2+\widetilde{X}_2^{\rm T}(\Xi_2\otimes I_p)\dot {\widetilde{X}_2} \end{align} (31)

 \begin{align} &S_2^{\rm T}M(X_2)\ddot {X}_{r2}\leq k_{\overline{m}}S_2^{\rm T}\ddot {X}_{r2}\leq\notag\\\\ &\qquad \alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{22})\|S_2\|\|\widetilde{X}_2\|~+\notag\\\\ &\qquad \alpha k_{\overline{m}}\sigma_{\max}(L_{22})\|S_2\|^2~+\notag\\\\ &\qquad \alpha^2k_{\overline{m}}\sigma_{\max}(L_{22}L_{21})\|S_2\|\|\widetilde{X}_1\|~+\notag\\\\ &\qquad \alpha k_{\overline{m}}\sigma_{\max}(L_{21})\|S_2\|\|\dot {X}_1\| \end{align} (32)

 \begin{align} \dot {\widetilde{X}}_2=&\ S_2-\alpha(L_{21}\otimes I_p)\widetilde{X}_1-\alpha(L_{22}\otimes I_p)\widetilde{X}_2~-\notag\\\\ &\ {\bf1}_{n_2}\otimes(\zeta^{\rm T}_2\otimes I_p)[{S}_1-\alpha(L_{21}\otimes I_p)\widetilde{X}_1] \end{align} (33)

 \begin{align} \dot {V}_2(t)=&-kS_2^{\rm T}S_2-S_2^{\rm T}M(X_2)\ddot {X}_{r2}~+\notag\\\\ &\ 2\widetilde{X}_2^{\rm T}[(\Xi_2-\zeta_2\zeta_2^{\rm T})\otimes I_p]\dot {X}_2~-\notag\\\\ &\ \alpha\widetilde{X}_2^{\rm T}(B_2\otimes I_p)\widetilde{X}_2~\leq\notag\\\\ &-k\|S_2\|^2+\alpha^2k_{\overline{m}}\sigma_{\max}(L_{22}L_{21})\|S_2\|\|\widetilde{X}_1\|~+\notag\\\\ &\ \alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{22})\|S_2\|\|\widetilde{X}_2\|~+\notag\\\\ &\ \alpha k_{\overline{m}}\sigma_{\max}(L_{22})\|S_2\|^2-a(L_{22})\|S_2\|^2~+\notag\\\\ &\ \alpha k_{\overline{m}}\sigma_{\max}(L_{21})\|S_2\|\|\dot {X}_1\|+\|\widetilde{X}_2\|\|S_2\|~+\notag\\\\ &\ \alpha\sigma_{\max}(L_{21})\|\widetilde{X}_2\|\|\widetilde{X}_1\| \end{align} (34)

 \begin{align} &\alpha k_{\overline{m}}\sigma_{\max}(L_{21})\|S_2\|\|\dot {X}_1\|\leq\notag\\\\ &\qquad \frac{[\alpha k_{\overline{m}}\sigma_{\max}(L_{21})]^2}{2}\|\dot {X}_1\|^2+\frac{1}{2}\|S_2\|^2 \end{align} (35)
 \begin{align} &\alpha^2k_{\overline{m}}\sigma_{\max}(L_{22}L_{21})\|S_2\|\|\widetilde{X}_1\|\leq\notag\\\\ &\qquad \frac{[\alpha^2k_{\overline{m}}\sigma_{\max}(L_{22}L_{21})]^2}{2}\|\widetilde{X}_1\|^2+\frac{1}{2}\|S_2\|^2 \end{align} (36)

 \begin{align} &[\alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{22})+1]\|S_2\|\|\widetilde{X}_2\|\leq\notag\\\\ &\qquad\frac{[\alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{22})+1]^2}{a(L_{22})}\|S_2\|^2~+\notag\\\\ &\qquad\frac{a(L_{22})}{4}\|\widetilde{X}_2\|^2\\ \end{align} (37)
 \begin{align} &\alpha\sigma_{\max}(L_{21})\|\widetilde{X}_2\|\|\widetilde{X}_1\|\leq\notag\\\\ &\qquad\frac{{\alpha\sigma_{\max}(L_{21})}^2}{a(L_{22})}\|\widetilde{X}_1\|^2 +\frac{1}{2}\|\widetilde{X}_2\|^2 \end{align} (38)

 \begin{align*} &w_1=\frac{[\alpha^2k_{\overline{m}}\sigma_{\max}(L_{22}L_{21})]^2}{2}+\frac{{\alpha\sigma_{\max}(L_{21})}^2}{a(L_{22})}\\ &w_2=\frac{[\alpha k_{\overline{m}}\sigma_{\max}(L_{21})]^2}{2}\\ &k_2=\alpha k_{\overline{m}}\sigma_{\max}(L_{21})+\frac{[\alpha^2k_{\overline{m}}\sigma_{\max}^2(L_{22})+1]^2}{a(L_{22})}+1\end{align*}

 \begin{align} \dot {V}_2(t)\leq&-k_0\|S_2\|^2-\frac{a(L_{22})}{2}\|\widetilde{X}_2\|^2~+\notag\\\\ &\ w_1\|\widetilde{X}_1\|^2+w_2\|\dot {X}_1\|^2 \end{align} (39)

 \begin{align} &V_2(t)-V_2(0)\leq\notag\\\\ &\qquad -k_0\int_0^t\|S_2\|^2{\rm d}\tau-\frac{a(L_{22})}{2}\int_0^t\|\widetilde{X}_2\|^2{\rm d}\tau~+\notag\\\\ &\qquad\ w_1\int_0^t\|\widetilde{X}_1\|^2{\rm d}\tau+w_2\int_0^t\|\dot {X}_1\|^2{\rm d}\tau \end{align} (40)

$S_1$ , $\widetilde {X}_1\in\mathbb{L}_2\bigcap\mathbb{L}_\infty$ 可以得到 $\dot {X}_1\in\mathbb{L}_2\bigcap\mathbb{L}_\infty$ .所以式(40)中最后两项都是有界的, 故 $V_2(t)$ 也是有界的.由式(28)和式(40)可知, $S_2$ , $\widetilde {X}_2\in$ $\mathbb{L}_2$ $\bigcap$ $\mathbb{L}_\infty$ , 再由式(29)可得 $\dot {X}_2\in\mathbb{L}_2\bigcap\mathbb{L}_\infty$ .根据式(33)可知 $\dot {\widetilde {X}}_1\in\mathbb{L}_\infty$ , 最后使用Barbalat引理可知, 当 $t$ $\rightarrow$ $\infty$ , 有 $\widetilde {X}_2\rightarrow{\bf0}_{n_2p}$ .

$S$ $s_i$ 的列堆栈向量, $i=1,\cdots,n$ .令 $\overline{Q}$ $={\rm diag}\{Q_1,Q_2,\cdots,Q_d\}$ .注意此时的 $\overline{Q}$ 仍然满足式(12)和式(13), 而由式(14)可知 $\overline{Q}^{\rm T}\overline{Q}=I_n$ $-$ ${\rm diag}\{\frac{1}{n_1}{\bf1}_{n_1}{\bf1}_{n_1}^{\rm T},\cdots,\frac{1}{n_d}{\bf1}_{n_d}{\bf1}_{n_d}^{\rm T}\}$ .定义 $\widehat{S}=(\overline{Q}\otimes I_p)S$ , $\widehat{X}=(\overline{Q}\otimes I_p)X$ .其中 $X$ 已在证明开始时给出定义.

 \begin{align}S=\dot X+\alpha(L\otimes I_p)X \end{align} (41)

 \begin{align}\widehat{S}=&\ \dot {\widehat{X}}+\alpha(\overline{Q}L\otimes I_p)X=\notag\\\\ &\ \dot {\widehat{X}}+\alpha(\overline{Q}L\overline{Q}^{\rm T}\overline{Q}\otimes I_p)X=\notag\\\\ &\ \dot {\widehat{X}}+\alpha(\overline{Q}L\overline{Q}^{\rm T}\otimes I_p)\widehat{X} \end{align} (42)

3 仿真分析

 图 1 智能体间的拓扑关系 Figure 1 The networked topology associated with the agents
 图 2 有向拓扑图下智能体位置状态信息 Figure 2 The position state of agents under the directed interaction graph
 图 3 有向拓扑图下智能体速度信息 Figure 3 The velocities of agents under the directed interaction graph
4 结论

 1 Oh K K, Park M C, Ahn H S. A survey of multi-agent formation control. Automatica, 2015, 53: 424-440. DOI:10.1016/j.automatica.2014.10.022 2 Su H S, Wang X F, Lin Z L. Flocking of multi-agents with a virtual leader. IEEE Transactions on Automatic Control, 2009, 54(2): 293-307. DOI:10.1109/TAC.2008.2010897 3 Yu H, Xia X H. Adaptive consensus of multi-agents in networks with jointly connected topologies. Automatica, 2012, 48(8): 1783-1790. DOI:10.1016/j.automatica.2012.05.068 4 Wu W, Zhou W J, Chen T P. Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Transactions on Circuits and Systems I:Regular Papers, 2009, 56(4): 829-839. DOI:10.1109/TCSI.2008.2003373 5 Chen T P, Liu X W, Lu W L. Pinning complex networks by a single controller. IEEE Transactions on Circuits and Systems I:Regular Papers, 2007, 54(6): 1317-1326. DOI:10.1109/TCSI.2007.895383 6 Xu C J, Zheng Y, Su H S, Chen M Z Q, Zhang C F. Cluster consensus for second-order mobile multi-agent systems via distributed adaptive pinning control under directed topology. Nonlinear Dynamics, 2016, 83(4): 1975-1985. DOI:10.1007/s11071-015-2459-5 7 Xia W G, Cao M. Clustering in diffusively coupled networks. Automatica, 2011, 47(11): 2395-2405. DOI:10.1016/j.automatica.2011.08.043 8 Yu J Y, Wang L. Group consensus in multi-agent systems with switching topologies and communication delays. Systems and Control Letters, 2010, 59(6): 340-348. DOI:10.1016/j.sysconle.2010.03.009 9 Qin J H, Yu C B. Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition. Automatica, 2013, 49(9): 2898-2905. DOI:10.1016/j.automatica.2013.06.017 10 Wen G G, Huang J, Wang C Y, Chen Z, Peng Z X. Group consensus control for heterogeneous multi-agent systems with fixed and switching topologies. International Journal of Control, 2016, 89(2): 259-269. DOI:10.1080/00207179.2015.1072876 11 Min Hai-Bo, Liu Yuan, Wang Shi-Cheng, Sun Fu-Chun. An overview on coordination control problem of multi-agent system. Acta Automatica Sinica, 2012, 38: 1557-1570.( 闵海波, 刘源, 王仕成, 孙富春. 多个体协调控制问题综述. 自动化学报, 2012, 38(10): 1557-1570.) 12 Cheng L, Hou Z G, Tan M. Decentralized adaptive consensus control for multi-manipulator system with uncertain dynamics. In: Proceedings of the 2008 IEEE International Conference on Systems, Man, and Cybernetics. Singapore:IEEE, 2008, 2712-2717. 13 Ren W. Distributed leaderless consensus algorithms for networked Euler-Lagrange systems. International Journal of Control, 2009, 82(11): 2137-2149. DOI:10.1080/00207170902948027 14 Mei Jie, Zhang Hai-Bo, Ma Guang-Fu. Adaptive coordinated tracking for networked Euler-Lagrange systems under a directed graph. Acta Automatica Sinica, 2011, 37: 596-603.( 梅杰, 张海博, 马广富. 有向图中网络Euler-Lagrange系统的自适应协调跟踪. 自动化学报, 2011, 37(5): 596-603.) 15 Mei J, Ren W, Ma G F. Distributed coordinated tracking with a dynamic leader for multiple Euler-Lagrange systems. IEEE Transactions on Automatic Control, 2011, 56(6): 1415-1421. DOI:10.1109/TAC.2011.2109437 16 Meng Z Y, Ren W, You Z. Distributed finite-time attitude containment control for multiple rigid bodies. Automatica, 2010, 46(12): 2092-2099. DOI:10.1016/j.automatica.2010.09.005 17 Mei J, Ren W, Ma G F. Distributed containment control for Lagrangian networks with parametric uncertainties under a directed graph. Automatica, 2012, 48(4): 653-659. DOI:10.1016/j.automatica.2012.01.020 18 Hu H X, Zhang Z, Yu L, Yu W W, Xie G M. Group consensus for multiple networked Euler-Lagrange systems with parametric uncertainties. Journal of Systems Science and Complexity, 2014, 27(4): 632-649. DOI:10.1007/s11424-014-2149-2 19 Liu J, Xiang L, Zhao L Y, Zhou J. Group consensus in uncertain networked Euler-Lagrange systems with acyclic interaction topology. In: Proceedings of the 34th Chinese Control Conference. Hangzhou, China:IEEE, 2015, 835-840. 20 Liu J, Ji J C, Zhou J, Xiang L, Zhao L Y. Adaptive group consensus in uncertain networked Euler-Lagrange systems under directed topology. Nonlinear Dynamics, 2015, 82(3): 1145-1157. DOI:10.1007/s11071-015-2222-y 21 Mei J, Ren W, Chen J, Ma G F. Distributed adaptive coordination for multiple Lagrangian systems under a directed graph without using neighbors' velocity information. Automatica, 2013, 49(6): 1723-1731. DOI:10.1016/j.automatica.2013.02.058 22 Mei J, Ren W, Chen J. Distributed consensus of second-order multi-agent systems with heterogeneous unknown inertias and control gains under a directed graph. IEEE Transactions on Automatic Control, 2016, 61(8): 2019-2034. DOI:10.1109/TAC.2015.2480336 23 Spong M W, Hutchinson S, Vidyasagar M. Robot Modeling and Control. New Jersey, USA: John Wiley and Sons, 2006. 24 Mesbahi M, Egerstedt M. Graph Theoretic Methods in Multiagent Networks. New Jersey, USA: Princeton University Press, 2010. 25 Ren W, Beard R W. Distributed Consensus in Multi-Vehicle Cooperative Control. London, Britain: Springer-Verlag, 2008. 26 Yu W W, Chen G R, Gao M, Kurths J. Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 2010, 40(3): 881-891. DOI:10.1109/TSMCB.2009.2031624 27 Scardovi L, Arcak M, Sontag E D. Synchronization of interconnected systems with applications to biochemical networks:an input-output approach. IEEE Transactions on Automatic Control, 2010, 55(6): 1367-1379. DOI:10.1109/TAC.2010.2041974 28 Mei J. Weighted consensus for multiple Lagrangian systems under a directed graph. In: Proceedings of the 2015 Chinese Automation Congress (CAC). Wuhan, China:IEEE, 2015, 1064-1068. 29 Horn R A, Johnson C R. Matrix Analysis. New York, USA: Cambridge University Press, 1985. 30 Kelly R, Sáñtibanez V, Loría A. Control of Robot Manipulators in Joint Space. London, Britain: Springer, 2005.