research article average consensus in multiagent systems with...

Research ArticleAverage Consensus in Multiagent Systems withthe Problem of Packet Losses When Using the Second-OrderNeighbors’ Information

Mei Yu,1 Lijuan Li,2 Guangming Xie,3 and Hong Shi4

1 School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, China2 Tianjin Electric Power Design Institute, Tianjin 300400, China3 State Key Laboratory for Turbulence and Complex Systems and College of Engineering, Peking University, Beijing 100871, China4Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, China

Correspondence should be addressed to Guangming Xie; [email protected]

Received 17 February 2014; Accepted 18 March 2014; Published 10 April 2014

Academic Editor: Housheng Su

Copyright © 2014 Mei Yu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper mainly investigates the average consensus of multiagent systems with the problem of packet losses when both the first-order neighbors’ information and the second-order neighbors’ information are used.The problem is formulated under the sampled-data framework by discretizing the first-order agent dynamics with a zero-order hold.The communication graph is undirected andthe loss of data across each communication link occurs at certain probability, which is governed by a Bernoulli process. It is foundthat the distributed average consensus speeds up by using the second-order neighbors’ informationwhenpackets are lost.Numericalexamples are given to demonstrate the effectiveness of the proposed methods.

1. Introduction

In recent years, there has been an increasing research incoordination control of multiagent systems. Informationconsensus has attractedmore andmore attentions frommanyengineering application fields, such as formation control,flocking, artificial intelligence, and automatic control [1–4]. A critical problem in distributed control is to developdistributed protocols under which agents can reach an agree-ment on a common decision.

An excellent protocol can reduce cost, increase efficiency,and can optimize performance. Convergence rate is animportant index to evaluate the performance of consensus.There has been much research interest in dealing with thisissue. In [5], the authors pointed out that the second smallesteigenvalue of its Laplacian matrix was a measure of speed ofsolving consensus problems. From [6], we know that the con-vergence speeds up by finding the optimal weight associatedwith each communication link, where the global structureof the network must be known beforehand. Reference [7]

accelerated the convergence rate by using the polynomialfiltering algorithms. In [8], the authors presented randomizedgossip algorithm on an arbitrary connected network andshowed its performance precisely in the terms of the secondlargest eigenvalue of an appropriate stochastic matrix. Theabove literatures all tried to seek a suitable topology commu-nication to achieve a fast convergence. However, in practice,it is more useful to design a protocol to obtain a betterconvergence performance under a given topology. In order toget a better convergence speedwithout changing the topologyand edge weights, the authors in [9] proposed a protocolin an unchanged topology network that each node got itsstate value updated by using the information of multihopcommunication and showed that the protocol increased theconvergence speed effectively for the first time. Then, in [10],the authors discussed that the node in the network topologyupdated its current state value not only from its immediateneighbors but also from its second-order neighbors for boththe discrete-time case and the continuous-time case. Further,the authors in [11] extended the systems to second-order

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 304126, 6 pageshttp://dx.doi.org/10.1155/2014/304126

2 Mathematical Problems in Engineering

case and made comparisons between the convergence rateof second-order neighbor protocol and the general protocol.What is more, the delay margins of general protocol andsecond-order neighbor protocol were derived.

It is noted that the literatures mentioned above mainlyfocus on consensus problem for agents under first-orderdynamics with time delay. In reality, the agents exchange dataover fading communication channels instead of ideal ones.In fact, in many practical applications, this data exchangebetween sensors is done by wireless communication, whichhas a possibility of packets lost. Thereby, the packet lossesshould be taken into consideration. Many related workshave been reported. Reference [12] dealt with consensuswith random delay and data losses. Reference [13] comparedthe memory and memoryless consensus protocols in thepresence of uniform packet losses. In [14, 15], the authorsdiscussed the average consensus in first-order agents andanalyzed the convergence speed under data losses. Further-more, [16] showed that packet dropouts can be treated asan absence of a communication link over time. In addition,[17–19] studied stochastic consensus subject to a randomprocess.

Inspired by the above references, we consider multiagentsystems with the problem of packet losses based on thesecond-order neighbors’ information. We construct a groupof agents, which can communicate with their second-orderneighbors and each communication link has a probabilityof failure. We assume that all channels are independent andsubject to a distributed random process. Thereby, they havethe same probability of data loss. Each agent is equipped witha sampler and a zero-order hold, which are synchronizedin time. Then, by converting the system to the equivalenterror dynamics, stochastic stability of the error dynamicsystem is studied. Here, a Lyapunov function is constructedand a sufficient condition is established to guarantee theaverage consensus in the form of linear matrix inequality(LMI). We are curious about whether the protocol based onthe second-order neighbors’ information can accelerate theconvergence speed with the problem of packet losses.Then, asimulation comparison of the convergence rate between theprotocol based on the second-order neighbors and the onein general linear is shown. Comparison of the convergencespeed between different probabilities of packet losses is alsosimulated.

The rest of this paper is organized as follows. Section 2provides some preliminaries on graph theory and gives thedesigned protocol. Section 3 analyses the average consensusand gives a sufficient condition. Section 4 includes somenumerical examples, which demonstrate the effectivenessof the proposed approach. Finally, Section 5 offers theconcluding remarks.

Notations. The set of real numbers is denoted by R. For anymatrix 𝑄 ∈ R𝑛×𝑛, sym(𝑄) = 𝑄 + 𝑄

𝑇. The index set Λ𝑛

=

{1, 2, . . . , 𝑛} is a group of consecutive integers from 1 to 𝑛.Thevector 1

𝑛= [1, 1, . . . , 1]

𝑇

∈ R𝑛 has all of its elements equal to1.Themathematical expectation is denoted by 𝐸{⋅} and 𝑃{⋅} isthe probability operator.

2. Problem Formation

2.1. Preliminaries on GraphTheory. In this paper, the interac-tion among 𝑛 agents is modeled by an undirected graph 𝐺 =

{], 𝜀, 𝐴}, where ] = {]1, ]2, . . . , ]

𝑛} is the node set. The edge

set 𝜀 ⊆ ] × ] contains ordered pairs of nodes. The neighborset of agent 𝑖 is denoted by 𝑁

𝑖, which includes agents from

which agent 𝑖 receives information. The adjacency matrix𝐴 = [𝑤

𝑖𝑗] ∈ R𝑛×𝑛 is a nonnegative matrix, where 𝑤

𝑖𝑗> 0

if and only if (V𝑗, V𝑖) ∈ 𝜀; otherwise, 𝑤

𝑖𝑗= 0. We assume

that there is no self-loop, so 𝑤𝑖𝑖

= 0. The Laplacian matrix𝐿 = [𝑙

𝑖𝑗] ∈ R𝑛×𝑛 is defined as

𝑙𝑖𝑗

= − 𝑤𝑖𝑗, if 𝑖 ̸= 𝑗;

𝑙𝑖𝑗

= ∑

𝑘∈𝑁𝑖

𝑤𝑖𝑘.

(1)

From the above definitions, we know some facts: 𝐴 and𝐿 determine each other uniquely, and 𝐿 has nonnegativeeigenvalues. Moreover, 𝐿 has at least one zero eigenvaluewith the associated eigenvector 1

𝑇

𝑛(𝐿1𝑇𝑛

= 0); that is,span{1𝑇

𝑛} ⊆ null{𝐿}, where null{𝐿} is the null space of 𝐿. For

the undirected graph, we further have 𝐿 = 𝐿𝑇, 1𝑇𝑛𝐿 = 0. From

[20], it is known that span{1𝑇𝑛} = null{𝐿} if and only if the

undirected graph 𝐺 is connected.

2.2. Protocols Based on Second-Order Neighbor with PacketLosses. Consider the following first-order dynamics:

�̇�𝑖= 𝑢𝑖, 𝑖 ∈ Λ

𝑛, (2)

where 𝑥𝑖, 𝑢𝑖

∈ R are the state and the input of agent 𝑖,respectively. With sampling period 𝑇 and a zero-order hold,the agent dynamics is discretized as

𝑥𝑖(𝑘 + 1) = 𝑥

𝑖(𝑘) + 𝑇𝑢

𝑖(𝑘) , 𝑖 ∈ Λ

𝑛. (3)

Considering the protocol based on second-order neigh-bors’ information, if there is no communication constrainttaken into account, the following control protocol can beused:

𝑢𝑖(𝑘) = −𝑟

𝑐∑

𝑗∈𝑁𝑖

𝑤𝑖𝑗

[

[

(𝑥𝑖(𝑘) − 𝑥

𝑗(𝑘))

+ ∑

ℎ∈𝑁𝑗

𝑤𝑗ℎ


ℎ(𝑘))]

]

,

(4)

where the control gain 𝑟𝑐is to be designed.

Mathematical Problems in Engineering 3

Next, we consider the packet losses among agents. Thefollowing control protocol is designed:

𝑢𝑖(𝑘) = −𝑟

𝑐

{

{

{

∑

𝑗∈𝑁𝑖

𝛾𝑖𝑗(𝑘) 𝑤𝑖𝑗

[

[


𝑗(𝑘))

+ ∑

ℎ∈𝑁𝑗

𝛾𝑗ℎ

(𝑘) 𝑤𝑗ℎ

× (𝑥𝑖(𝑘) − 𝑥

ℎ(𝑘)) ]

]

}

}

}

,

(5)

where 𝛾𝑖𝑗(𝑘) = 1, if there is no packet loss between agents 𝑖

and 𝑗; 𝛾𝑖𝑗(𝑘) = 0, otherwise.

Furthermore, we assume that the occurrence of packetloss is governed by a Bernoulli process with uniform prob-ability 𝑝 satisfying 0 < 𝑝 < 1; that is,

𝑃 {𝛾𝑖𝑗(𝑘) = 1} = 𝑝, 𝑃 {𝛾

𝑖𝑗(𝑘) = 0} = 1 − 𝑝,

∀𝑖 ̸= 𝑗.

(6)

As a result, we have 𝐸{𝛾𝑖𝑗(𝑘)} = 𝑝.

Assumption 1. The undirected topology is coupled; that is,for any pair of agents 𝑖 and 𝑗, the communication channelsbetween them exist or vanish simultaneously.

Assumption 1 ensures that the communication topologyis always symmetric, so the average of agents’ states can beretained during dynamic evolution.

We define two sets of matrices 𝐿1(𝑘) = [𝑙

1𝑖𝑗(𝑘)] ∈ R𝑛×𝑛

and 𝐿2(𝑘) = [𝑙

2𝑖𝑗(𝑘)] ∈ R𝑛×𝑛 as follows:

𝑙1𝑖𝑗

(𝑘) = − 𝛾𝑖𝑗(𝑘) 𝑤𝑖𝑗,

𝑙2𝑖𝑗

(𝑘) = − ∑

ℎ∈𝑁𝑖

𝑗∈𝑁ℎ

𝛾𝑖ℎ

(𝑘) 𝑤𝑖ℎ𝛾ℎ𝑗

(𝑘) 𝑤ℎ𝑗, 𝑖 ̸= 𝑗,

𝑙1𝑖𝑖

(𝑘) = ∑

𝑗∈𝑁𝑖

𝛾𝑖𝑗(𝑘) 𝑤𝑖𝑗,

𝑙2𝑖𝑖

(𝑘) = ∑

ℎ∈𝑁𝑖

𝛾𝑖ℎ

(𝑘) 𝑤𝑖ℎ

∑

𝑙∈𝑁ℎ

𝛾ℎ𝑙

(𝑘) 𝑤ℎ𝑙.

(7)

Denote the vectors 𝑥(𝑘) and 𝑢(𝑘) by

𝑥 (𝑘) = [𝑥1(𝑘) , 𝑥

2(𝑘) , . . . , 𝑥

𝑛(𝑘)]𝑇

,

𝑢 (𝑘) = [𝑢1(𝑘) , 𝑢

2(𝑘) , . . . , 𝑢

𝑛(𝑘)]𝑇

.

(8)

Then, the control protocol can be rewritten as

𝑢 (𝑘) = −𝑟𝑐𝐿 (𝑘) 𝑥 (𝑘) , (9)

where 𝐿(𝑘) = 𝐿1(𝑘) + 𝐿

2(𝑘).

So, the system dynamics can be written as

𝑥 (𝑘 + 1) = 𝑥 (𝑘) − 𝑟𝑐𝑇𝐿 (𝑘) 𝑥 (𝑘) = [𝐼 − 𝑟

𝑐𝑇𝐿 (𝑘)] 𝑥 (𝑘) .

(10)

By taking the mathematical expectation of 𝐿1(𝑘) and 𝐿

2(𝑘),

we have 𝐸{𝐿1(𝑘)} = 𝑝×L(1), 𝐸{𝐿

2(𝑘)} = 𝑝

2

×𝐿(2), where 𝐿

(1)

is the nominal Laplacian matrix of full weights where there isno packet loss and 𝐿

(2) is the nominal Laplacian matrix of thesystem which is only based on the second-order neighbors’information with full weights and without packet loss.

Assumption 2. The nominal communication topologies 𝐺(1)

associated with 𝐿(1) and 𝐺

(2) associated with 𝐿(2) are all

connected.

The above assumption is necessary for consensus becauseif the undirected graph is not connected, then it does not havea spanning tree. From [21, Lemma 1] and [22,Theorem 5], weknow that there exist two nonempty, disjoint groups of agentsthat have no communication with each other at any time. Inthis case, consensus cannot be reached.

3. Consensus Analysis

The average states of the agents

𝛼 = Ave (𝑥 (𝑘)) =1

𝑛

𝑛

∑

𝑖=1

𝑥𝑖(𝑘) =

1

𝑛1𝑇

𝑛𝑥 (𝑘) (11)

are invariant.We say the average consensus problem is solved,if

lim𝑘→+∞

𝑥𝑖(𝑘) = 𝛼, 𝑖 = 1, . . . , 𝑛. (12)

Each agent state can be presented by the form

𝑥 (𝑘) = 𝛼1𝑛+ 𝛿 (𝑘) , (13)

where the variable 𝛿(𝑘) = [𝛿1(𝑘), 𝛿2(𝑘), . . . , 𝛿

𝑛(𝑘)]𝑇 satisfies

1𝑇

𝑛𝛿 = 0. (14)

The following error dynamics are obtained:

𝛿 (𝑘 + 1) = 𝛿 (𝑘) − 𝑟𝑐𝑇𝐿 (𝑘) 𝛿 (𝑘) = [𝐼 − 𝑟

𝑐𝑇𝐿 (𝑘)] 𝛿 (𝑘) .

(15)

Obviously, the stability of (15) is equivalent to the consensusin (2). Then, we introduce the following lemma, which playsan important role in the stability analysis of (15).

Lemma 3 (see [16]). For an undirected graph, given theLaplacian matrix 𝐿

1(𝑘), 𝐿

2(𝑘), and a symmetric matrix


𝑄 = 𝑄𝑇, 𝐸 {(𝐿

1(𝑘) + 𝐿

2(𝑘)) 𝑄(𝐿

1(𝑘) + 𝐿

2(𝑘))} can be

calculated as follows:

𝐸 {(𝐿1(𝑘) + 𝐿

2(𝑘)) 𝑄 (𝐿

1(𝑘) + 𝐿

2(𝑘))}

= 𝐸{Ι [𝐿1(𝑘) 0

0 𝐿2(𝑘)

] Ι𝑇

𝑄Ι [𝐿1(𝑘) 0

0 𝐿2(𝑘)

] Ι𝑇

}

= Ι𝐸{[𝐿1(𝑘) 0

0 𝐿2(𝑘)

] Γ [𝐿1(𝑘) 0

0 𝐿2(𝑘)

]} Ι𝑇

= Ι {∧

𝑝 (𝐿(0)

)∧

𝑝 (𝐿(0)

) 𝐿(0)

Γ𝐿(0)

+∧

𝑝 (𝐿(0)

) (𝐼−∧

𝑝 (𝐿(0)

)) Ξ (Γ)} Ι𝑇

,

(16)

where 𝐿(0)

= [ 𝐿(1)0

0 𝐿(2)

], Ι𝑇 = [ 𝐼𝐼]2𝑛∗𝑛

,∧

𝑝 (L(0)) = [𝑝 0

0 𝑝2 ], Γ =

[𝑄 𝑄

𝑄 𝑄], and Ξ(Γ) is a function of 𝑄, defined as

Ξ (Γ)

=

𝑛

∑𝑚=1

𝑛

∑𝑞=1

[𝐸𝑇

1(𝑚,𝑞)0

0 𝐸𝑇

2(𝑚,𝑞)

] Γ [𝐸1(𝑚,𝑞) 0

0 𝐸2(𝑚,𝑞)

]

× [𝑎2

1(𝑚,𝑞)0

0 𝑎2

2(𝑚,𝑞)

]

+

𝑛

∑𝑚=1

𝑛

∑𝑞=1

[𝐸𝑇

1(𝑚,𝑚)0

0 𝐸𝑇

2(𝑚,𝑚)

] Γ [𝐸1(𝑚,𝑚)

0

0 𝐸2(𝑚,𝑚)

]

× [𝑎2

1(𝑚,𝑞)0

0 𝑎2

2(𝑚,𝑞)

]

−

𝑛

∑𝑚=1

𝑛

∑𝑞=1

sym ([𝐸𝑇

1(𝑚,𝑞)0

0 𝐸𝑇

2(𝑚,𝑞)

] Γ [𝐸1(𝑚,𝑚)

0

0 𝐸2(𝑚,𝑚)

])

× [𝑎2

1(𝑚,𝑞)0

0 𝑎2

2(𝑚,𝑞)]

+

𝑛

∑

𝑗=1

𝑛

∑

𝑚=𝑗+1

sym (2[𝐸𝑇

1(𝑗,𝑗)0

0 𝐸𝑇

2(𝑗,𝑗)

] Γ [𝐸1(𝑚,𝑚)

0

0 𝐸2(𝑚,𝑚)

]

− [𝐸𝑇

1(𝑗,𝑗)0

0 𝐸𝑇

2(𝑗,𝑗)

] Γ [𝐸1(𝑚,𝑗)

0

0 𝐸2(𝑚,𝑗)

]

− [𝐸𝑇

1(𝑗,𝑚)0

0 𝐸𝑇

2(𝑗,𝑚)

]

× Γ [𝐸1(𝑚,𝑚)

0

0 𝐸2(𝑚,𝑚)

])

× [𝑎2

1(𝑚,𝑞)0

0 𝑎2

2(𝑚,𝑞)

] .

(17)

The following theorem gives a sufficient condition on theaverage consensus of the system (2).

Theorem 4. Given the scalar 𝑟𝑐, the average consensus of the

system (2) is achieved if there exists a matrix 𝑄 > 0, such thatthe following LMI holds:

− [𝐿(1)

𝑄 + 𝑄𝐿(1)

+ 𝑝 (𝐿(2)

𝑄 + 𝑄𝐿(2)

)]

+ 𝑟𝑐𝑇Ι {∧

𝑝 (𝐿(0)

)∧

𝑝 (𝐿(0)

) 𝐿(0)

Γ𝐿(0)

+∧

𝑝 (𝐿(0)

) (𝐼−∧

𝑝 (𝐿(0)

)) Ξ (Γ)} ΙT< 0.

(18)

Proof. Construct the candidate Lyapunov function as𝑉(𝑘) =

𝛿𝑇

(𝑘)𝑄𝛿(𝑘). We have

𝐸 {Δ𝑉 (𝑘)}

= 𝐸 {𝑉 (𝑘 + 1) − 𝑉 (𝑘)}

= 𝐸 {𝛿𝑇

(𝑘 + 1)𝑄𝛿 (𝑘 + 1) − 𝛿𝑇

(𝑘) 𝑄𝛿 (𝑘)}

= 𝐸 {𝛿𝑇

(𝑘) [𝐼 − 𝑟𝑐𝑇𝐿 (𝑘)]𝑄 [𝐼 − 𝑟

𝑐𝑇𝐿 (𝑘)] 𝛿 (𝑘)

−𝛿𝑇

(𝑘) 𝑄𝛿 (𝑘)}

= 𝐸 {𝛿𝑇

(𝑘) [ − 𝑟𝑐𝑇𝐿 (𝑘)𝑄 − 𝑟

𝑐𝑇𝑄𝐿 (𝑘)

+𝑟2

𝑐𝑇𝐿 (𝑘)𝑄𝐿 (𝑘)] 𝛿 (𝑘)}

= −𝑟𝑐𝑇𝛿𝑇

(𝑘) (𝑝𝐿(1)

+ 𝑝2

𝐿(2)

)𝑄𝛿 (𝑘)

− 𝑟𝑐𝑇𝛿𝑇

(𝑘) 𝑄 (𝑝𝐿(1)

+ 𝑝2

𝐿(2)

) 𝛿 (𝑘)

+ 𝑟2

𝑐𝑇2

𝛿𝑇

(𝑘) 𝐸 {𝐿 (𝑘)𝑄𝐿 (𝑘)} 𝛿 (𝑘)

= −𝑟𝑐𝑇𝑝𝛿𝑇

(𝑘) [𝐿(1)

𝑄 + 𝑄𝐿(1)

+𝑝 (𝐿(2)

𝑄 + 𝑄𝐿(2)

)] 𝛿 (𝑘)

+ 𝑟2

𝑐𝑇2

𝛿𝑇

(𝑘) Ι {∧

𝑝 (𝐿(0)

)∧

𝑝 (𝐿(0)

) 𝐿(0)

Γ𝐿(0)

+∧

𝑝 (𝐿(0)

) (𝐼−∧

𝑝 (𝐿(0)

)) Ξ (Γ)}

× Ι𝑇

𝛿 (𝑘) .

(19)

Thus, from the Lyapunov stability theory, we know thatif 𝐸{Δ𝑉(𝑘)} is negative, then (15) is asymptotically stable.Thereby, the states of all agents will converge to their averagestate; that is, the average consensus of the system (2) isachieved.

Mathematical Problems in Engineering 5

1

2

34

5

Figure 1: Nominal communication topology and topology based onthe second-order neighbors’ information.

4. Simulations and Analyses

To illustrate the average consensus of the system (2) underthe condition of packet losses and fast convergence rate ofthe protocol based on second-order neighbors’ information,a numerical example is provided. The nominal interactiontopology 𝐺

(1) and topology only based on second-orderneighbors’ information 𝐺

(2) among five agents are shown inFigure 1.

The weights are set to unity for simplicity here. We set thecorresponding Laplacian matrices 𝐿(1) and 𝐿

(2) as follows:

𝐿(1)

=

[[[[[

[

2 −1 0 0 −1

−1 2 −1 0 0

0 −1 2 −1 0

0 0 −1 2 −1

−1 0 0 −1 2

]]]]]

]

,

𝐿(2)

=

[[[[[

[

2 0 −1 −1 0

0 2 0 −1 −1

−1 0 2 0 −1

−1 −1 0 2 0

0 −1 −1 0 2

]]]]]

]

.

(20)

We choose the sampling period as 𝑇 = 0.05 sec, controlgain as 𝑟

𝑐= 0.5, and the probability of successfully receiving

information as 𝑃 = 0.9. The initial condition is set to be𝑥(0) = [1, 2, 3, 4, 5]

𝑇, and it will be shown that the agents’states finally converge to the average value 𝛼 = Ave(𝑥(0)) =

(1+2+3+4+5)/5 = 3.Then, by solving the LMI inTheorem4,the result shows that it is feasible. Thus, consensus will beachieved. The time history of the Bernoulli variable 𝛾

𝑎𝑏(𝑘) is

shown in Figure 2. Figure 3 compares the convergence speedof the nominal communication and the topology based onsecond-order neighbors’ information with 𝑃 = 0.9, fromwhich we can see that the protocol we designed is moreeffective. Figure 4 compares the convergence speed basedon second-order neighbors’ information with 𝑃 = 0.9 and𝑃 = 0.5, fromwhich we can see the influence of packet losses.

0 5 10 15 20 25 30 35 40 45 50

0

0.5

1

1.5

Time (s)

−0.5

𝛾ab(k)

Figure 2: Time history of the Bernoulli variable 𝛾𝑎𝑏(𝑘).

0 1 2 3 4 5 6 7 8 912345

Time (s)

Stat

es

0 1 2 3 4 5 6 7 8 912345

Time (s)

Stat

es

Figure 3: Nominal communication and the topology based onsecond-order neighbors’ information with 𝑃 = 0.9.

0 1 2 3 4 5 6 7 8 912345

Time (s)

Stat

es

(a) 𝑃 = 0.9

0 1 2 3 4 5 6 7 8 912345

Time (s)

Stat

es

(b) 𝑃 = 0.5

Figure 4: Comparison of the convergence speed based on second-order neighbors’ informationwith different package loss probability.

5. Conclusions

In this paper, we have investigated the average consensusin multiagent systems with the problem of packet losseswhen second-order neighbors’ information was used. Theconvergence rates of general protocol and second-orderneighbor protocol with packet losses have been comparedand it is concluded that second-order neighbor protocolspeeds up the consensus rate. What is more, we can see


the influence of packet losses. Future work will extend theagent dynamics to second-order or higher-order dynamicswith data loss and time-varying delay.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural ScienceFoundation (NNSF) of China under Grants no. 61004031,no. 61174096, and no. 61104141, Beijing Municipal Com-mission of Education Science and Technology Program(KM201310017006), and the Fundamental Research Funds forthe Central Universities.

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