Fast second-order distributed consensus with adaptive quantization

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

penghx@niit.edu.cn

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

wangwk@niit.edu.cn

Abstract—In order to improve the accuracy and the convergence rate of distributed consensus under quantized communication, in the paper, based on adaptive quantization scheme, we propose the fast second-order distributed consensus algorithm. We analyze the convergence performance. The fast second-order distributed consensus with adaptive quantization achieves a consensus in a mean square sense, and the consensus is equal to the average of the initial states. Simultaneously, Simulations are done about the fast second-order distributed consensus based on adaptive quantization. Results show that the fast second-order distributed consensus algorithm based on adaptive quantization can reach an average consensus, and its convergence rate is higher than that of the first-order adaptive quantized distributed consensus algorithm, moreover, the mean square errors are smaller within the finite steps.

Keywords-distributed consensus; adaptive quantization; quantized communication; second-order algorithm

I. INTRODUCTION In the past few years, distributed consensus problems have

attracted much attention due to broad application in many areas. In the practical applications, for the limit bandwidth and energy constraint, the impact of the quantization errors on the distributed consensus has to be considered. The impact of communicational noise on distributed average consensus is considered firstly in [1], but the quantized error is different from the communication noise in the strict sense. The definition of quantized consensus is proposed in [2], but quantized consensus isn’t a consensus in the strict sense. A uniformly quantized distributed consensus algorithm is proposed in [3], but the uniformly quantized distributed consensus algorithm usually can’t reach a consensus. A dithered quantization scheme is proposed in [4, 5], a distributed consensus algorithm with dithered quantization can reach a consensus, but the consensus is random and not equivalent to the average of initial states of nodes. A probabilistic quantization scheme is proposed in [6]. Based on the probabilistic quantization scheme, a uniform probabilistically quantized distributed consensus algorithm is proposed in [7]. Every node based on uniformly probabilistic quantization can reach a common value, and the common value is random and not equivalent to the average of the initial states. An adaptive quantization (AQ) scheme is proposed in [8], in which the quantization step-sizes are adjusted adaptively by learning from previous states, the quantization step sizes asymptotically

decrease to zero when the values of nodes convergence to zero. For fixed topology, Distributed consensus with adaptive quantization can reach an average consensus.

The above mentioned distributed consensus algorithms with quantization communication mainly focus on whether the nodes can reach a consensus or not, and whether the accuracy of the algorithm is high or not. For large-scale complex networks, the convergence rate of the distributed consensus with quantized communication has to be considered. A fast second-order distributed consensus (the pseudo two-hop distributed consensus algorithm) is proposed in [9]. By utilizing the previous information of the non-adjacency nodes based on single-hop communication, the convergence rate of the pseudo two-hop distributed consensus algorithm is accelerated dramatically. Based on the above mentioned analyses, in order to reach an average consensus, and accelerate the convergence rate of distributed consensus with quantized communication, on the basis of adaptive quantization, we adopt the fast second-order distributed consensus algorithm (the pseudo two-hop distributed consensus algorithm) to update the state of every node. We analyze the convergence performance of the fast second-order distributed consensus with adaptive quantization.

II. PSEUDO TWO-HOP DISTRIBUTED CONSENSUS ALGORITHM

We introduce some notations and concepts that will be used through this paper. A undirected graph

1 1 1( , )G V E= represents the single-hop communication topology in a undirected networked system with N nodes, where 1V is a set of vertices, and 1E is a set of edges. An

adjacency matrix for graph 1G is denoted as 1 { }ij N NA a ×= ,

where 1

1

0 ( , )

0 ( , )ij

i j Ea

i j E

≠ ∈=

∉⎧ ⎫⎨ ⎬⎩ ⎭

. A seti

N is denoted as the set

of vertices that can send directly information to vertex i .A path is a set of vertices, in which the edge formed by adjacent vertices is an element of the set 1E . A path is an m-hop path if

it has m non-intersected edges. A graph ( , )m m mG V E= is defined as m-hop topology graph, where all elements of the edge set mE are m-hop paths of the graph 1G , and mVV =1 .

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

An adjacency matrix of mG is denoted as { }m il N NA β ×= ,

where* * ( , ) ( , )

0

ij st

mil

a a j i and and t s E

otherwiseβ

∈=⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

. A

single-hop Laplacian matrix is denoted as 1 1 1{ }ijL l A= = Δ − ,

where 1 11 1

{ ..... .}N N

j Njj j

diag a a= =

Δ = ∑ ∑ . An m-hop Laplacian

matrix is { }m ij m mL l A= = Δ − ,

where1

1 1

{ ..... .}N N

m j Nj

j j

diag β β= =

Δ = ∑ ∑ ..

The pseudo multi-hop distributed consensus algorithm in [9] can be written as:

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

i i ij i j jj N

x k x k a x k x k y kε∈

+ = − − +∑

{( 1) 1))}}j

jl i jl N

a x k x k∈

+ − − −∑ （ （

( 1) ( ( ) ( ))i

i ij i jj N

y k a x k x k∈

+ = −∑ (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. Each node transmits the information received in the previous m step to adjacent-nodes.

III. FAST SECOND DISTRIBUTED CONSENSUS ALGORITHM WITH ADAPTIVE QUANTIZATION

In the paper, we adopt the adaptive quantized scheme proposed in [8]. We consider a set of nodes of a network, each with an initial real valued scalar ],[)( UUkxi −∈ , where ni ,,1= . We define )(kiτ as the threshold in step k , and )(kiΔ as the quantized step in step k . The initial threshold and initial quantized step is respectively 0)0( =iτ and

Uiii =Δ=Δ=Δ )2()1()0( . We define ))()(sgn()( kkxkb iii τ−= , )()()()1( kkbkk iiii Δ+=+ ττ ,

and ( ) ( 1)( 1) ( ) i ib k b ki ik k K −Δ + = Δ ( 2≥k ), where the real

parameter 1>K . We define the quantized value of node i in step k as )()( kkqi τ= .

Based on adaptive quantized scheme, we adopt the pseudo two-hop algorithm to update the states of nodes. For node i , we define )(kvi as quantization error, then, the quantization error can be written by

))(()( kxQkq ii = )()()( kxkqkv iii −= (2)

Where ( )Q ⋅ denotes quantization operation. We update the state of every node by its state and the present quantized

values and the previous quantized values of the non-adjacency nodes. The fast second-order distributed consensus with adaptive quantization can be written by

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

i i ij i jj N

x k x k a q k q kε∈

+ = − −∑

( ( )) ( ( 1) ( 1)))j

j jl i jl N

n q y k a q k q kγ ε γ∈

+ + − − −∑

1

( 1) ( ( ) ( ))i

i ij i jj N

y k a q k q kn ∈

+ = −∑ ,

( ( )) ( ) ( )j j jq y k y k vy k= + . (3)

Where ( ( ))jq y k is the quantized value of the ( )jy k , and

( )jvy k is the corresponding quantization error. The collective form of (3) is written by：

1 2 1( 1) ( ) ( ) ( 1) ( )x k x k L q k L q k n L vy kε γε εγ+ = − − − −

1 2( ) ( ) ( 1)x k L x k L x kε εγ= − − −

1 2 1( ) ( 1) ( )L v k L v k n L vy kε γε εγ− − − − (4)

where 1( ) [ ( ) ( )]T

nx k x k x k= , and 1( ) [ ( ) ( )]T

nq k q k q k= .

We define 1( ) [ ( ) ( )]T

nvy k vy k vy k= ,

1( ) [ ( ) ( )]T

nv k v k v k= , and { ( )}v k , { ( )}vy k have zero mean. For simplicity, we suppose the ( )v k , ( )vy k are

independent and irrelevant, i.e. ( ( ) ( )) 0TE v i v j = ,

( ( ) ( )) 0TE vy i vy j = . ( ( ) ( )) 0TE v i vy j = . Then, (4) can be represented by

1 2( 1) ( )

0( ) ( 1)n

n

I L Lx k x k

Ix k x k

ε εγ− −+=

−

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

1 2 1( ) ( 1) ( )

0

L v k L v k n L vy kε γε εγ+ − +−⎡ ⎤⎢ ⎥⎣ ⎦

(5)

We define 1 2

0n

n

I L LH

I

ε εγ− −=⎡ ⎤⎢ ⎥⎣ ⎦

,0

0

KJ

K=⎡ ⎤⎢ ⎥⎣ ⎦

,

( 1)( 1)

( )

x kX k

x k

++ =

⎡ ⎤⎢ ⎥⎣ ⎦

，1 TKn

= 11 ,

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

0

L v k L v k n L vy kV k

ε γε εγ+ − +=⎡ ⎤⎢ ⎥⎣ ⎦

， where

[1 1]T=1 is a n dimensional vector. We rewrite (5)

( 1) ( ) ( )X k HX k V k+ = − (6)

The initial condition is (0) [ (0), ( 1)]T T TX x x= − . We have the following lemma.

Lemma1[9]. For the initial condition )1()0( −= xx , if

norm. 2|| || 1H J− < , the pseudo two-hop algorithm without distortion can reach an average consensus.

If norm. 2|| || 1H J− < , where 2|| . || is denoted by Euclidean norm. or inductive norm., matrix H has only one eigenvalue ( ) 1Hλ = , the corresponding left eigenvector

is [ , ]T T T

lw = 1 0 , and right eigenvector1

[ , ]T T T

rwn

= 1 1 . We

get 1 2lim lim0

k

nk T

r lk k

n

I L LH w w J

I

ε εγ→∞ →∞

− −= = =

⎡ ⎤⎢ ⎥⎣ ⎦

. The other

eigenvalues of matrix H stay inside the unit circle. From (6), we have:

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

X k HX k V k→∞ →∞

+ = −

1lim { (0) ( )}0

kk k jH X H V j

k jε+ −= −

→∞ =∑

0

(0) lim ( )k

k j

kj

JX H V jε −

→∞=

= − ∑

Obviously, there is any finite positive integer m , lim k m

kH J−

→∞= . So, we have:

0 1

lim ( 1) (0) lim ( ) lim ( )m k

k j k j

k k kj j m

X k JX H V j H V jε ε− −

→∞ →∞ →∞= = +

+ = − −∑ ∑

0 1

(0) ( ) lim ( )m k

k j

kj j m

JX JV j H V jε ε −

→∞= = +

= − −∑ ∑

Noting that 0T L =1 , then ( ) 0JV j = . The above equation is represented by

1

( 1)lim (0) lim ( )

( )

kk j

k kj m

x kJX H V j

x kε −

→∞ →∞= +

+= −

⎡ ⎤⎢ ⎥⎣ ⎦

∑ (7)

Obviously, in (7), the quantization errors incurred at each step before step m will eventually vanish as the system evolves over time. Suppose that the quantization errors after step 1+m converge to zero, then above iteration (7) achieves an average consensus.

Theorem1. Suppose that the quantization errors )}({ kv , { ( )}vy k have zero mean and their mean variance convergence

to zero, i.e. 2lim ( ( )) 0k

E v k→∞

= , 2lim ( ( )) 0k

E vy k→∞

= and

2|| ( || 1H J− < , then the fast second-order distributed consensus algorithm with adaptive quantization achieves a consensus in a mean square sense, i.e.

1 1lim ( ( ) (0)) ( ( ) (0)) 0T T T

kE x k x x k x

n n→∞− − =11 11 ..

Proof. From the above iteration (6), we can

get 1

0

( 1) ( ) ( ) (0) ( )k

k k i

i

X k HX k V k H X H V i+ −

=

+ = − = −∑ , then

2 1 2

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

0 0

( ( )) ( ( ))k k

k i T k i

i i

E H V i H V i− −

= =

+ ∑ ∑ .

for HJ J= , and 2J J= , then we can get 2( )( )H J H J H J− − = − , so, we have

1 1( )k kH J H J+ +− = − . Noting that 0T L =1 , then ( ) 0JV i = ,. 2 1 2

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

2

20

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

k i T

i

H J E V i V i−

=

+ −∑

1 2 2 2

2 2 20

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

k k i T

i

H J X H J E V i V i+ −

=

≤ − + −∑ ,

When 2|| || 1H J− < , Obviously, 2

2lim || ) || 0k

kH J

→∞− = ,

lim ( ( ) ( )) 0T

kE V k V k

→∞= , we have:

2 2 2 2

2 2 2lim || ( 1) (0) || lim || || || (0) ||k

k kE X k JX H J X+

→∞ →∞+ − ≤ −

2

20

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

k i T

ki

H J E V i V i−

→∞=

+ − =∑

Then, we have

1 1

lim ( ( ) (0)) ( ( ) (0)) 0T T T

kE x k x x k x

n n→∞− − =11 11

IV. SIMULATIONS AND ANALYSES We present simulation results to illustrate the performance

of our proposed the fast second-order distributed consensus algorithm with adaptive quantization. Figure 1 shows undirected regular networks with 25 nodes. We choose the step size 2.00=ε (second), the parameter 1.2K = , and weight }),(|{ Ejiaij ∈ is equivalent to 1. The state of every node is respectively a random variable between 1 and 25. After running 1000 Monte-Carlo simulations, we have the following simulating results about the fast second-order distributed consensus algorithm with adaptive quantization. The mean square errors in every iteration is defined as

2

2

1( ) || ( ) (0) ||TMSE k x k x

n= − 11 .

We can see that the mean square errors in the fast second- order adaptive quantized distributed consensus are smaller than those in the first-order adaptive quantized distributed consensus from figure 2. Figure 3 shows the performance of two distributed consensus algorithms, obviously, the

Figure 1. Undirected regular topology

0 2 4 6 8 10 12 14 1610

−3

10−2

10−1

100

101

102

103

104

adaptive quantization distributed consensus

time(s)

MS

E

first order q=1fast second order q=1

Figure 2. Comparison of MSE about two quantized distributed consensus

based on undirected regular graph.

0 5 10 15 200

10

20

30first order algorithm with q=1

Sta

tes

Time(s)

0 5 10 15 200

10

20

30fast second order algorithm with q=1

Sta

tes

Time(s) Figure 3. Comparison of the convergence rates of two quantized

consensus based on undirected regular graph.

0 10 20 30 40 50 60 70 8010

−25

10−20

10−15

10−10

10−5

100

Time(/s)

MS

E

second−order adaptive quantization

k=1.2k=1.4k=1.8

Figure 4. Comparison of MSE of the fast second-order algorithm with different K based on undirected regular graph

convergence rate in the fast second-order algorithm with quantization is bigger than that in the fast-order algorithm with quantization communication, and the average consensus can be reached in the fast second-order algorithm with quantization. Figure 4 shows the mean square errors of the fast second-order adaptive quantized distributed consensus algorithm vary with different K values, and the impact of different K on the algorithm is negligible.

V. CONCLUSIONS Under the quantization communication, the adaptive

quantization scheme can adjust adaptively the quantization step-sizes by learning from previous states. The quantization step sizes asymptotically decreases to zero when the values of nodes convergence to zero. For fixed topology, Distributed consensus with adaptive quantization can reach an average consensus in a mean square sense. By utilizing the previous quantized values of the non-adjacency nodes, the fast second-order adaptively quantized distributed consensus algorithm can reach an average consensus, and the convergence rate of the fast second-order adaptively quantized distributed consensus is improved greatly. Moreover, within the finite steps, the mean square errors of the fast second-order adaptively quantized distributed consensus algorithm are smaller than those of the first-order adaptively quantized distributed consensus algorithm.

REFERENCES [1] L. Xiao, S. Boyd, S. J. Kim, “Distributed average consensus with least

mean square deviation,” Journal of parallel and distributed computing, Vol. 67, No. 1, pp. 33-46, 2007.

[2] A. Kashyap, T. Basar, R. Srikant, “Quantized consensus,” Automatica, Vol. 43, pp. 1192-1203, 2007.

[3] M. E. Yildiz, A. Scaglione, “Differential nested lattice encoding for consensus problems”. Proceedings of the information processing in sensor networks, Cambridge, MA: IEEE, 2007, pp. 89-98.

[4] T. C. Aysal, M. J. Coates, M. G. Rabbat, “Distributed average consensus with dithered quantization,” IEEE transaction on signal processing, Vol. 56, No. 10, pp. 4905-4918, 2008.

[5] S. Kar, J. M. F. Moura, “Distributed consensus algorithms in sensor networks: quantized data and random link failure,” IEEE transaction on signal processing, Vol. 58, No. 3, pp. 1383-1400, 2010.

[6] J. J. Xiao, Z. Q. Luo, “Decentralized estimation in an inhomogeneous sensing environment,” IEEE transaction on information theory, Vol. 51, No. 10, pp. 3564-3575, 2005.

[7] T. C. Aysal, M. J. Coates, M. G. Rabbat, “Distributed average consensus using probabilistic quantization,” IEEE\SP 14th workshop on statistical signal processing. Madison, Wisconsin: IEEE, 2007, pp.640-644.

[8] Jun Fang, Hongbin Li, “An adaptive quantization scheme for distributed consensus,” in proceedings of 2009 IEEE international conference on Acoustics, Speech and Signal Processing.Taipei, Taiwan: IEEE, 2009, pp. 2777-2780.

[9] Peng Huanxin, Qi Guoqing, Sheng Andong. “Pseudo two-hop distributed consensus algorithm” Journal of Control Theory and Applications, unpublished.