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A control strategy for maitaining controllability and observability of a multi-agent system with the second-order neighborhood protocol
WANG Kang, JI Zhijian, CHAO Yongcui
School of Automation Engineering, Qingdao University, Qingdao 266071, China
Abstract: In order to study the characteristics of the consensus, controllability and observability of multi-agent systems, we analyze the consensus speed of a multi-agent system with time-varying topologies under first-order and second-order neighborhood protocols. By utilizing the properties of the structural controllability and the relationship between the second-smallest eigenvalue of the Laplacian matrix and the consensus speed, we designed a control strategy to maintain both controllability and observability. In addition, we concluded that the multi-agent system had a faster consensus speed under the second-order neighborhood protocol. Using examples and simulations, we verified the two main theorems proposed in this paper, with our observed results in full agreement with the conclusions of our theoretical analysis.
Key words: multi-agent system     second-order neighborhood protocol     time varying topologies     structural controllability     controllability     observability     graph theory

1 预备知识

1) 对于一个所有元素均为1的列向量，L与该列向量的乘积为零矩阵;

2) 令λ1, λ2, …, λN为拉普拉斯矩阵的特征值，则0=λ1λ2≤…≤λN;

3) 第1个非零特征值 (第2个最小的特征值λ2) 称为代数连通度。

2 系统模型

 (1)

Ni为智能体i的邻居集，即Ni={jV(G)|(vi, vj)∈E(G)}。那么在一阶邻居协议下,

 (2)

Ni2为智能体i的二阶邻居集，即Ni2={kv|kNj, jNi, ki, kNi }。所以各智能体之间通过二阶邻居一致性协议进行下式连接：

 (3)

 (4)

wij=wik=1，则

 (5)

λ2(G)‖x2xTLx，故

 (6)

 (7)

 图 1 一阶邻居协议下各智能体的状态轨迹 Fig. 1 The state trajectory of each agent under the first-order neighbors protocol
 图 2 二阶邻居协议下各智能体的状态轨迹 Fig. 2 The state trajectory of each agent under the second-order neighbors protocol

 (8)

 (9)

3 实例分析

 图 3 具有5个节点的拓扑图 Fig. 3 Topology with 5 nodes

 (10)

 (11)

 (12)

4 对结构能控性维持策略的研究

4.1 结构能控性

4.2 实例分析

 图 4 权图 Fig. 4 Weight graph

 图 5 多智能体系统能控性和结构能控性的关系 Fig. 5 Relationship between controllability and structural controllability of multi-agent system
3.3 结构能控性的保持方法

 图 6 时变拓扑结构 Fig. 6 Time varying topology

 (13)

 (14)

 (15)

1) ∀λ2(·)>0，能量函数连续可微；

2) 能量函数是非负的。

 (16)

 (17)

 (18)

 (19)

 (20)

 (21)

 (22)

 (23)

 图 7 能量函数Ε的图像 Fig. 7 Image of energy function E

4 结束语

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DOI: 10.11992/tis.201601022

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#### 文章信息

WANG Kang, JI Zhijian, CHAO Yongcui

A control strategy for maitaining controllability and observability of a multi-agent system with the second-order neighborhood protocol

CAAI Transactions on Intelligent Systems, 2017, 12(2): 213-220
http://dx.doi.org/10.11992/tis.201601022