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Finite-time containment control of second-order multi-agent systems with jointly connected topologies
ZHUANG Hao, YANG Hongyong
School of Information and Electrical Engineering, Ludong University, Yantai 264025, China
Abstract: In this paper, we propose a containment control algorithm with finite-time convergence for a second-order networked system flocking with multiple leaders. By applying modern control theory, matrix theory, and algebraic graph theory, we theoretically analyzed our proposed control algorithm; by doing so, we identified the convergence conditions required for a second-order networked system to realize flocking within finite time when the communication topology applies a dynamic joint connection. Through our containment control algorithm, the networked systems converge to object regions in finite time given the circumstances of static and jointly connected topologies. Finally, we verified the effectiveness of our proposed system via simulation examples.
Key words: multiple leaders     flocking     finite time     jointly-connected     containment control

1 代数图论

G=(V, E, A) 是n个节点的权重无向图，V={1，2, …, n}为一个顶点 (或节点) 集合，EV×V为一个边的集合，A=[aij] ∈ Rn×n为权重邻接矩阵。对于∀iV, aii=0；对于∀i, jVij，若 (i, j)∈ω，则aij>0，否则，aij=0。节点i的邻居集合定义为Ni={jV|(i, j)∈E}。定义D=diag (d1, d2, …, dn)∈Rn×n为图xL(t) 的度矩阵，其中。权重图G的Laplacian矩阵定义为：L=D-ARn×n

2 二阶多自主体系统的包容控制

 (1)

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3 联合连通下的控制算法分析

 (10)
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xF(t) 表示跟随者的位置，xL(t) 表示领航者的位置，vF(t) 表示跟随者的速度，vL(t) 表示领航者的速度。其中：

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4 仿真验证 4.1 静态拓扑仿真

 图 1 跟随者与领航者静态拓扑图 Fig. 1 The static topological graph of followers and leaders

 图 2 控制器1跟随者位置横坐标与时间关系 Fig. 2 The relationship between the abscissa of followers' position under the action of controller1 and time
 图 3 控制器2跟随者位置横坐标与时间关系 Fig. 3 The relationship between the abscissa of followers' position under the action of controller2 and time
 图 4 控制器1跟随者速度横坐标与时间关系 Fig. 4 The relationship between the abscissa of followers' speed under the action of controller1 and time
 图 5 控制器2跟随者速度横坐标与时间关系 Fig. 5 The relationship between the abscissa of followers' speed under the action of controller2 and time

 图 6 控制器1跟随者与领航者位置关系 Fig. 6 The positional relationship between followers and leaders under the action of controller1

4.2 动态拓扑仿真

 图 7 跟随者与领航者拓扑图 Fig. 7 The topological graph of followers and leaders

 图 8 控制器1跟随者位置横坐标与时间关系 Fig. 8 The relationship between the abscissa of followers' position under the action of controller1 and time
 图 9 控制器2跟随者位置横坐标与时间关系 Fig. 9 The relationship between the abscissa of followers' position under the action of controller2 and time
 图 10 控制器1跟随者速度纵坐标与时间关系 Fig. 10 The relationship between the ordinate of followers' speed under the action of controller1 and time
 图 11 控制器2跟随者速度纵坐标与时间关系 Fig. 11 The relationship between the ordinate of followers' speed under the action of controller2 and time
 图 12 控制器1跟随者与领航者位置关系 Fig. 12 The positional relationship between followers and leaders under the action of controller1

5 结论

1) 本文分别针对静态拓扑和动态拓扑的二阶多自主体系统提出一般性的包容控制算法，并运用现代控制理论及矩阵论等理论工具分析了该算法的有限时间收敛问题，给出了二阶系统在动态联合连通拓扑条件下的有限时间收敛条件，并给予仿真验证。

2) 本文研究的是连续条件下的有限时间收敛问题，为了贴近实际应用，下一步将继续研究离散条件下的有限时间收敛问题。

3) 通过本文设计的包容控制算法，可以使网络化系统快速达到收敛，大大减少收敛时间，提高了系统收敛效率。

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DOI: 10.11992/tis.201605013

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#### 文章信息

ZHUANG Hao, YANG Hongyong

Finite-time containment control of second-order multi-agent systems with jointly connected topologies

CAAI Transactions on Intelligent Systems, 2017, 12(2): 188-195
http://dx.doi.org/10.11992/tis.201605013