Pseudo two-hop distributed consensus algorithm with communication noise

WANG Wenkai School of Mechanical Engineering, Nanjing Institute of industry Technology, Nanjing Jiangsu 210046, China

wangwk@niit.edu.cn

PENG Huanxin, School of Mechanical Engineering, Nanjing Institute of industry Technology, Nanjing Jiangsu 210046, China

penghx@niit.edu.cn,

Abstract—In this paper, we analyze the pseudo two-hop distributed consensus algorithm with communication noise. If there is communication noise among agents, the convergence performance of distributed consensus algorithms degrades. Supposing that the communication noises are zero-mean Gaussian white noise, and the noises are irrelevant and independent in time and spatial, we analyze the convergence performance of the pseudo two-hop distributed consensus algorithm with noise, and calculate mean square errors and the upper bound of mean-square error of the pseudo two-hop distributed consensus algorithm. Finally, simulation results are provided to verify these analytical results.

Keywords--distributed average consensus; communication noise; convergence performance; pseudo two-hop algorithm.

I. INTRODUCTION In the last few years, for the broad application in many

fields, such as load balancing for parallel processors, distributed coordination of mobile autonomous agents, cooperative control of unmanned aerial vehicles (UAVs), formation control, swarming\flocking, wireless sensor networks, load balance, oscillator synchronization, attitude alignment of clusters of satellites, congestion control in communication networks, and network synchronization and so on, distributed average consensus problems have attracted much attention. Distributed average consensus means that each node can reach an agreement based on the information of local nodes. In [1, 2], Olfati-Saber and Murray establish a theoretical framework for the analysis of consensus based on first order algorithms. In the past years, many distributed consensus algorithms are proposed [3-10].

Convergence rates for distributed average consensus algorithms are very important, especially for large scale complex systems. In order to accelerate the convergence rate of distributed consensus, in [10], the pseudo two-hop distributed consensus algorithm is proposed. The pseudo two-hop distributed consensus algorithm accelerates dramatically the convergence rate by utilizing the previous information of non-adjacency nodes based on single-hop communication. The convergence rate of the pseudo two-hop algorithm is much bigger than those of the other algorithms.

In the paper, we analyze the pseudo two-hop algorithm with communication noise. We calculate the upper bound of mean-square error of the pseudo two-hop algorithm. For simplicity’s sake, we suppose that the communication is zero-

mean Gaussian white noise, and irrelevant and independent in time and spatial.

The remainder of this paper is organized as follows: In Section II, we propose the pseudo two-hop distributed consensus algorithm. We explicitly analyze the convergence performance and calculate the upper bound of mean-square error in Section III. Examples and simulation results are provided in Section IV, and conclusions are summarized in Section V.

II. PSEUDO TWO-HOP DISTRIBUTED CONSENSUS ALGORITHM WITH COMMUNICATION NOISE

We introduce some notations and concepts that will be used through this paper. A graph ( , )G V E= represents the communication topology in a networked multi-agent system with N agents, where V is a set of vertices, and E is a set of edges. Each edge in the graph is denoted by ),( ji . An undirected graph is denoted by ( , )i j E∈ ⇔ ( , )j i E∈ . An adjacency matrix for graph G is denoted as NNijaA ×= }{ ,

where⎪⎩

⎪⎨⎧

⎭⎬⎫

∉∈

=EjiEji

aij ),(0),(1

. A set of vertices iN is denoted as

the set of vertices that can send directly information to

vertex i . In-degree of agent i is ∑=

=N

jjiin ai

1)(deg , and out-

degree of agent i is ∑=

=N

jijout ai

1)(deg . A path is a set of

vertices, in which the edge formed by adjacent vertices is an element of the set E . A path is an m-hop path if it has m edges. A graph is connected if any two vertices in the graph can be joined by a path. A graph is balanced if in-degree of any agent is equal to out-degree of itself. A graph )~,(~ EVG = is defined as two-hop topology graph, and all elements of the edge set E~ are two-hop paths of the graph G . An adjacency matrix of G~ is denoted as NNilA ×= }{~ β ,

where ⎪⎩

⎪⎨⎧

⎭⎬⎫∈×

=otherwise

Ejlandijaa jlijil 0

),(),(β . A Laplacian

matrix is denoted as AlL ij −Δ== }{ ,

Suppose by Precision manufacturing engineering technology research center foundation of Nanjing Institute of Industry Technology (ZK11-01-04)

978-1-4577-1600-3/12/$26.00 © 2012 IEEE

where 11 1

{ ..... .}N N

j Njj j

diag a a= =

Δ = ∑ ∑ . A two-hop Laplacian

matrix is denoted as { }ijL l A= = Δ − ,

where 11 1

{ ..... .}N N

j Njj j

diag β β= =

Δ = ∑ ∑ .

The pseudo two-hop distributed consensus algorithm in [10] can be written as:

( 1) ( ) {( ( ) ( ))i

i i ij i jj N

x k x k a x k x kε∈

+ = − −∑

{ ( ) 1) ( 1))}}j

j jl i jl N

y k a x k x k∈

+ + − − −∑ （ （

))()(()1( kxkxaky jiNj

ijii

−=+ ∑∈

(1)

where )(kxi is the state of node i in step k , )(kyi is the state derivative of node i in step k , ε is a constant step size.

For simplicity’s sake, we suppose that the communication is zero-mean Gaussian white noise, and irrelevant and independent in time and spatial. The pseudo two-hop algorithm with communication noise can be written by:

)())()({()()1( kvkxkxakxkx jNj

jiijiii

+−−=+ ∑∈

ε

{ ( ) 1) ( 1) ( 1))}}j

j jl i j jl N

y k a x k x k v k∈

+ + − − − + −∑ （ （

))()()(()1( kvkxkxaky jjiNj

ijii

+−=+ ∑∈

(2)

Where )(kv j is zero-mean square Gaussian white noise,

0))(( =kvE j , ⎪⎩

⎪⎨⎧

⎭⎬⎫

=≠

=jikji

kvkvEvi

ji )(0

))()(( 2σ, and

⎪⎩

⎪⎨⎧

⎭⎬⎫

=≠

=kmkkm

mvkvEvi

ii )(0

))()(( 2σ

III. DYNAMIC ANALYSES FOR PSEUDO TWO-HOP CONSENSUS ALGORITHM WITH COMMUNICATION NOISE

From (2), we can get an equivalent equation:

)())()({()()1( kvkxkxakxkx jNj

jiijiii

+−−=+ ∑∈

ε

{ 1) ( 1) ( 1) ( 1))}}j

jl i j j ll N

a x k x k v k v k∈

+ − − − + − + −∑ （ （ (3)

The collective dynamics of the group of nodes can be written as

1 1 1 2( 1) ( ) ( ) ( ) ( 1)x k x k L x k A v k A A v kε ε ε+ = − − − −

2 2( 1) ( 1)L x k A v kε ε− − − − (4)

from (4), we can get:

1 2( 1) ( )

( ) 0 ( 1)

x k I L L x k

x k I x k

ε ε+ − −=

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 2 1 2( ) ( 1) ( 1)

0

A v k A v k A A v kε

+ − + −−⎡ ⎤⎢ ⎥⎣ ⎦

(5)

We define:

1 2

0

I L LH

I

ε ε− −=⎡ ⎤⎢ ⎥⎣ ⎦

,( 1)

( 1)( )

x kX k

x k

++ =

⎡ ⎤⎢ ⎥⎣ ⎦

, 0

0

KJ

K=⎡ ⎤⎢ ⎥⎣ ⎦

,

and 1 2 1 2( ) ( 1) ( 1)

0( )

A v k A v k A A kk

vV =

+ − + −⎡ ⎤⎢ ⎥⎣ ⎦

, where

1 TKn

= 11 , then (5) can be written by:

( 1) ( ) ( )X k HX k V kε+ = − (6)

From (6), we can get:

1

0

( 1) (0) ( )k

k k j

j

X k H X H V jε+ −

=

+ = − ∑ (7)

We have the following lemma.

Lemma1[10]. If the spectral radius of JH − is less than 1, i.e. 1)( <− JHρ , under the condition that )1()0( −= xx , the pseudo two-hop algorithm without distortion convergences to the average of initial states. Theorem1. For the pseudo two-hop distributed consensus algorithm with communication noise in (7), when spectral radius of the matrix JH − is lees than 1, i.e., 1)( <− JHρ , then:

1

{lim( ( 1))} (0)T

kE x k x

n→∞+ = 11

Proof. When 1)( <− JHρ , then lim k

kH J

→∞= , noting that

0))(( =kvE j , we can get :

{lim( ( ( 1))} (lim( ( 1)) (0)k k

E X k E X k JX→∞ →∞

+ = + = ,

so we have:

1

{lim( ( 1))} (0)T

kE x k x

n→∞+ = 11

Remark 1. If the graph G is undirected and connected, and 1)( <− JHρ , then , the asymptotic expectation of the states

of nodes convergence to the average of initial states as system evolves over time.

In order to analyze the convergence accuracy of the pseudo two-hop distributed consensus algorithm with communication noise, we calculate the mean-square error of disagreement.

Supposing that the communication is zero-mean Gaussian white noise, and irrelevant and independent in time and spatial, from (6), we can get:

2 1 2

2 2|| ( 1) (0)) || || (0) (0) ||kE X k JX H X JX++ − = −

2

20

( || ( ) ||k

k j

j

E H V jε −

=

+ ∑ (8)

Noting that 2( )( )TH J H J H J− − = − , then, we can

get 1 1( )k kH J H J+ +− = − . Noting that 0T L =1 , then, we can

get ( ) 0JV j = , and ( ) ( ) ( )k j k jH V j H J V j− −= − . The mean-square error is equal to:

2 1 2

2 2|| ( 1) (0) || || ( ) (0)) ||kE X k JX H J X++ − = −

2

20

( (|| ( ) ( ) || )k

k j

j

E H J V j−

=

+ −∑

2 2 2 2

2 20

( ) || (0) || ( (|| ( ) ( ) || )k

k k j

j

H J X E H J V jρ + −

=

≤ − + −∑

where 2|| . || is denoted by Euclidean norm or inductive norm. Supposing that the communication noise is zero-mean Gaussian white noise, and irrelevant and independent in time and spatial, we can get:

2 2

2 20 0

( (|| ( ) ( ) || ) ( ( ( ) || ( ) || )k k

k j k j

j j

E H J V j H J E V jρ− −

= =

− ≤ −∑ ∑ 2 2

1 2 2( ( ) ( )) || || || ( ) ||TE V j V j A E v j=

2 2

1 1 2 2 2|| || || ( 1) ||)A A A E v j+ + −

The mean-square error of disagreement meets the following inequality:

2 2 2 2

2 2|| ( 1) (0) || ( ) || (0) ||kE X k JX H J Xρ ++ − ≤ −

2 22 2

1 2 1 1 2 2 max2

1 ( )(|| || || ||

1 ( ))

k H JA A A A R

H J

ρρ

+− −+ + +

− −

Where 2

max 2max{ || ( ) || }R E v k= . Based on the above mentioned analysis, we have the following result.

Theorem2. For the pseudo two-hop distributed consensus algorithm with communication noise in (7), when spectral radius of the matrix JH − is lees than 1, i.e., 1)( <− JHρ , then the upper bound of mean-square error is:

22 2 max2 2

|| ||lim( || ( 1) (0) || )

1 ( )k

A RE X k JX

H Jρ→∞+ − ≤

− −

where 22

2

2

1 2 1 1 2 2| || || | || | || | AA A A A+ +=

Proof. Based on the above mentioned analysis, we can get the results.

IV. SIMULATION RESULTS In order to verify the results of the pseudo two-hop

distributed consensus algorithm with communication noise, we test it on two different networks listed in figure 1, figure 2, denoted as 1G and 2G . Topology 1G is an undirected regular

graph, and 2G is an undirected random network with 25 nodes.

For the sake of simplicity, we assume the weights 1=ija for any links.

Figure3, 4 respectively show the convergence performance of pseudo two-hop algorithm with noise based on graph 1G

and graph 2G . Figure 5 shows the simulation results of the mean-square error of the pseudo two-hop distributed consensus algorithm under 1G with the random initial state from 1 to 12. Figure 6 show the simulation results of the mean-square error of the pseudo two-hop distributed consensus algorithm under 2G with the random initial state from 1 to 25, and step size 02.0=ε . Obviously, the convergence performance of the pseudo two-hop distributed consensus algorithm degrades when variance of communication noise increases, and the mean-square error of pseudo two-hop distributed consensus algorithm increases when variance of communication noise increases.

Figure1. Undirected regular topology graph

Figure2. Undirected random topology graph

0 100 200 300 4000

5

10

15σ2=25

Sta

tes

iterations)0 100 200 300 400

0

5

10

15σ2=9

Sta

tes

iterations)

0 100 200 300 4000

5

10

15σ2=4

Sta

tes

iterations)0 100 200 300 400

0

5

10

15σ2=1

Sta

tes

iterations)

Figure3. The convergence performance about the pseudo two-hop

distributed consensus algorithm with noise under graph 1G

0 100 200 300 4000

10

20

30σ2=25

Sta

tes

iterations0 100 200 300 400

0

10

20

30σ2=9

Sta

tes

iterations

0 100 200 300 4000

10

20

30σ2=4

Sta

tes

iterations0 100 200 300 400

0

10

20

30σ2=1

Sta

tes

iterations

Figure4. The convergence performance about the pseudo two-hop

distributed consensus algorithm with noise under graph 2G

0 2 4 6 810

0

101

102

103

σ2=1

MS

E

Time(s)0 2 4 6 8

100

101

102

103

σ2=4

MS

E

Time(s)

0 2 4 6 810

0

101

102

103

σ2=9

MS

E

Time(s)0 2 4 6 8

101

102

103

σ2=25

MS

E

Time(s)

Figure5. MSE of the consensus algorithm with noise under graph 1G

0 2 4 6 810

0

102

104

σ2=1

MS

E

Time(s)0 2 4 6 8

101

102

103

104

σ2=4

MS

E

Time(s)

0 2 4 6 810

1

102

103

104

σ2=9

MS

E

Time(s)0 2 4 6 8

102

103

104

σ2=25

MS

E

Time(s)

Figure6. MSE of the consensus algorithm with noise under graph 2G

V. CONCLUSIONS In this paper, we analyze the convergence performance of

the pseudo two-hop distributed consensus algorithm with communication noise. When there is communication noise among agents, the performance of distributed consensus algorithm degrades. We proof the expectation value of agents can reach consensus in the pseudo two-hop algorithm with noise, and we also calculate the upper bounded of mean-square error in the pseudo two-hop consensus algorithm.

REFERENCES [1] R. Olfati-Saber, R. M. Murray, “Consensus problems in networks of

agents with switching topology and time-delays,” IEEE Trans. Automat. Contr.,2004, vol. 49, no. 9, pp. 1520–1533.

[2] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and Cooperation in Networked Multi-Agent Systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007.

[3] L. Xiao, S. Boyd, “Fast linear iterations for distributed averaging,” Systems and Control Letters, vol. 53, pp. 65–78, 2004.

[4] L. Xiao, S. Boyd, and S. J. Kim, “Distributed Average Consensus with Least-Mean-Square Deviation,” Journal of Parallel and Distributed Computing, vol. 67(1):33-46, 2007.

[5] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Gossip algorithms: Design, analysis and applications,” in Proceedings of IEEE INFOCOM 2005, vol. 3, 2005.

[6] Y. Kim and M. Mesbahi, “on maximizing the second smallest Eigen value of a state-dependent graph laplacian,” IEEE Trans. Automat. Contr., vol. 51, no. 1, pp. 116–120, Jan. 2006.

[7] T. C. Aysal, B. N. Oreshkin, and M. J. Coates, “Accelerated Distributed Average Consensus via Localized Node State Prediction,” IEEE Transactions on signal processing, VOL. 57, NO. 4, APRIL 2009. pp. 1563–1576.

[8] Gang Xiong, S. Kishore, “Linear high-order distributed average consensus algorithm in wireless sensor networks,” IEEE/SP 15th Workshop on Statistical Signal Processing, 2009. pp. 529-532.

[9] Zhipu Jin, R.M. Murray, “Multi-Hop Relay Protocols for Fast Consensus Seeking,” in Proceedings of 45th IEEE Conference on Decision and Control, 2006, 1001 – 1006.

[10] Peng Huanxin, Qi Guoqing, Sheng Andong. “Pseudo multi-hop distributed consensus algorithm,” unpublished.