Research Article Average Consensus Analysis of Distributed...

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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 505848, 7 pages http://dx.doi.org/10.1155/2013/505848 Research Article Average Consensus Analysis of Distributed Inference with Uncertain Markovian Transition Probability Won Il Kim, 1,2 Rong Xiong, 1 Qiuguo Zhu, 1 and Jun Wu 1 1 State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China 2 Kimchaek Industry University, Pyongyang 999093, Democratic People’s Republic of Korea Correspondence should be addressed to Rong Xiong; [email protected] Received 18 June 2013; Revised 9 October 2013; Accepted 29 October 2013 Academic Editor: Shuli Sun Copyright © 2013 Won Il Kim et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e average consensus problem of distributed inference in a wireless sensor network under Markovian communication topology of uncertain transition probability is studied. A sufficient condition for average consensus of linear distributed inference algorithm is presented. Based on linear matrix inequalities and numerical optimization, a design method of fast distributed inference is provided. 1. Introduction During the past few decades, consensus problems of multia- gent systems by information exchange have been extensively studied by many researchers, due to their widespread appli- cations in autonomous spacecraſt, unmanned air vehicles, mobile robots, and distributed sensor networks. Olfati-Saber and Murray introduced in [1, 2] a theoretical framework for solving consensus problems. In [3, 4], consensus problems of first-order integrator systems were proposed based on algebra graph theory. In [5, 6], consensus problems of directed second-order systems were presented. In [5], the authors pro- vided necessary and sufficient condition for reaching mean square consensus of discrete-time second order systems. Consensus conditions were studied in [6] of continuous- time second order systems by Linear Matrix Inequality (LMI) approach. Among consensus problems, the average consensus prob- lem is challenging which requires distributed computation of the average of the initial state of a network [1, 2]. For a strongly connected network, [1] proved that the average consensus problem is solvable if and only if the network is balanced. e discrete-time average consensus plays a key role in distributed inference in sensor networks. In networks of fixed topology, [7] gave necessary and sufficient conditions for linear distributed inference to achieve aver- age consensus. A design method was presented in [7] to implement linear distributed inference of fastest consensus. Because of noisy communication channels, link failures oſten occur in a real network. erefore, it is meaningful to study distributed inference in networks of swing topology. rough a common Lyapunov function, a result of [1] stated that distributed inference reaches average consensus in a network of swing topology if the network holds strongly connected and balanced topology. Reference [8] modeled a network of swing topology using a Bernoulli process and established a necessary and sufficient condition for average consensus of distributed inference. e condition is related to a mean Laplacian matrix. e Bernoulli process in [8] means that the network link failure events are temporally independent. From the viewpoint of engineering, it is more reasonable to consider network link failures of temporal independence. e most famous and most tractable stochastic process of temporal independence is Markovian chain in which any future event is independent of the past events and depends only on the present event. is motivates us to model a network of swing topology using a Markovian chain and hence to study distributed inference using Markovian jump linear system method [9–12]. In practice, transition probabilities of a Markovian chain are not known precisely a priori, and only estimated values of transition probabilities are available. Hence, this paper thinks of networks with Markovian com- munication of uncertain transition probability.

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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 505848 7 pageshttpdxdoiorg1011552013505848

Research ArticleAverage Consensus Analysis of Distributed Inference withUncertain Markovian Transition Probability

Won Il Kim12 Rong Xiong1 Qiuguo Zhu1 and Jun Wu1

1 State Key Laboratory of Industrial Control Technology Zhejiang University Hangzhou 310027 China2 Kimchaek Industry University Pyongyang 999093 Democratic Peoplersquos Republic of Korea

Correspondence should be addressed to Rong Xiong rxiongiipczjueducn

Received 18 June 2013 Revised 9 October 2013 Accepted 29 October 2013

Academic Editor Shuli Sun

Copyright copy 2013 Won Il Kim et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The average consensus problem of distributed inference in a wireless sensor network underMarkovian communication topology ofuncertain transition probability is studied A sufficient condition for average consensus of linear distributed inference algorithm ispresented Based on linearmatrix inequalities andnumerical optimization a designmethod of fast distributed inference is provided

1 Introduction

During the past few decades consensus problems of multia-gent systems by information exchange have been extensivelystudied by many researchers due to their widespread appli-cations in autonomous spacecraft unmanned air vehiclesmobile robots and distributed sensor networks Olfati-Saberand Murray introduced in [1 2] a theoretical framework forsolving consensus problems In [3 4] consensus problems offirst-order integrator systemswere proposed based on algebragraph theory In [5 6] consensus problems of directedsecond-order systems were presented In [5] the authors pro-vided necessary and sufficient condition for reaching meansquare consensus of discrete-time second order systemsConsensus conditions were studied in [6] of continuous-time second order systems by LinearMatrix Inequality (LMI)approach

Among consensus problems the average consensus prob-lem is challenging which requires distributed computationof the average of the initial state of a network [1 2] Fora strongly connected network [1] proved that the averageconsensus problem is solvable if and only if the networkis balanced The discrete-time average consensus plays akey role in distributed inference in sensor networks Innetworks of fixed topology [7] gave necessary and sufficientconditions for linear distributed inference to achieve aver-age consensus A design method was presented in [7] to

implement linear distributed inference of fastest consensusBecause of noisy communication channels link failures oftenoccur in a real network Therefore it is meaningful to studydistributed inference in networks of swing topologyThrougha common Lyapunov function a result of [1] stated thatdistributed inference reaches average consensus in a networkof swing topology if the network holds strongly connectedand balanced topology Reference [8] modeled a networkof swing topology using a Bernoulli process and establisheda necessary and sufficient condition for average consensusof distributed inference The condition is related to a meanLaplacian matrix

The Bernoulli process in [8] means that the networklink failure events are temporally independent From theviewpoint of engineering it is more reasonable to considernetwork link failures of temporal independence The mostfamous and most tractable stochastic process of temporalindependence is Markovian chain in which any future eventis independent of the past events and depends only onthe present event This motivates us to model a networkof swing topology using a Markovian chain and hence tostudy distributed inference using Markovian jump linearsystem method [9ndash12] In practice transition probabilitiesof a Markovian chain are not known precisely a priori andonly estimated values of transition probabilities are availableHence this paper thinks of networks with Markovian com-munication of uncertain transition probability

2 Mathematical Problems in Engineering

In fact in the research on networked control systemsMarkovian chain has been used by several authors todescribe random communication in networks Reference[13] provided packet-loss model by Markovian chain in119867infin

networked control Under network communication ofupdate times driven by Markovian chain [14] gave stabil-ity conditions of model-based networked control systemsNetworked control systems with bounded packet losses andtransmission delays aremodeled throughMarkovian chain in[15]The networked predictive control system in [16] adopted2 Markovian chains to express date transmission in boththe controller-actuator channel and the sensor-controllerchannel

In this paperZ is used to denote the set of all nonnegativeintegersThe real identitymatrix of 119899times119899 is denoted by 119868

119899 Let 1

be the vector whose elements are all equal to 1The Euclideannorm is denoted by ∙ If a matrix 119875 is positive (negative)definite it is denoted by 119875 gt 0(lt0) The notation ⋇ withina matrix represents the symmetric term of the matrix Theexpected value is represented by 119864[∙]

The paper is organized as follows Section 2 containsa description of the network and linear distributed infer-ence Section 3 presents average consensus conditions and adesignmethodNumerical simulation results are in Section 4Finally Section 5 draws conclusions

2 Network Description

Consider distributed inference in a wireless sensor networkconsisting of 119899 sensor Each sensor 119894 isin 119873 ≜ 1 2 119899

collects a local measurement 119910119894isin R about the situation of

environment It is assumed that these local measurements1199101 1199102 119910

119899are independent and identically distributed

random variables The goal of inference is for all sensors toreach the global measurement

119910 =1

119899

119899

sum

119894=1

119910119894

(1)

such that the true situation of environment can be monitoredconvincingly

This paper studies iterative distributed inference Define

119880 = (119904 119905) | 119904 lt 119905 119904 isin 119873 119905 isin 119873 (2)

which includes all realizable undirected links in the wirelesssensor network At the 119896th iteration 119896 isin Z the successfulcommunication links in the wireless sensor network aredescribed by the set

119864 (119896) sub 119880 (3)

A pair (1199041 1199051) isin 119864(119896) means that the sensors 119904

1and 119905

1

communicate with each other at 119896 A pair (1199042 1199052) isin 119880

but (1199042 1199052) notin 119864(119896) means that there is no communication

link between the sensors 1199042and 119905

2at 119896 Due to noisy

communication channels and limited network power budget119864(119896) is assumed to be random and to be modeled as followsGiven 119898 distinct subsets 119865

1 1198652 119865

119898sub 119880 Let 120579

119896be

a stochastic process taking values in 119872 = 1 2 119898 anddriven by aMarkov chain with a transition probability matrixΓ = [120574

ℎ119897] isin R119898times119898 where 120574

ℎ119897= Pr(120579

119896+1= 119897 | 120579

119896= ℎ) for all

ℎ isin 119872 for all 119897 isin 119872 However these 120574ℎ119897s are not known

precisely Each 120574ℎ119897is expressed as

120574ℎ119897= 120574ℎ119897+ Δ120574ℎ119897 (4)

with a known 120574ℎ119897and a unknownΔ120574

ℎ119897whose absolute value is

less than a given positive constant 2120587ℎ119897 For any for all ℎ isin 119872

sum119898

119897=1120574ℎ119897= 1 and sum119898

119897=1Δ120574ℎ119897= 0 This paper models

119864 (119896) = 119865120579119896 (5)

The neighborhood of sensor 119894 at 119896 is denoted by

Ω119894(119896) = 119895 isin 119873 | (119894 119895) isin 119865

120579119896or (119895 119894) isin 119865

120579119896 (6)

and the element number of set Ω119894(119896) is denoted by 119889

119894(119896)

For sensor 119894 set its initial state 119909119894(0) = 119910

119894 At the

119896th iteration each sensor 119894 obtains its neighborsrsquo states andupdates its state using the following linear iteration law

119909119894(119896 + 1) = 119909

119894(119896) + 120572 sum

119895isinΩ119894(119896)

(119909119895(119896) minus 119909

119894(119896))

119894 isin 119873

(7)

where 120572 gt 0 is the weight parameter which is assignedby designers Our study on the above distributed inferencehas two objectives one is to derive a condition on theconvergence of 119909

119894(119896) to 119910 in the sense of mean square the

other is how to find 120572 such that a fast convergence is achieved

3 Average Consensus Analysis

31 Convergence Condition Denote

119909 (119896) =

[[[[

[

1199091(119896)

1199092(119896)

119909119899(119896)

]]]]

]

isin R119899

(8)

The system in Section 2 can be described as

119909 (119896 + 1) = 119882(120572 120579119896) 119909 (119896) 119896 isin Z

119909 (0) = [11991011199102sdot sdot sdot 119910119899]119879

(9)

where119882(120572 120579119896) is a 119899 times 119899 matrix with entries 119908

119894119895(120572 120579119896) For

119894 = 119895

119908119894119895(120572 120579119896) =

120572 when 119895 isin Ω119894(119896)

0 when 119895 notin Ω119894(119896) (10)

For 119894 = 119895

119908119894119894(120572 120579119896) = 1 minus 120572119889

119894(119896) (11)

Mathematical Problems in Engineering 3

1 2 3 4 51 2 3 4 5

678910 678910

1 2 3 4 5

678910

(1) (2) (3)

Figure 1 Three communication situations

05

04

03

02

01

0

minus01

minus02

0 01 02 03 04 05 06

120572

120588

Figure 2 Graph of 120588(120572)

0 10 20 30 40 50142

144

146

148

15

152

154

156

158

k

State

Figure 3 State curve when 120572opt = 04812

From (2)sim(6) it is seen that 119894 isin Ω119895(119896) if and only if 119895 isin Ω

119894(119896)

and hence

119882(120572 120579119896) = 119882

119879

(120572 120579119896) (12)

Furthermore (10)sim(12) imply that for all 119896 isin Z

1119879119882(120572 120579119896) = 1119879

119882 (120572 120579119896) 1 = 1

(13)

142

144

146

148

15

152

154

156

158

State

0 10 20 30 40 50k

Figure 4 State curve when 1205721= 05528

0 10 20 30 40 50142

144

146

148

15

152

154

156

158

k

State

Figure 5 State curve when 1205722= 05359

Then we have

119882(120572 120579119896) 1199101 = 119910119882(120572 120579

119896) 1 = 1199101 (14)

1

11989911119879119909 (119896 + 1) = 1

11989911119879119882(120572 120579

119896) 119909 (119896) =

1

11989911119879119909 (119896)

(15)

which means that for all 119896 isin Z1

11989911119879119909 (119896) = 1

11989911119879119909 (119896 minus 1) = sdot sdot sdot = 1

11989911119879119909 (0) = 1199101

(16)

4 Mathematical Problems in Engineering

Denote 119890(119896) = 119909(119896) minus 1199101 The iterative distributed inferenceis said to be average consensus in mean square sense iflim119896rarrinfin

119864[119890(119896)2

] = 0 for any initial condition 119909(0) isin R119899

and 1205790isin 119872

Theorem 1 The linear distributed inference (7) reaches aver-age consensus by choice of 120572 if there exist m positive definitematrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎlt 0

(17)

with 119876(120572 ℎ) = 119882(120572 ℎ) minus 111989911119879

Proof Assume 120579119896= ℎ isin 119872 at time step 119896 From (9) (14) and

(16) one has

119890 (119896 + 1) = 119909 (119896 + 1) minus 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896) + 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896)

+1

119899111198791199101

= (119882(120572 ℎ) minus1

11989911119879) (119909 (119896) minus 1199101)

= 119876 (120572 ℎ) 119890 (119896)

(18)

We now consider the stochastic Lyapunov function

119881 (119890 (119896) 120579119896) = 119890119879

(119896) 119875120579119896119890 (119896) (19)

Then for all 120579119896= ℎ isin 119872 we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) minus 119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

(120574ℎ119897+Δ120574ℎ119897) 119875119897119876 (120572 ℎ)minus119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times (

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875119897+ Δ120574ℎℎ119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1119897 = ℎ

Δ120574ℎ119897119875119897minus

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897(119875119897minus 119875ℎ))

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1

4Δ1205742

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1205872

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+119876119879

(120572 ℎ)

119898

sum

119897=1

(119875119897minus 119875ℎ)2

119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

(20)

Denote

Θℎ= 119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎ

(21)

When the conditions inTheorem 1 are satisfied we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

le minus120582min (minusΘℎ) 119890 (119896)2

le minus120578119890 (119896)2

(22)

where 120582min(minusΘℎ) denotes theminimal eigenvalue ofminusΘℎand

120578 = inf 120582min (minusΘℎ) ℎ isin 119872 (23)

Mathematical Problems in Engineering 5

Therefore for all 119890(0) isin R119899 for all 1205790isin 119872 for all 119879 isin Z

119879

sum

119896=0

119864 [119890 (119896)2

] le1

120578

119879

sum

119896=0

(119864 [119881 (119890 (119896) 120579119896)]

minus119864 [119881 (119890 (119896 + 1) 120579119896+1)])

le1

120578(119881 (119890 (0) 120579

0) minus 119864 [119881 (119890 (119879 + 1) 120579

119879+1)])

le1

120578119881 (119890 (0) 120579

0)

(24)

This means lim119896rarrinfin

119864[119890(119896)2

] = 0

32 Optimal Design From the above proof it is seen thatthe conditions in Theorem 1 result in not only lim

119896rarrinfin119864 [

119890(119896)2

] = 0 but also decreasing 119864[119881(119890(119896) 120579119896)] Moreover

lim119896rarrinfin

119864[ 119890(119896)2

] = 0 implies lim119896rarrinfin

119864[119881(119890(119896) 120579119896)] = 0

that is 119864[119881(119890(119896) 120579119896)] also converges to zero Therefore the

decrease rate of 119864[119881(119890(119896) 120579119896)] can express the convergence

speed of distributed inference The following theorem isabout the decrease rate of 119864[119881(119890(119896) 120579

119896)]

Theorem 2 Given 120572 gt 0 and 120588 isin R if there exist m positivedefinite matrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

[[[[[[[

[

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) minus (1 + 120588) 119875

ℎ⋇

(119875ℎminus 1198751) 119876 (120572 ℎ)

minus119868119898119899

(119875ℎminus 119875119898) 119876 (120572 ℎ)

]]]]]]]

]

lt 0 (25)

then in linear distributed inference (7) for any nonzero 119890(119896) isinR119899

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (26)

Proof According to Schur complement [17] condition (25)can be rewritten as

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)2

)119876 (120572 ℎ) minus (1 + 120588) 119875ℎlt 0

(27)

From (27) for any nonzero 119890(119896) isin R119899 one has

119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)

2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

lt 120588119890119879

(119896) 119875ℎ119890 (119896)

(28)

for all 120579119896= ℎ isin 119872 and for all 120579

119896+1= 119897 isin 119872 From (28) (19)

and (20) it is known that119864 [119881 (119890 (119896 + 1) 120579

119896+1| 119890 (119896) 120579

119896)]

minus 119881 (119890 (119896) 120579119896) lt 120588119881 (119890 (119896) 120579

119896)

(29)

and hence

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (30)

Condition (25) in Theorem 2 is a LMI We denote condi-tion (25) as Ξ(120572 120588) lt 0 and for any 120572 gt 0 define

120588 (120572) = inf 120588 | 120588 isin R Ξ (120572 120588) lt 0 has solutions (31)

Using the LMI toolbox of MATLAB 120588(120572) can be computedby Algorithm 1

For a 120572 gt 0with 120588 lt 0 it is known fromTheorems 1 and 2that linear distributed inference (7) is average consensus andthat 120588 is a bound of convergence speed Since a less value of120588 lt 0 gives a faster convergence speed the fast distributedinference problem is addressed as

120583 = inf120572gt0

120588 (120572) (32)

which is an unconstrainted optimization problem of only onevariable Many existing numerical optimizationmethods [18]can be utilized to solve this problem efficiently When 120583 lt 0the optimal parameter

120572opt = arginf120572gt0

120588 (120572) (33)

provides a fast linear distributed inference which reachesaverage consensus

6 Mathematical Problems in Engineering

Choose TOL gt 0Choose enough large 120588

1isin R such that Ξ(120572 120588

1) has solutions

Choose enough small 1205882isin R such that Ξ(120572 120588

2) has no solution

repeat until 1205881minus 1205882lt TOL

1205880larr (1205881+ 1205882)2

Solve Ξ(120572 1205880) lt 0

if Ξ(120572 1205880) lt 0 has solutions

1205881larr 1205880

else1205882larr 1205880

end (repeat)Set 120588(120572) larr 120588

1

Algorithm 1

4 Numerical Example

In this section we present simulation results for averageconsensus of distributed inference in a simple sensor net-work The network has 10 sensor nodes and switched inthree possible communication situations Figure 1 illustrates3 communication situations The estimated transition proba-bilities of 120579

119896is

Γ = [120574ℎ119897] = [

[

07 02 01

045 03 025

05 01 04

]

]

(34)

The estimate error Δ120574ℎ119897satisfies

1003816100381610038161003816Δ120574ℎ1198971003816100381610038161003816 le 2120587ℎ119897 = 01 forallℎ isin 1 2 3 forall119897 isin 1 2 3 (35)

Using the computation procedure in Section 3 the optimiza-tion problem (32) is solved Graph of 120588(120572) is displayed inFigure 2 The result is 120583 = minus02407 lt 0 and 120572opt = 04812For the communication situation in Figure 1(1) we use thedesign method in [7] of minimizing asymptotic convergencefactor and obtain 120572

1= 05528 The design method in [7]

is also applied to the other 2 situations in Figure 1 and get1205722= 1205723= 05359

In order to compare our method with that in [7] theinitial states of each sensor node is selected as

119909 (0) = [11991011199102119910311991041199105119910611991071199108119910911991010]119879

= [152 146 143 156 153 145 157 149 151 148]119879

(36)

Thus from (1) we have 119910 = 15 The real transition probabilitymatrix is set as

Γ = [120574ℎ119897] = [

[

068 024 008

04 027 033

053 011 036

]

]

(37)

Figures 3 4 and 5 show state curves of all sensor nodesunder 120572opt 1205721 and 1205722 respectively It can be seen that allsensor states convergence to 119910 = 15 and that our 120572opt hasfaster convergence rate than 120572

1or 1205722has Achieving faster

convergence is because our method considers the randomswitching among the 3 communication situations while [7]rsquosmethod considers only 1 communication situation

5 Conclusion

The distributed average consensus problem in sensor net-works has been studied under a Markovian switching com-munication topology of uncertain transition probabilities

Stochastic Lyapunov functions have been employed to inves-tigate average consensus of linear distributed inference Asufficient condition of average consensus has been proposedbased on feasibility of a set of coupled LMIs The designproblem of fast distributed inference has been solved bynumerical optimization techniques

Acknowledgment

This work was supported by 973 Program of China (Grant2009CB320603)

References

[1] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[2] R Olfati-Saber and R M Murray ldquoConsensus protocols fornetworks of dynamic agentsrdquo in Proceedings of the AmericanControl Conference pp 951ndash956 June 2003

Mathematical Problems in Engineering 7

[3] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[4] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[5] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[6] YHatano andMMesbahi ldquoAgreement over randomnetworksrdquoIEEE Transactions on Automatic Control vol 50 no 11 pp1867ndash1872 2005

[7] L Xiao and S Boyd ldquoFast linear iterations for distributedaveragingrdquo Systems amp Control Letters vol 53 no 1 pp 65ndash782004

[8] S Kar and J M F Moura ldquoSensor networks with random linkstopology design for distributed consensusrdquo IEEE Transactionson Signal Processing vol 56 no 7 part 2 pp 3315ndash3326 2008

[9] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991

[10] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[11] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[12] L Zhang and E-K Boukas ldquo119867infin

control for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo International Journal of Robust and NonlinearControl vol 19 no 8 pp 868ndash883 2009

[13] P Seiler and R Sengupta ldquoAn 119867infin

approach to networkedcontrolrdquo IEEE Transactions on Automatic Control vol 50 no3 pp 356ndash364 2005

[14] L A Montestruque and P Antsaklis ldquoStability of model-basednetworked control systems with time-varying transmissiontimesrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1562ndash1572 2004

[15] J Yu L Wang and M Yu ldquoSwitched system approach to sta-bilization of networked control systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 17 pp 1925ndash19462011

[16] Y Xia G-P Liu M Fu and D Rees ldquoPredictive control ofnetworked systems with random delay and data dropoutrdquo IETControl Theory and Applications vol 3 no 11 pp 1476ndash14862009

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer New York NY USA1999

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Stochastic AnalysisInternational Journal of

Page 2: Research Article Average Consensus Analysis of Distributed ...downloads.hindawi.com/journals/mpe/2013/505848.pdf · Research Article Average Consensus Analysis of Distributed Inference

2 Mathematical Problems in Engineering

In fact in the research on networked control systemsMarkovian chain has been used by several authors todescribe random communication in networks Reference[13] provided packet-loss model by Markovian chain in119867infin

networked control Under network communication ofupdate times driven by Markovian chain [14] gave stabil-ity conditions of model-based networked control systemsNetworked control systems with bounded packet losses andtransmission delays aremodeled throughMarkovian chain in[15]The networked predictive control system in [16] adopted2 Markovian chains to express date transmission in boththe controller-actuator channel and the sensor-controllerchannel

In this paperZ is used to denote the set of all nonnegativeintegersThe real identitymatrix of 119899times119899 is denoted by 119868

119899 Let 1

be the vector whose elements are all equal to 1The Euclideannorm is denoted by ∙ If a matrix 119875 is positive (negative)definite it is denoted by 119875 gt 0(lt0) The notation ⋇ withina matrix represents the symmetric term of the matrix Theexpected value is represented by 119864[∙]

The paper is organized as follows Section 2 containsa description of the network and linear distributed infer-ence Section 3 presents average consensus conditions and adesignmethodNumerical simulation results are in Section 4Finally Section 5 draws conclusions

2 Network Description

Consider distributed inference in a wireless sensor networkconsisting of 119899 sensor Each sensor 119894 isin 119873 ≜ 1 2 119899

collects a local measurement 119910119894isin R about the situation of

environment It is assumed that these local measurements1199101 1199102 119910

119899are independent and identically distributed

random variables The goal of inference is for all sensors toreach the global measurement

119910 =1

119899

119899

sum

119894=1

119910119894

(1)

such that the true situation of environment can be monitoredconvincingly

This paper studies iterative distributed inference Define

119880 = (119904 119905) | 119904 lt 119905 119904 isin 119873 119905 isin 119873 (2)

which includes all realizable undirected links in the wirelesssensor network At the 119896th iteration 119896 isin Z the successfulcommunication links in the wireless sensor network aredescribed by the set

119864 (119896) sub 119880 (3)

A pair (1199041 1199051) isin 119864(119896) means that the sensors 119904

1and 119905

1

communicate with each other at 119896 A pair (1199042 1199052) isin 119880

but (1199042 1199052) notin 119864(119896) means that there is no communication

link between the sensors 1199042and 119905

2at 119896 Due to noisy

communication channels and limited network power budget119864(119896) is assumed to be random and to be modeled as followsGiven 119898 distinct subsets 119865

1 1198652 119865

119898sub 119880 Let 120579

119896be

a stochastic process taking values in 119872 = 1 2 119898 anddriven by aMarkov chain with a transition probability matrixΓ = [120574

ℎ119897] isin R119898times119898 where 120574

ℎ119897= Pr(120579

119896+1= 119897 | 120579

119896= ℎ) for all

ℎ isin 119872 for all 119897 isin 119872 However these 120574ℎ119897s are not known

precisely Each 120574ℎ119897is expressed as

120574ℎ119897= 120574ℎ119897+ Δ120574ℎ119897 (4)

with a known 120574ℎ119897and a unknownΔ120574

ℎ119897whose absolute value is

less than a given positive constant 2120587ℎ119897 For any for all ℎ isin 119872

sum119898

119897=1120574ℎ119897= 1 and sum119898

119897=1Δ120574ℎ119897= 0 This paper models

119864 (119896) = 119865120579119896 (5)

The neighborhood of sensor 119894 at 119896 is denoted by

Ω119894(119896) = 119895 isin 119873 | (119894 119895) isin 119865

120579119896or (119895 119894) isin 119865

120579119896 (6)

and the element number of set Ω119894(119896) is denoted by 119889

119894(119896)

For sensor 119894 set its initial state 119909119894(0) = 119910

119894 At the

119896th iteration each sensor 119894 obtains its neighborsrsquo states andupdates its state using the following linear iteration law

119909119894(119896 + 1) = 119909

119894(119896) + 120572 sum

119895isinΩ119894(119896)

(119909119895(119896) minus 119909

119894(119896))

119894 isin 119873

(7)

where 120572 gt 0 is the weight parameter which is assignedby designers Our study on the above distributed inferencehas two objectives one is to derive a condition on theconvergence of 119909

119894(119896) to 119910 in the sense of mean square the

other is how to find 120572 such that a fast convergence is achieved

3 Average Consensus Analysis

31 Convergence Condition Denote

119909 (119896) =

[[[[

[

1199091(119896)

1199092(119896)

119909119899(119896)

]]]]

]

isin R119899

(8)

The system in Section 2 can be described as

119909 (119896 + 1) = 119882(120572 120579119896) 119909 (119896) 119896 isin Z

119909 (0) = [11991011199102sdot sdot sdot 119910119899]119879

(9)

where119882(120572 120579119896) is a 119899 times 119899 matrix with entries 119908

119894119895(120572 120579119896) For

119894 = 119895

119908119894119895(120572 120579119896) =

120572 when 119895 isin Ω119894(119896)

0 when 119895 notin Ω119894(119896) (10)

For 119894 = 119895

119908119894119894(120572 120579119896) = 1 minus 120572119889

119894(119896) (11)

Mathematical Problems in Engineering 3

1 2 3 4 51 2 3 4 5

678910 678910

1 2 3 4 5

678910

(1) (2) (3)

Figure 1 Three communication situations

05

04

03

02

01

0

minus01

minus02

0 01 02 03 04 05 06

120572

120588

Figure 2 Graph of 120588(120572)

0 10 20 30 40 50142

144

146

148

15

152

154

156

158

k

State

Figure 3 State curve when 120572opt = 04812

From (2)sim(6) it is seen that 119894 isin Ω119895(119896) if and only if 119895 isin Ω

119894(119896)

and hence

119882(120572 120579119896) = 119882

119879

(120572 120579119896) (12)

Furthermore (10)sim(12) imply that for all 119896 isin Z

1119879119882(120572 120579119896) = 1119879

119882 (120572 120579119896) 1 = 1

(13)

142

144

146

148

15

152

154

156

158

State

0 10 20 30 40 50k

Figure 4 State curve when 1205721= 05528

0 10 20 30 40 50142

144

146

148

15

152

154

156

158

k

State

Figure 5 State curve when 1205722= 05359

Then we have

119882(120572 120579119896) 1199101 = 119910119882(120572 120579

119896) 1 = 1199101 (14)

1

11989911119879119909 (119896 + 1) = 1

11989911119879119882(120572 120579

119896) 119909 (119896) =

1

11989911119879119909 (119896)

(15)

which means that for all 119896 isin Z1

11989911119879119909 (119896) = 1

11989911119879119909 (119896 minus 1) = sdot sdot sdot = 1

11989911119879119909 (0) = 1199101

(16)

4 Mathematical Problems in Engineering

Denote 119890(119896) = 119909(119896) minus 1199101 The iterative distributed inferenceis said to be average consensus in mean square sense iflim119896rarrinfin

119864[119890(119896)2

] = 0 for any initial condition 119909(0) isin R119899

and 1205790isin 119872

Theorem 1 The linear distributed inference (7) reaches aver-age consensus by choice of 120572 if there exist m positive definitematrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎlt 0

(17)

with 119876(120572 ℎ) = 119882(120572 ℎ) minus 111989911119879

Proof Assume 120579119896= ℎ isin 119872 at time step 119896 From (9) (14) and

(16) one has

119890 (119896 + 1) = 119909 (119896 + 1) minus 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896) + 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896)

+1

119899111198791199101

= (119882(120572 ℎ) minus1

11989911119879) (119909 (119896) minus 1199101)

= 119876 (120572 ℎ) 119890 (119896)

(18)

We now consider the stochastic Lyapunov function

119881 (119890 (119896) 120579119896) = 119890119879

(119896) 119875120579119896119890 (119896) (19)

Then for all 120579119896= ℎ isin 119872 we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) minus 119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

(120574ℎ119897+Δ120574ℎ119897) 119875119897119876 (120572 ℎ)minus119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times (

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875119897+ Δ120574ℎℎ119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1119897 = ℎ

Δ120574ℎ119897119875119897minus

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897(119875119897minus 119875ℎ))

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1

4Δ1205742

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1205872

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+119876119879

(120572 ℎ)

119898

sum

119897=1

(119875119897minus 119875ℎ)2

119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

(20)

Denote

Θℎ= 119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎ

(21)

When the conditions inTheorem 1 are satisfied we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

le minus120582min (minusΘℎ) 119890 (119896)2

le minus120578119890 (119896)2

(22)

where 120582min(minusΘℎ) denotes theminimal eigenvalue ofminusΘℎand

120578 = inf 120582min (minusΘℎ) ℎ isin 119872 (23)

Mathematical Problems in Engineering 5

Therefore for all 119890(0) isin R119899 for all 1205790isin 119872 for all 119879 isin Z

119879

sum

119896=0

119864 [119890 (119896)2

] le1

120578

119879

sum

119896=0

(119864 [119881 (119890 (119896) 120579119896)]

minus119864 [119881 (119890 (119896 + 1) 120579119896+1)])

le1

120578(119881 (119890 (0) 120579

0) minus 119864 [119881 (119890 (119879 + 1) 120579

119879+1)])

le1

120578119881 (119890 (0) 120579

0)

(24)

This means lim119896rarrinfin

119864[119890(119896)2

] = 0

32 Optimal Design From the above proof it is seen thatthe conditions in Theorem 1 result in not only lim

119896rarrinfin119864 [

119890(119896)2

] = 0 but also decreasing 119864[119881(119890(119896) 120579119896)] Moreover

lim119896rarrinfin

119864[ 119890(119896)2

] = 0 implies lim119896rarrinfin

119864[119881(119890(119896) 120579119896)] = 0

that is 119864[119881(119890(119896) 120579119896)] also converges to zero Therefore the

decrease rate of 119864[119881(119890(119896) 120579119896)] can express the convergence

speed of distributed inference The following theorem isabout the decrease rate of 119864[119881(119890(119896) 120579

119896)]

Theorem 2 Given 120572 gt 0 and 120588 isin R if there exist m positivedefinite matrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

[[[[[[[

[

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) minus (1 + 120588) 119875

ℎ⋇

(119875ℎminus 1198751) 119876 (120572 ℎ)

minus119868119898119899

(119875ℎminus 119875119898) 119876 (120572 ℎ)

]]]]]]]

]

lt 0 (25)

then in linear distributed inference (7) for any nonzero 119890(119896) isinR119899

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (26)

Proof According to Schur complement [17] condition (25)can be rewritten as

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)2

)119876 (120572 ℎ) minus (1 + 120588) 119875ℎlt 0

(27)

From (27) for any nonzero 119890(119896) isin R119899 one has

119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)

2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

lt 120588119890119879

(119896) 119875ℎ119890 (119896)

(28)

for all 120579119896= ℎ isin 119872 and for all 120579

119896+1= 119897 isin 119872 From (28) (19)

and (20) it is known that119864 [119881 (119890 (119896 + 1) 120579

119896+1| 119890 (119896) 120579

119896)]

minus 119881 (119890 (119896) 120579119896) lt 120588119881 (119890 (119896) 120579

119896)

(29)

and hence

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (30)

Condition (25) in Theorem 2 is a LMI We denote condi-tion (25) as Ξ(120572 120588) lt 0 and for any 120572 gt 0 define

120588 (120572) = inf 120588 | 120588 isin R Ξ (120572 120588) lt 0 has solutions (31)

Using the LMI toolbox of MATLAB 120588(120572) can be computedby Algorithm 1

For a 120572 gt 0with 120588 lt 0 it is known fromTheorems 1 and 2that linear distributed inference (7) is average consensus andthat 120588 is a bound of convergence speed Since a less value of120588 lt 0 gives a faster convergence speed the fast distributedinference problem is addressed as

120583 = inf120572gt0

120588 (120572) (32)

which is an unconstrainted optimization problem of only onevariable Many existing numerical optimizationmethods [18]can be utilized to solve this problem efficiently When 120583 lt 0the optimal parameter

120572opt = arginf120572gt0

120588 (120572) (33)

provides a fast linear distributed inference which reachesaverage consensus

6 Mathematical Problems in Engineering

Choose TOL gt 0Choose enough large 120588

1isin R such that Ξ(120572 120588

1) has solutions

Choose enough small 1205882isin R such that Ξ(120572 120588

2) has no solution

repeat until 1205881minus 1205882lt TOL

1205880larr (1205881+ 1205882)2

Solve Ξ(120572 1205880) lt 0

if Ξ(120572 1205880) lt 0 has solutions

1205881larr 1205880

else1205882larr 1205880

end (repeat)Set 120588(120572) larr 120588

1

Algorithm 1

4 Numerical Example

In this section we present simulation results for averageconsensus of distributed inference in a simple sensor net-work The network has 10 sensor nodes and switched inthree possible communication situations Figure 1 illustrates3 communication situations The estimated transition proba-bilities of 120579

119896is

Γ = [120574ℎ119897] = [

[

07 02 01

045 03 025

05 01 04

]

]

(34)

The estimate error Δ120574ℎ119897satisfies

1003816100381610038161003816Δ120574ℎ1198971003816100381610038161003816 le 2120587ℎ119897 = 01 forallℎ isin 1 2 3 forall119897 isin 1 2 3 (35)

Using the computation procedure in Section 3 the optimiza-tion problem (32) is solved Graph of 120588(120572) is displayed inFigure 2 The result is 120583 = minus02407 lt 0 and 120572opt = 04812For the communication situation in Figure 1(1) we use thedesign method in [7] of minimizing asymptotic convergencefactor and obtain 120572

1= 05528 The design method in [7]

is also applied to the other 2 situations in Figure 1 and get1205722= 1205723= 05359

In order to compare our method with that in [7] theinitial states of each sensor node is selected as

119909 (0) = [11991011199102119910311991041199105119910611991071199108119910911991010]119879

= [152 146 143 156 153 145 157 149 151 148]119879

(36)

Thus from (1) we have 119910 = 15 The real transition probabilitymatrix is set as

Γ = [120574ℎ119897] = [

[

068 024 008

04 027 033

053 011 036

]

]

(37)

Figures 3 4 and 5 show state curves of all sensor nodesunder 120572opt 1205721 and 1205722 respectively It can be seen that allsensor states convergence to 119910 = 15 and that our 120572opt hasfaster convergence rate than 120572

1or 1205722has Achieving faster

convergence is because our method considers the randomswitching among the 3 communication situations while [7]rsquosmethod considers only 1 communication situation

5 Conclusion

The distributed average consensus problem in sensor net-works has been studied under a Markovian switching com-munication topology of uncertain transition probabilities

Stochastic Lyapunov functions have been employed to inves-tigate average consensus of linear distributed inference Asufficient condition of average consensus has been proposedbased on feasibility of a set of coupled LMIs The designproblem of fast distributed inference has been solved bynumerical optimization techniques

Acknowledgment

This work was supported by 973 Program of China (Grant2009CB320603)

References

[1] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[2] R Olfati-Saber and R M Murray ldquoConsensus protocols fornetworks of dynamic agentsrdquo in Proceedings of the AmericanControl Conference pp 951ndash956 June 2003

Mathematical Problems in Engineering 7

[3] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[4] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[5] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[6] YHatano andMMesbahi ldquoAgreement over randomnetworksrdquoIEEE Transactions on Automatic Control vol 50 no 11 pp1867ndash1872 2005

[7] L Xiao and S Boyd ldquoFast linear iterations for distributedaveragingrdquo Systems amp Control Letters vol 53 no 1 pp 65ndash782004

[8] S Kar and J M F Moura ldquoSensor networks with random linkstopology design for distributed consensusrdquo IEEE Transactionson Signal Processing vol 56 no 7 part 2 pp 3315ndash3326 2008

[9] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991

[10] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[11] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[12] L Zhang and E-K Boukas ldquo119867infin

control for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo International Journal of Robust and NonlinearControl vol 19 no 8 pp 868ndash883 2009

[13] P Seiler and R Sengupta ldquoAn 119867infin

approach to networkedcontrolrdquo IEEE Transactions on Automatic Control vol 50 no3 pp 356ndash364 2005

[14] L A Montestruque and P Antsaklis ldquoStability of model-basednetworked control systems with time-varying transmissiontimesrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1562ndash1572 2004

[15] J Yu L Wang and M Yu ldquoSwitched system approach to sta-bilization of networked control systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 17 pp 1925ndash19462011

[16] Y Xia G-P Liu M Fu and D Rees ldquoPredictive control ofnetworked systems with random delay and data dropoutrdquo IETControl Theory and Applications vol 3 no 11 pp 1476ndash14862009

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer New York NY USA1999

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Stochastic AnalysisInternational Journal of

Page 3: Research Article Average Consensus Analysis of Distributed ...downloads.hindawi.com/journals/mpe/2013/505848.pdf · Research Article Average Consensus Analysis of Distributed Inference

Mathematical Problems in Engineering 3

1 2 3 4 51 2 3 4 5

678910 678910

1 2 3 4 5

678910

(1) (2) (3)

Figure 1 Three communication situations

05

04

03

02

01

0

minus01

minus02

0 01 02 03 04 05 06

120572

120588

Figure 2 Graph of 120588(120572)

0 10 20 30 40 50142

144

146

148

15

152

154

156

158

k

State

Figure 3 State curve when 120572opt = 04812

From (2)sim(6) it is seen that 119894 isin Ω119895(119896) if and only if 119895 isin Ω

119894(119896)

and hence

119882(120572 120579119896) = 119882

119879

(120572 120579119896) (12)

Furthermore (10)sim(12) imply that for all 119896 isin Z

1119879119882(120572 120579119896) = 1119879

119882 (120572 120579119896) 1 = 1

(13)

142

144

146

148

15

152

154

156

158

State

0 10 20 30 40 50k

Figure 4 State curve when 1205721= 05528

0 10 20 30 40 50142

144

146

148

15

152

154

156

158

k

State

Figure 5 State curve when 1205722= 05359

Then we have

119882(120572 120579119896) 1199101 = 119910119882(120572 120579

119896) 1 = 1199101 (14)

1

11989911119879119909 (119896 + 1) = 1

11989911119879119882(120572 120579

119896) 119909 (119896) =

1

11989911119879119909 (119896)

(15)

which means that for all 119896 isin Z1

11989911119879119909 (119896) = 1

11989911119879119909 (119896 minus 1) = sdot sdot sdot = 1

11989911119879119909 (0) = 1199101

(16)

4 Mathematical Problems in Engineering

Denote 119890(119896) = 119909(119896) minus 1199101 The iterative distributed inferenceis said to be average consensus in mean square sense iflim119896rarrinfin

119864[119890(119896)2

] = 0 for any initial condition 119909(0) isin R119899

and 1205790isin 119872

Theorem 1 The linear distributed inference (7) reaches aver-age consensus by choice of 120572 if there exist m positive definitematrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎlt 0

(17)

with 119876(120572 ℎ) = 119882(120572 ℎ) minus 111989911119879

Proof Assume 120579119896= ℎ isin 119872 at time step 119896 From (9) (14) and

(16) one has

119890 (119896 + 1) = 119909 (119896 + 1) minus 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896) + 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896)

+1

119899111198791199101

= (119882(120572 ℎ) minus1

11989911119879) (119909 (119896) minus 1199101)

= 119876 (120572 ℎ) 119890 (119896)

(18)

We now consider the stochastic Lyapunov function

119881 (119890 (119896) 120579119896) = 119890119879

(119896) 119875120579119896119890 (119896) (19)

Then for all 120579119896= ℎ isin 119872 we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) minus 119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

(120574ℎ119897+Δ120574ℎ119897) 119875119897119876 (120572 ℎ)minus119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times (

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875119897+ Δ120574ℎℎ119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1119897 = ℎ

Δ120574ℎ119897119875119897minus

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897(119875119897minus 119875ℎ))

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1

4Δ1205742

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1205872

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+119876119879

(120572 ℎ)

119898

sum

119897=1

(119875119897minus 119875ℎ)2

119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

(20)

Denote

Θℎ= 119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎ

(21)

When the conditions inTheorem 1 are satisfied we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

le minus120582min (minusΘℎ) 119890 (119896)2

le minus120578119890 (119896)2

(22)

where 120582min(minusΘℎ) denotes theminimal eigenvalue ofminusΘℎand

120578 = inf 120582min (minusΘℎ) ℎ isin 119872 (23)

Mathematical Problems in Engineering 5

Therefore for all 119890(0) isin R119899 for all 1205790isin 119872 for all 119879 isin Z

119879

sum

119896=0

119864 [119890 (119896)2

] le1

120578

119879

sum

119896=0

(119864 [119881 (119890 (119896) 120579119896)]

minus119864 [119881 (119890 (119896 + 1) 120579119896+1)])

le1

120578(119881 (119890 (0) 120579

0) minus 119864 [119881 (119890 (119879 + 1) 120579

119879+1)])

le1

120578119881 (119890 (0) 120579

0)

(24)

This means lim119896rarrinfin

119864[119890(119896)2

] = 0

32 Optimal Design From the above proof it is seen thatthe conditions in Theorem 1 result in not only lim

119896rarrinfin119864 [

119890(119896)2

] = 0 but also decreasing 119864[119881(119890(119896) 120579119896)] Moreover

lim119896rarrinfin

119864[ 119890(119896)2

] = 0 implies lim119896rarrinfin

119864[119881(119890(119896) 120579119896)] = 0

that is 119864[119881(119890(119896) 120579119896)] also converges to zero Therefore the

decrease rate of 119864[119881(119890(119896) 120579119896)] can express the convergence

speed of distributed inference The following theorem isabout the decrease rate of 119864[119881(119890(119896) 120579

119896)]

Theorem 2 Given 120572 gt 0 and 120588 isin R if there exist m positivedefinite matrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

[[[[[[[

[

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) minus (1 + 120588) 119875

ℎ⋇

(119875ℎminus 1198751) 119876 (120572 ℎ)

minus119868119898119899

(119875ℎminus 119875119898) 119876 (120572 ℎ)

]]]]]]]

]

lt 0 (25)

then in linear distributed inference (7) for any nonzero 119890(119896) isinR119899

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (26)

Proof According to Schur complement [17] condition (25)can be rewritten as

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)2

)119876 (120572 ℎ) minus (1 + 120588) 119875ℎlt 0

(27)

From (27) for any nonzero 119890(119896) isin R119899 one has

119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)

2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

lt 120588119890119879

(119896) 119875ℎ119890 (119896)

(28)

for all 120579119896= ℎ isin 119872 and for all 120579

119896+1= 119897 isin 119872 From (28) (19)

and (20) it is known that119864 [119881 (119890 (119896 + 1) 120579

119896+1| 119890 (119896) 120579

119896)]

minus 119881 (119890 (119896) 120579119896) lt 120588119881 (119890 (119896) 120579

119896)

(29)

and hence

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (30)

Condition (25) in Theorem 2 is a LMI We denote condi-tion (25) as Ξ(120572 120588) lt 0 and for any 120572 gt 0 define

120588 (120572) = inf 120588 | 120588 isin R Ξ (120572 120588) lt 0 has solutions (31)

Using the LMI toolbox of MATLAB 120588(120572) can be computedby Algorithm 1

For a 120572 gt 0with 120588 lt 0 it is known fromTheorems 1 and 2that linear distributed inference (7) is average consensus andthat 120588 is a bound of convergence speed Since a less value of120588 lt 0 gives a faster convergence speed the fast distributedinference problem is addressed as

120583 = inf120572gt0

120588 (120572) (32)

which is an unconstrainted optimization problem of only onevariable Many existing numerical optimizationmethods [18]can be utilized to solve this problem efficiently When 120583 lt 0the optimal parameter

120572opt = arginf120572gt0

120588 (120572) (33)

provides a fast linear distributed inference which reachesaverage consensus

6 Mathematical Problems in Engineering

Choose TOL gt 0Choose enough large 120588

1isin R such that Ξ(120572 120588

1) has solutions

Choose enough small 1205882isin R such that Ξ(120572 120588

2) has no solution

repeat until 1205881minus 1205882lt TOL

1205880larr (1205881+ 1205882)2

Solve Ξ(120572 1205880) lt 0

if Ξ(120572 1205880) lt 0 has solutions

1205881larr 1205880

else1205882larr 1205880

end (repeat)Set 120588(120572) larr 120588

1

Algorithm 1

4 Numerical Example

In this section we present simulation results for averageconsensus of distributed inference in a simple sensor net-work The network has 10 sensor nodes and switched inthree possible communication situations Figure 1 illustrates3 communication situations The estimated transition proba-bilities of 120579

119896is

Γ = [120574ℎ119897] = [

[

07 02 01

045 03 025

05 01 04

]

]

(34)

The estimate error Δ120574ℎ119897satisfies

1003816100381610038161003816Δ120574ℎ1198971003816100381610038161003816 le 2120587ℎ119897 = 01 forallℎ isin 1 2 3 forall119897 isin 1 2 3 (35)

Using the computation procedure in Section 3 the optimiza-tion problem (32) is solved Graph of 120588(120572) is displayed inFigure 2 The result is 120583 = minus02407 lt 0 and 120572opt = 04812For the communication situation in Figure 1(1) we use thedesign method in [7] of minimizing asymptotic convergencefactor and obtain 120572

1= 05528 The design method in [7]

is also applied to the other 2 situations in Figure 1 and get1205722= 1205723= 05359

In order to compare our method with that in [7] theinitial states of each sensor node is selected as

119909 (0) = [11991011199102119910311991041199105119910611991071199108119910911991010]119879

= [152 146 143 156 153 145 157 149 151 148]119879

(36)

Thus from (1) we have 119910 = 15 The real transition probabilitymatrix is set as

Γ = [120574ℎ119897] = [

[

068 024 008

04 027 033

053 011 036

]

]

(37)

Figures 3 4 and 5 show state curves of all sensor nodesunder 120572opt 1205721 and 1205722 respectively It can be seen that allsensor states convergence to 119910 = 15 and that our 120572opt hasfaster convergence rate than 120572

1or 1205722has Achieving faster

convergence is because our method considers the randomswitching among the 3 communication situations while [7]rsquosmethod considers only 1 communication situation

5 Conclusion

The distributed average consensus problem in sensor net-works has been studied under a Markovian switching com-munication topology of uncertain transition probabilities

Stochastic Lyapunov functions have been employed to inves-tigate average consensus of linear distributed inference Asufficient condition of average consensus has been proposedbased on feasibility of a set of coupled LMIs The designproblem of fast distributed inference has been solved bynumerical optimization techniques

Acknowledgment

This work was supported by 973 Program of China (Grant2009CB320603)

References

[1] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[2] R Olfati-Saber and R M Murray ldquoConsensus protocols fornetworks of dynamic agentsrdquo in Proceedings of the AmericanControl Conference pp 951ndash956 June 2003

Mathematical Problems in Engineering 7

[3] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[4] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[5] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[6] YHatano andMMesbahi ldquoAgreement over randomnetworksrdquoIEEE Transactions on Automatic Control vol 50 no 11 pp1867ndash1872 2005

[7] L Xiao and S Boyd ldquoFast linear iterations for distributedaveragingrdquo Systems amp Control Letters vol 53 no 1 pp 65ndash782004

[8] S Kar and J M F Moura ldquoSensor networks with random linkstopology design for distributed consensusrdquo IEEE Transactionson Signal Processing vol 56 no 7 part 2 pp 3315ndash3326 2008

[9] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991

[10] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[11] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[12] L Zhang and E-K Boukas ldquo119867infin

control for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo International Journal of Robust and NonlinearControl vol 19 no 8 pp 868ndash883 2009

[13] P Seiler and R Sengupta ldquoAn 119867infin

approach to networkedcontrolrdquo IEEE Transactions on Automatic Control vol 50 no3 pp 356ndash364 2005

[14] L A Montestruque and P Antsaklis ldquoStability of model-basednetworked control systems with time-varying transmissiontimesrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1562ndash1572 2004

[15] J Yu L Wang and M Yu ldquoSwitched system approach to sta-bilization of networked control systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 17 pp 1925ndash19462011

[16] Y Xia G-P Liu M Fu and D Rees ldquoPredictive control ofnetworked systems with random delay and data dropoutrdquo IETControl Theory and Applications vol 3 no 11 pp 1476ndash14862009

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer New York NY USA1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Average Consensus Analysis of Distributed ...downloads.hindawi.com/journals/mpe/2013/505848.pdf · Research Article Average Consensus Analysis of Distributed Inference

4 Mathematical Problems in Engineering

Denote 119890(119896) = 119909(119896) minus 1199101 The iterative distributed inferenceis said to be average consensus in mean square sense iflim119896rarrinfin

119864[119890(119896)2

] = 0 for any initial condition 119909(0) isin R119899

and 1205790isin 119872

Theorem 1 The linear distributed inference (7) reaches aver-age consensus by choice of 120572 if there exist m positive definitematrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎlt 0

(17)

with 119876(120572 ℎ) = 119882(120572 ℎ) minus 111989911119879

Proof Assume 120579119896= ℎ isin 119872 at time step 119896 From (9) (14) and

(16) one has

119890 (119896 + 1) = 119909 (119896 + 1) minus 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896) + 1199101

= 119882(120572 ℎ) 119909 (119896) minus 119882 (120572 ℎ) 1199101 minus 111989911119879119909 (119896)

+1

119899111198791199101

= (119882(120572 ℎ) minus1

11989911119879) (119909 (119896) minus 1199101)

= 119876 (120572 ℎ) 119890 (119896)

(18)

We now consider the stochastic Lyapunov function

119881 (119890 (119896) 120579119896) = 119890119879

(119896) 119875120579119896119890 (119896) (19)

Then for all 120579119896= ℎ isin 119872 we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) minus 119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

(120574ℎ119897+Δ120574ℎ119897) 119875119897119876 (120572 ℎ)minus119875

ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ) + 119876

119879

(120572 ℎ)

times (

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875119897+ Δ120574ℎℎ119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1119897 = ℎ

Δ120574ℎ119897119875119897minus

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897119875ℎ)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

= 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)(

119898

sum

119897=1 119897 = ℎ

Δ120574ℎ119897(119875119897minus 119875ℎ))

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1

4Δ1205742

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)

119898

sum

119897=1

120574ℎ119897119875119897119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1 119897 = ℎ

(1205872

ℎ119897119868119899+ (119875119897minus 119875ℎ)2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

le 119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+119876119879

(120572 ℎ)

119898

sum

119897=1

(119875119897minus 119875ℎ)2

119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

(20)

Denote

Θℎ= 119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ)

+ 119876119879

(120572 ℎ)

119898

sum

119897=1

(119875ℎminus 119875119897)2

119876 (120572 ℎ) minus 119875ℎ

(21)

When the conditions inTheorem 1 are satisfied we have

119864 [119881 (119890 (119896 + 1) 120579119896+1| 119890 (119896) 120579

119896)] minus 119881 (119890 (119896) 120579

119896)

le minus120582min (minusΘℎ) 119890 (119896)2

le minus120578119890 (119896)2

(22)

where 120582min(minusΘℎ) denotes theminimal eigenvalue ofminusΘℎand

120578 = inf 120582min (minusΘℎ) ℎ isin 119872 (23)

Mathematical Problems in Engineering 5

Therefore for all 119890(0) isin R119899 for all 1205790isin 119872 for all 119879 isin Z

119879

sum

119896=0

119864 [119890 (119896)2

] le1

120578

119879

sum

119896=0

(119864 [119881 (119890 (119896) 120579119896)]

minus119864 [119881 (119890 (119896 + 1) 120579119896+1)])

le1

120578(119881 (119890 (0) 120579

0) minus 119864 [119881 (119890 (119879 + 1) 120579

119879+1)])

le1

120578119881 (119890 (0) 120579

0)

(24)

This means lim119896rarrinfin

119864[119890(119896)2

] = 0

32 Optimal Design From the above proof it is seen thatthe conditions in Theorem 1 result in not only lim

119896rarrinfin119864 [

119890(119896)2

] = 0 but also decreasing 119864[119881(119890(119896) 120579119896)] Moreover

lim119896rarrinfin

119864[ 119890(119896)2

] = 0 implies lim119896rarrinfin

119864[119881(119890(119896) 120579119896)] = 0

that is 119864[119881(119890(119896) 120579119896)] also converges to zero Therefore the

decrease rate of 119864[119881(119890(119896) 120579119896)] can express the convergence

speed of distributed inference The following theorem isabout the decrease rate of 119864[119881(119890(119896) 120579

119896)]

Theorem 2 Given 120572 gt 0 and 120588 isin R if there exist m positivedefinite matrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

[[[[[[[

[

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) minus (1 + 120588) 119875

ℎ⋇

(119875ℎminus 1198751) 119876 (120572 ℎ)

minus119868119898119899

(119875ℎminus 119875119898) 119876 (120572 ℎ)

]]]]]]]

]

lt 0 (25)

then in linear distributed inference (7) for any nonzero 119890(119896) isinR119899

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (26)

Proof According to Schur complement [17] condition (25)can be rewritten as

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)2

)119876 (120572 ℎ) minus (1 + 120588) 119875ℎlt 0

(27)

From (27) for any nonzero 119890(119896) isin R119899 one has

119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)

2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

lt 120588119890119879

(119896) 119875ℎ119890 (119896)

(28)

for all 120579119896= ℎ isin 119872 and for all 120579

119896+1= 119897 isin 119872 From (28) (19)

and (20) it is known that119864 [119881 (119890 (119896 + 1) 120579

119896+1| 119890 (119896) 120579

119896)]

minus 119881 (119890 (119896) 120579119896) lt 120588119881 (119890 (119896) 120579

119896)

(29)

and hence

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (30)

Condition (25) in Theorem 2 is a LMI We denote condi-tion (25) as Ξ(120572 120588) lt 0 and for any 120572 gt 0 define

120588 (120572) = inf 120588 | 120588 isin R Ξ (120572 120588) lt 0 has solutions (31)

Using the LMI toolbox of MATLAB 120588(120572) can be computedby Algorithm 1

For a 120572 gt 0with 120588 lt 0 it is known fromTheorems 1 and 2that linear distributed inference (7) is average consensus andthat 120588 is a bound of convergence speed Since a less value of120588 lt 0 gives a faster convergence speed the fast distributedinference problem is addressed as

120583 = inf120572gt0

120588 (120572) (32)

which is an unconstrainted optimization problem of only onevariable Many existing numerical optimizationmethods [18]can be utilized to solve this problem efficiently When 120583 lt 0the optimal parameter

120572opt = arginf120572gt0

120588 (120572) (33)

provides a fast linear distributed inference which reachesaverage consensus

6 Mathematical Problems in Engineering

Choose TOL gt 0Choose enough large 120588

1isin R such that Ξ(120572 120588

1) has solutions

Choose enough small 1205882isin R such that Ξ(120572 120588

2) has no solution

repeat until 1205881minus 1205882lt TOL

1205880larr (1205881+ 1205882)2

Solve Ξ(120572 1205880) lt 0

if Ξ(120572 1205880) lt 0 has solutions

1205881larr 1205880

else1205882larr 1205880

end (repeat)Set 120588(120572) larr 120588

1

Algorithm 1

4 Numerical Example

In this section we present simulation results for averageconsensus of distributed inference in a simple sensor net-work The network has 10 sensor nodes and switched inthree possible communication situations Figure 1 illustrates3 communication situations The estimated transition proba-bilities of 120579

119896is

Γ = [120574ℎ119897] = [

[

07 02 01

045 03 025

05 01 04

]

]

(34)

The estimate error Δ120574ℎ119897satisfies

1003816100381610038161003816Δ120574ℎ1198971003816100381610038161003816 le 2120587ℎ119897 = 01 forallℎ isin 1 2 3 forall119897 isin 1 2 3 (35)

Using the computation procedure in Section 3 the optimiza-tion problem (32) is solved Graph of 120588(120572) is displayed inFigure 2 The result is 120583 = minus02407 lt 0 and 120572opt = 04812For the communication situation in Figure 1(1) we use thedesign method in [7] of minimizing asymptotic convergencefactor and obtain 120572

1= 05528 The design method in [7]

is also applied to the other 2 situations in Figure 1 and get1205722= 1205723= 05359

In order to compare our method with that in [7] theinitial states of each sensor node is selected as

119909 (0) = [11991011199102119910311991041199105119910611991071199108119910911991010]119879

= [152 146 143 156 153 145 157 149 151 148]119879

(36)

Thus from (1) we have 119910 = 15 The real transition probabilitymatrix is set as

Γ = [120574ℎ119897] = [

[

068 024 008

04 027 033

053 011 036

]

]

(37)

Figures 3 4 and 5 show state curves of all sensor nodesunder 120572opt 1205721 and 1205722 respectively It can be seen that allsensor states convergence to 119910 = 15 and that our 120572opt hasfaster convergence rate than 120572

1or 1205722has Achieving faster

convergence is because our method considers the randomswitching among the 3 communication situations while [7]rsquosmethod considers only 1 communication situation

5 Conclusion

The distributed average consensus problem in sensor net-works has been studied under a Markovian switching com-munication topology of uncertain transition probabilities

Stochastic Lyapunov functions have been employed to inves-tigate average consensus of linear distributed inference Asufficient condition of average consensus has been proposedbased on feasibility of a set of coupled LMIs The designproblem of fast distributed inference has been solved bynumerical optimization techniques

Acknowledgment

This work was supported by 973 Program of China (Grant2009CB320603)

References

[1] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[2] R Olfati-Saber and R M Murray ldquoConsensus protocols fornetworks of dynamic agentsrdquo in Proceedings of the AmericanControl Conference pp 951ndash956 June 2003

Mathematical Problems in Engineering 7

[3] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[4] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[5] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[6] YHatano andMMesbahi ldquoAgreement over randomnetworksrdquoIEEE Transactions on Automatic Control vol 50 no 11 pp1867ndash1872 2005

[7] L Xiao and S Boyd ldquoFast linear iterations for distributedaveragingrdquo Systems amp Control Letters vol 53 no 1 pp 65ndash782004

[8] S Kar and J M F Moura ldquoSensor networks with random linkstopology design for distributed consensusrdquo IEEE Transactionson Signal Processing vol 56 no 7 part 2 pp 3315ndash3326 2008

[9] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991

[10] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[11] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[12] L Zhang and E-K Boukas ldquo119867infin

control for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo International Journal of Robust and NonlinearControl vol 19 no 8 pp 868ndash883 2009

[13] P Seiler and R Sengupta ldquoAn 119867infin

approach to networkedcontrolrdquo IEEE Transactions on Automatic Control vol 50 no3 pp 356ndash364 2005

[14] L A Montestruque and P Antsaklis ldquoStability of model-basednetworked control systems with time-varying transmissiontimesrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1562ndash1572 2004

[15] J Yu L Wang and M Yu ldquoSwitched system approach to sta-bilization of networked control systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 17 pp 1925ndash19462011

[16] Y Xia G-P Liu M Fu and D Rees ldquoPredictive control ofnetworked systems with random delay and data dropoutrdquo IETControl Theory and Applications vol 3 no 11 pp 1476ndash14862009

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer New York NY USA1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Average Consensus Analysis of Distributed ...downloads.hindawi.com/journals/mpe/2013/505848.pdf · Research Article Average Consensus Analysis of Distributed Inference

Mathematical Problems in Engineering 5

Therefore for all 119890(0) isin R119899 for all 1205790isin 119872 for all 119879 isin Z

119879

sum

119896=0

119864 [119890 (119896)2

] le1

120578

119879

sum

119896=0

(119864 [119881 (119890 (119896) 120579119896)]

minus119864 [119881 (119890 (119896 + 1) 120579119896+1)])

le1

120578(119881 (119890 (0) 120579

0) minus 119864 [119881 (119890 (119879 + 1) 120579

119879+1)])

le1

120578119881 (119890 (0) 120579

0)

(24)

This means lim119896rarrinfin

119864[119890(119896)2

] = 0

32 Optimal Design From the above proof it is seen thatthe conditions in Theorem 1 result in not only lim

119896rarrinfin119864 [

119890(119896)2

] = 0 but also decreasing 119864[119881(119890(119896) 120579119896)] Moreover

lim119896rarrinfin

119864[ 119890(119896)2

] = 0 implies lim119896rarrinfin

119864[119881(119890(119896) 120579119896)] = 0

that is 119864[119881(119890(119896) 120579119896)] also converges to zero Therefore the

decrease rate of 119864[119881(119890(119896) 120579119896)] can express the convergence

speed of distributed inference The following theorem isabout the decrease rate of 119864[119881(119890(119896) 120579

119896)]

Theorem 2 Given 120572 gt 0 and 120588 isin R if there exist m positivedefinite matrices 119875

1 1198752 119875

119898isin R119899times119899 such that for all ℎ isin 119872

[[[[[[[

[

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1 119897 = ℎ

1205872

ℎ119897119868119899)119876 (120572 ℎ) minus (1 + 120588) 119875

ℎ⋇

(119875ℎminus 1198751) 119876 (120572 ℎ)

minus119868119898119899

(119875ℎminus 119875119898) 119876 (120572 ℎ)

]]]]]]]

]

lt 0 (25)

then in linear distributed inference (7) for any nonzero 119890(119896) isinR119899

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (26)

Proof According to Schur complement [17] condition (25)can be rewritten as

119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)2

)119876 (120572 ℎ) minus (1 + 120588) 119875ℎlt 0

(27)

From (27) for any nonzero 119890(119896) isin R119899 one has

119890119879

(119896)(119876119879

(120572 ℎ)(

119898

sum

119897=1

120574ℎ119897119875119897+

119898

sum

119897=1119897 = ℎ

1205872

ℎ119897119868119899

+

119898

sum

119897=1

(119875ℎminus 119875119897)

2

)

times119876 (120572 ℎ) minus 119875ℎ)119890 (119896)

lt 120588119890119879

(119896) 119875ℎ119890 (119896)

(28)

for all 120579119896= ℎ isin 119872 and for all 120579

119896+1= 119897 isin 119872 From (28) (19)

and (20) it is known that119864 [119881 (119890 (119896 + 1) 120579

119896+1| 119890 (119896) 120579

119896)]

minus 119881 (119890 (119896) 120579119896) lt 120588119881 (119890 (119896) 120579

119896)

(29)

and hence

119864 [119881 (119890 (119896 + 1) 120579119896+1)] minus 119864 [119881 (119890 (119896) 120579

119896)]

119864 [119881 (119890 (119896) 120579119896)]

lt 120588 (30)

Condition (25) in Theorem 2 is a LMI We denote condi-tion (25) as Ξ(120572 120588) lt 0 and for any 120572 gt 0 define

120588 (120572) = inf 120588 | 120588 isin R Ξ (120572 120588) lt 0 has solutions (31)

Using the LMI toolbox of MATLAB 120588(120572) can be computedby Algorithm 1

For a 120572 gt 0with 120588 lt 0 it is known fromTheorems 1 and 2that linear distributed inference (7) is average consensus andthat 120588 is a bound of convergence speed Since a less value of120588 lt 0 gives a faster convergence speed the fast distributedinference problem is addressed as

120583 = inf120572gt0

120588 (120572) (32)

which is an unconstrainted optimization problem of only onevariable Many existing numerical optimizationmethods [18]can be utilized to solve this problem efficiently When 120583 lt 0the optimal parameter

120572opt = arginf120572gt0

120588 (120572) (33)

provides a fast linear distributed inference which reachesaverage consensus

6 Mathematical Problems in Engineering

Choose TOL gt 0Choose enough large 120588

1isin R such that Ξ(120572 120588

1) has solutions

Choose enough small 1205882isin R such that Ξ(120572 120588

2) has no solution

repeat until 1205881minus 1205882lt TOL

1205880larr (1205881+ 1205882)2

Solve Ξ(120572 1205880) lt 0

if Ξ(120572 1205880) lt 0 has solutions

1205881larr 1205880

else1205882larr 1205880

end (repeat)Set 120588(120572) larr 120588

1

Algorithm 1

4 Numerical Example

In this section we present simulation results for averageconsensus of distributed inference in a simple sensor net-work The network has 10 sensor nodes and switched inthree possible communication situations Figure 1 illustrates3 communication situations The estimated transition proba-bilities of 120579

119896is

Γ = [120574ℎ119897] = [

[

07 02 01

045 03 025

05 01 04

]

]

(34)

The estimate error Δ120574ℎ119897satisfies

1003816100381610038161003816Δ120574ℎ1198971003816100381610038161003816 le 2120587ℎ119897 = 01 forallℎ isin 1 2 3 forall119897 isin 1 2 3 (35)

Using the computation procedure in Section 3 the optimiza-tion problem (32) is solved Graph of 120588(120572) is displayed inFigure 2 The result is 120583 = minus02407 lt 0 and 120572opt = 04812For the communication situation in Figure 1(1) we use thedesign method in [7] of minimizing asymptotic convergencefactor and obtain 120572

1= 05528 The design method in [7]

is also applied to the other 2 situations in Figure 1 and get1205722= 1205723= 05359

In order to compare our method with that in [7] theinitial states of each sensor node is selected as

119909 (0) = [11991011199102119910311991041199105119910611991071199108119910911991010]119879

= [152 146 143 156 153 145 157 149 151 148]119879

(36)

Thus from (1) we have 119910 = 15 The real transition probabilitymatrix is set as

Γ = [120574ℎ119897] = [

[

068 024 008

04 027 033

053 011 036

]

]

(37)

Figures 3 4 and 5 show state curves of all sensor nodesunder 120572opt 1205721 and 1205722 respectively It can be seen that allsensor states convergence to 119910 = 15 and that our 120572opt hasfaster convergence rate than 120572

1or 1205722has Achieving faster

convergence is because our method considers the randomswitching among the 3 communication situations while [7]rsquosmethod considers only 1 communication situation

5 Conclusion

The distributed average consensus problem in sensor net-works has been studied under a Markovian switching com-munication topology of uncertain transition probabilities

Stochastic Lyapunov functions have been employed to inves-tigate average consensus of linear distributed inference Asufficient condition of average consensus has been proposedbased on feasibility of a set of coupled LMIs The designproblem of fast distributed inference has been solved bynumerical optimization techniques

Acknowledgment

This work was supported by 973 Program of China (Grant2009CB320603)

References

[1] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[2] R Olfati-Saber and R M Murray ldquoConsensus protocols fornetworks of dynamic agentsrdquo in Proceedings of the AmericanControl Conference pp 951ndash956 June 2003

Mathematical Problems in Engineering 7

[3] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[4] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[5] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[6] YHatano andMMesbahi ldquoAgreement over randomnetworksrdquoIEEE Transactions on Automatic Control vol 50 no 11 pp1867ndash1872 2005

[7] L Xiao and S Boyd ldquoFast linear iterations for distributedaveragingrdquo Systems amp Control Letters vol 53 no 1 pp 65ndash782004

[8] S Kar and J M F Moura ldquoSensor networks with random linkstopology design for distributed consensusrdquo IEEE Transactionson Signal Processing vol 56 no 7 part 2 pp 3315ndash3326 2008

[9] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991

[10] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[11] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[12] L Zhang and E-K Boukas ldquo119867infin

control for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo International Journal of Robust and NonlinearControl vol 19 no 8 pp 868ndash883 2009

[13] P Seiler and R Sengupta ldquoAn 119867infin

approach to networkedcontrolrdquo IEEE Transactions on Automatic Control vol 50 no3 pp 356ndash364 2005

[14] L A Montestruque and P Antsaklis ldquoStability of model-basednetworked control systems with time-varying transmissiontimesrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1562ndash1572 2004

[15] J Yu L Wang and M Yu ldquoSwitched system approach to sta-bilization of networked control systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 17 pp 1925ndash19462011

[16] Y Xia G-P Liu M Fu and D Rees ldquoPredictive control ofnetworked systems with random delay and data dropoutrdquo IETControl Theory and Applications vol 3 no 11 pp 1476ndash14862009

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer New York NY USA1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Average Consensus Analysis of Distributed ...downloads.hindawi.com/journals/mpe/2013/505848.pdf · Research Article Average Consensus Analysis of Distributed Inference

6 Mathematical Problems in Engineering

Choose TOL gt 0Choose enough large 120588

1isin R such that Ξ(120572 120588

1) has solutions

Choose enough small 1205882isin R such that Ξ(120572 120588

2) has no solution

repeat until 1205881minus 1205882lt TOL

1205880larr (1205881+ 1205882)2

Solve Ξ(120572 1205880) lt 0

if Ξ(120572 1205880) lt 0 has solutions

1205881larr 1205880

else1205882larr 1205880

end (repeat)Set 120588(120572) larr 120588

1

Algorithm 1

4 Numerical Example

In this section we present simulation results for averageconsensus of distributed inference in a simple sensor net-work The network has 10 sensor nodes and switched inthree possible communication situations Figure 1 illustrates3 communication situations The estimated transition proba-bilities of 120579

119896is

Γ = [120574ℎ119897] = [

[

07 02 01

045 03 025

05 01 04

]

]

(34)

The estimate error Δ120574ℎ119897satisfies

1003816100381610038161003816Δ120574ℎ1198971003816100381610038161003816 le 2120587ℎ119897 = 01 forallℎ isin 1 2 3 forall119897 isin 1 2 3 (35)

Using the computation procedure in Section 3 the optimiza-tion problem (32) is solved Graph of 120588(120572) is displayed inFigure 2 The result is 120583 = minus02407 lt 0 and 120572opt = 04812For the communication situation in Figure 1(1) we use thedesign method in [7] of minimizing asymptotic convergencefactor and obtain 120572

1= 05528 The design method in [7]

is also applied to the other 2 situations in Figure 1 and get1205722= 1205723= 05359

In order to compare our method with that in [7] theinitial states of each sensor node is selected as

119909 (0) = [11991011199102119910311991041199105119910611991071199108119910911991010]119879

= [152 146 143 156 153 145 157 149 151 148]119879

(36)

Thus from (1) we have 119910 = 15 The real transition probabilitymatrix is set as

Γ = [120574ℎ119897] = [

[

068 024 008

04 027 033

053 011 036

]

]

(37)

Figures 3 4 and 5 show state curves of all sensor nodesunder 120572opt 1205721 and 1205722 respectively It can be seen that allsensor states convergence to 119910 = 15 and that our 120572opt hasfaster convergence rate than 120572

1or 1205722has Achieving faster

convergence is because our method considers the randomswitching among the 3 communication situations while [7]rsquosmethod considers only 1 communication situation

5 Conclusion

The distributed average consensus problem in sensor net-works has been studied under a Markovian switching com-munication topology of uncertain transition probabilities

Stochastic Lyapunov functions have been employed to inves-tigate average consensus of linear distributed inference Asufficient condition of average consensus has been proposedbased on feasibility of a set of coupled LMIs The designproblem of fast distributed inference has been solved bynumerical optimization techniques

Acknowledgment

This work was supported by 973 Program of China (Grant2009CB320603)

References

[1] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[2] R Olfati-Saber and R M Murray ldquoConsensus protocols fornetworks of dynamic agentsrdquo in Proceedings of the AmericanControl Conference pp 951ndash956 June 2003

Mathematical Problems in Engineering 7

[3] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[4] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[5] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[6] YHatano andMMesbahi ldquoAgreement over randomnetworksrdquoIEEE Transactions on Automatic Control vol 50 no 11 pp1867ndash1872 2005

[7] L Xiao and S Boyd ldquoFast linear iterations for distributedaveragingrdquo Systems amp Control Letters vol 53 no 1 pp 65ndash782004

[8] S Kar and J M F Moura ldquoSensor networks with random linkstopology design for distributed consensusrdquo IEEE Transactionson Signal Processing vol 56 no 7 part 2 pp 3315ndash3326 2008

[9] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991

[10] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[11] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[12] L Zhang and E-K Boukas ldquo119867infin

control for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo International Journal of Robust and NonlinearControl vol 19 no 8 pp 868ndash883 2009

[13] P Seiler and R Sengupta ldquoAn 119867infin

approach to networkedcontrolrdquo IEEE Transactions on Automatic Control vol 50 no3 pp 356ndash364 2005

[14] L A Montestruque and P Antsaklis ldquoStability of model-basednetworked control systems with time-varying transmissiontimesrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1562ndash1572 2004

[15] J Yu L Wang and M Yu ldquoSwitched system approach to sta-bilization of networked control systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 17 pp 1925ndash19462011

[16] Y Xia G-P Liu M Fu and D Rees ldquoPredictive control ofnetworked systems with random delay and data dropoutrdquo IETControl Theory and Applications vol 3 no 11 pp 1476ndash14862009

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer New York NY USA1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Average Consensus Analysis of Distributed ...downloads.hindawi.com/journals/mpe/2013/505848.pdf · Research Article Average Consensus Analysis of Distributed Inference

Mathematical Problems in Engineering 7

[3] L Moreau ldquoStability of multiagent systems with time-depend-ent communication linksrdquo IEEE Transactions on AutomaticControl vol 50 no 2 pp 169ndash182 2005

[4] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[5] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[6] YHatano andMMesbahi ldquoAgreement over randomnetworksrdquoIEEE Transactions on Automatic Control vol 50 no 11 pp1867ndash1872 2005

[7] L Xiao and S Boyd ldquoFast linear iterations for distributedaveragingrdquo Systems amp Control Letters vol 53 no 1 pp 65ndash782004

[8] S Kar and J M F Moura ldquoSensor networks with random linkstopology design for distributed consensusrdquo IEEE Transactionson Signal Processing vol 56 no 7 part 2 pp 3315ndash3326 2008

[9] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991

[10] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[11] J Xiong J Lam H Gao and D W C Ho ldquoOn robust stabi-lization of Markovian jump systems with uncertain switchingprobabilitiesrdquo Automatica vol 41 no 5 pp 897ndash903 2005

[12] L Zhang and E-K Boukas ldquo119867infin

control for discrete-timeMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo International Journal of Robust and NonlinearControl vol 19 no 8 pp 868ndash883 2009

[13] P Seiler and R Sengupta ldquoAn 119867infin

approach to networkedcontrolrdquo IEEE Transactions on Automatic Control vol 50 no3 pp 356ndash364 2005

[14] L A Montestruque and P Antsaklis ldquoStability of model-basednetworked control systems with time-varying transmissiontimesrdquo IEEE Transactions on Automatic Control vol 49 no 9pp 1562ndash1572 2004

[15] J Yu L Wang and M Yu ldquoSwitched system approach to sta-bilization of networked control systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 17 pp 1925ndash19462011

[16] Y Xia G-P Liu M Fu and D Rees ldquoPredictive control ofnetworked systems with random delay and data dropoutrdquo IETControl Theory and Applications vol 3 no 11 pp 1476ndash14862009

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer New York NY USA1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Average Consensus Analysis of Distributed ...downloads.hindawi.com/journals/mpe/2013/505848.pdf · Research Article Average Consensus Analysis of Distributed Inference

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Distributed consensus revisedDistributed consensus revised Heidi Howard Summary We depend upon distributed systems in every aspect of life. Distributed consensus, the abil-ity to reach

Distributed consensus revisedDistributed consensus revised Heidi Howard Summary We depend upon distributed systems in every aspect of life. Distributed consensus, the abil-ity to reach

ZCloud Consensus on Hardware for Distributed Systems

ZCloud Consensus on Hardware for Distributed Systems

Consensus in Mobile Dynamic Distributed Systemsfaculty.cse.tamu.edu › hlee › Research › fomc2014-invited-hyun... · 2014-08-28 · Consensus in Mobile Dynamic Distributed Systems

Consensus in Mobile Dynamic Distributed Systemsfaculty.cse.tamu.edu › hlee › Research › fomc2014-invited-hyun... · 2014-08-28 · Consensus in Mobile Dynamic Distributed Systems

Distributed Consensus in Multivehicle Cooperative Control ...ren/book/consensus/RenBeard08...Distributed Consensus in Multivehicle Cooperative Control: Theory and Applications Wei

Distributed Consensus in Multivehicle Cooperative Control ...ren/book/consensus/RenBeard08...Distributed Consensus in Multivehicle Cooperative Control: Theory and Applications Wei

Distributed Consensus Protocols and Algorithms · fault-tolerant consensus in distributed system and two practical consensus pro-tocols for distributed computing. Section 2.3 presents

Distributed Consensus Protocols and Algorithms · fault-tolerant consensus in distributed system and two practical consensus pro-tocols for distributed computing. Section 2.3 presents

Consensus-based Distributed Estimation in Camera Networks

Consensus-based Distributed Estimation in Camera Networks

1 Distributed Average Consensus with Dithered Quantization

1 Distributed Average Consensus with Dithered Quantization

Ch11 Distributed Agreement. Outline Distributed Agreement Adversaries Byzantine Agreement Impossibility of Consensus Randomized Distributed Agreement.

Ch11 Distributed Agreement. Outline Distributed Agreement Adversaries Byzantine Agreement Impossibility of Consensus Randomized Distributed Agreement.

Distributed Consensus: Making Impossible Possiblehh360.user.srcf.net/slides/impossible_consensus.pdf · Distributed Consensus: Making Impossible Possible Heidi Howard ... A general

Distributed Consensus: Making Impossible Possiblehh360.user.srcf.net/slides/impossible_consensus.pdf · Distributed Consensus: Making Impossible Possible Heidi Howard ... A general

Distributed Consensus with Limited Communication Data Rate

Distributed Consensus with Limited Communication Data Rate

Distributed Algorithms Raft Consensus - disi.unitn.itdisi.unitn.it/~montreso/ds/handouts/14-raft.pdf · Distributed Algorithms Raft Consensus Alberto Montresor University of Trento,

Distributed Algorithms Raft Consensus - disi.unitn.itdisi.unitn.it/~montreso/ds/handouts/14-raft.pdf · Distributed Algorithms Raft Consensus Alberto Montresor University of Trento,

Research Article Distributed Consensus Control of ...downloads.hindawi.com/journals/mpe/2015/963282.pdfResearch Article Distributed Consensus Control of Networked Control Systems with

Research Article Distributed Consensus Control of ...downloads.hindawi.com/journals/mpe/2015/963282.pdfResearch Article Distributed Consensus Control of Networked Control Systems with

1 Distributed Detection via Gaussian Running Consensus ...users.isr.ist.utl.pt/~jxavier/ieeetsp2011a.pdf · Distributed Detection via Gaussian Running Consensus: Large Deviations

1 Distributed Detection via Gaussian Running Consensus ...users.isr.ist.utl.pt/~jxavier/ieeetsp2011a.pdf · Distributed Detection via Gaussian Running Consensus: Large Deviations

Distributed Average Consensus Algorithm for Damage ... · the damage model (connectivity) of a power distribution system ... service restoration. Commonly, the communication system

Distributed Average Consensus Algorithm for Damage ... · the damage model (connectivity) of a power distribution system ... service restoration. Commonly, the communication system

Distributed systems II Replication –Quorum Consensus

Distributed systems II Replication –Quorum Consensus

Information Weighted Consensus for Distributed ... - Tashrif

Information Weighted Consensus for Distributed ... - Tashrif

Distributed Stochastic Consensus Optimization with ...

Distributed Stochastic Consensus Optimization with ...

Distributed Consensus in Multivehicle Cooperative Control ...ee.ucr.edu/~ren/book/consensus/RenBeard08_Springer.pdf · Distributed Consensus in Multivehicle Cooperative Control: Theory

Distributed Consensus in Multivehicle Cooperative Control ...ee.ucr.edu/~ren/book/consensus/RenBeard08_Springer.pdf · Distributed Consensus in Multivehicle Cooperative Control: Theory

Finite-Time Weighted Average Consensus and Generalized ......Y. Shang: Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset Corollary1:Ifat’saretakenasinTheorem1,thenthefact

Finite-Time Weighted Average Consensus and Generalized ......Y. Shang: Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset Corollary1:Ifat’saretakenasinTheorem1,thenthefact

Distributed consensus ledgers for payments

Distributed consensus ledgers for payments