Research Article Event-Based Control for Average Consensus of Wireless Sensor Networks with Stochastic Communication Noises

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1 Discree Dynamics in Naure and Sociey Volume 2013, Aricle ID , 8 pages hp://dx.doi.org/ /2013/ Research Aricle Even-Based Conrol for Average Consensus of Wireless Sensor Neworks wih Sochasic Communicaion Noises Chuan Ji, Wuneng Zhou, and Huashan Liu College of Informaion Sciences and Technology, Donghua Universiy, Shanghai , China Correspondence should be addressed o Wuneng Zhou; zhouwuneng@163.com Received 1 November 2013; Acceped 20 November 2013 Academic Edior: Guoliang Wei Copyrigh 2013 Chuan Ji e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Thispaperfocusesonheaverageconsensusproblemforhewirelesssensorneworks(WSNs)wihfixedandMarkovianswiching, undireced and conneced nework opologies in he noise environmen. Even-based proocol is applied o each sensor node o reach he consensus. An even riggering sraegy is designed based on a Lyapunov funcion. Under he even rigger condiion, some sufficien condiions for average consensus in mean square are obained. Finally, some numerical simulaions are given o illusrae he effeciveness of he resuls derived in his paper. 1. Inroducion Wireless sensor nework (WSN) has araced significan aenion as an emerging communicaion archiecure. I has many pracical applicaions in such areas as roboics, surveillance and environmen monioring, and informaion collecion. A WSN can be viewed as a muliagen sysem (MAS) from a nework-heoreic perspecive. Each node represens a sensor and each edge performs informaion exchange beween sensors. In some cases, he agreemen is a common value which may be he average of he iniial saes of he sysem, is ofen called average consensus, and has wide applicaion background in he areas such as formaion conrol [1], disribued filering [2], and disribued compuaion [3]. I means o achieve he accordance of he saes of MAS. In [4], Olfai-Saber and Murray consider he average consensus conrol for he direced and undireced neworks wih fixed and swiching opologies. In [5], Kingson and Beard exend he resuls of [4] o he discree-ime models and weakened he condiion of insananeous srong conneciviy. In [6], Xiao and Boyd consider he disribued averaging consensus of he neworks wih fixed and undireced opologies. In [7], Q. Zhang and J. Zhang design a disribued consensus proocol o analyze he muliagen sysems in uncerain communicaion environmens including he communicaion noises and Markov opology swiches. In [8], Wang e al. invesigae he H consensus conrol problem for a class of discree ime-varying muliagen sysems wih boh missing measuremens and parameer uncerainies. Also, he disribued esimaion problems over sensor neworks have been widely discussed in [9 11]. For he node of he WSNs wih limied energy, conrol over neworks wih limied resources is a challenging ask. Consequenly, he mos imporan problem in WSN is he energy consumpion, which is direcly proporional o he ransmi power of he informaion exchange beween he sensors. The even-based conrol can faciliae he efficien usage of he resources. Based on he even-based conrol mechanism, i can reduce he useless communicaion beween neighboring agens along wih he energy consumpion. In [12],WangandLemmondiscussheeven-riggereddaa ransmission in disribued neworked conrol sysems wih packe loss and ransmission delays. In [13], Mazo and Tabuada focus on reducing he number of messages from sensors o conrollers and from conrollers o acuaors by he useofhedecenralizedeven-riggeredmechanism.in[14], Seyboh e al. propose a novel conrol sraegy for muliagen coordinaion wih even-based broadcasing. The proposed conrol sraegy guaranees eiher asympoic convergence o

2 2 Discree Dynamics in Naure and Sociey average consensus or convergence o a ball cenered a he average consensus. In [15], Meng and Chen sudy an average consensus problem for muliagen sysems by even-based conrol,whichisusedoneachagenodrivehesaeoheir iniial average evenually. The main conribuion of his paper is sudying average consensus problem for he WSN wih fixed opology and Markovian swiching opology in he noise environmen; he even-based average consensus conrol proocol is designed forhewsnbasedonalyapunovfuncion.underheeven rigger condiion, sufficien condiions for average consensus in mean square are obained. The remainder of he paper is organized as follows. In Secion 2, some conceps in graph heory are described, and he problem o be invesigaed is formulaed. In Secion 3, he main resuls are presened. In Secion 4,somenumerical examples show he reliabiliy of he main resuls. In Secion 5, some conclusions are given. 2. Problem Formulaion and Preliminaries 2.1. Conceps in Graph Theory. Le G = {V, E, A} be an undireced graph, where V ={1,2,...,n}is he se of nodes, node i represens he ih sensor node, E is he se of edges, and an edge in G is denoed by an ordered pair (j, i). (j, i) E if and only if he jh sensor node can send informaion o he ih sensor node direcly. The neighborhood of he ih sensor node is denoed by N i ={j V (j, i) E}. A =[a ij ] R n n is called he adjacency marix of G.For any i, j V, a ij 0,anda ij > 0 j N i. deg in (i) = n j=1 a ij is called he in-degree of i; deg ou (i) = n j=1 a ij is called he ou-degree of i; L=D A is called he Laplacian marix of G, whered = diag(deg in (1),..., deg in (n)). Is eigenvalues are real and can be ordered as λ 1 λ 2 λ n (1) wih λ 1 =0and λ 2 being he smalles nonzero eigenvalue for conneced graphs Average Consensus for WSNs. In his paper, we sudy he average consensus conrol for a WSN wih dynamics x i () =u i (), (2) where x i () R is he sae of he ih sensor and u i () R is he conrol inpu. The iniial sae x i (0) is deerminisic. Each sensor includes a digial microprocessor and dynamics. The microprocessor of sensor i moniors is own measuremen value x i () coninuously and decides when o communicae wih he neighboring sensors by broadcasing he acual measuremen value. Therefore he laes broadcased value of sensor i can be described by he piecewise consan funcion x i () =x i ( i k ), i k <i k+1, (3) where i 0,i 1,...is sequence of even-imes of sensor i.wecan see ha he discree-ime signal x i ( i k ) is convered ino he coninuous-ime signal x i (). The ih sensor can receive informaion from is neighbors y ji () = x j () +σ ji ω ji (), (4) where y ji () denoes he measuremen of he jh sensor s sae x j () by he ih sensors, {ω ji () i,j = 1,2,...,n} are he communicaion noises, and σ ji is he noise inensiy funcion. For he dynamic nework (G,X),we give he even-based consensus proocl. u i () = j N i a ij ( y ji () x i ()). The dynamics of agen i for [ i k +lh,i k +lh+h)is given by x i () = a ij [x j ( j ) x k i ( i k )+σ jiω ji ()] where j k is defined as = a ij [x j ( i k +lh) x i ( i k +lh)] + a ij [x i ( i k +lh) x i ( i k )] + a ij [x j ( j ) x k j ( i k +lh)] + a ij σ ji ω ji () = a ij [x j ( i k +lh) x i ( i k +lh)] + a ij [e j ( i k +lh) e i ( i k +lh)] + a ij σ ji ω ji (), (5) (6) j = max { { j k k,k=0,1,...}, i k +lh}. (7) Subsiuing he proocol (5)inohesysem(2)leadso dx () = [L (x () +e())] d + GdW (), (8) where L is he Laplacian marix of G, G is he noise inensiy marix, and W() = (W 1 (),...,W n ()) T is an n-dimensional Brownian moion. The even condiion for agen i has he following form: λ n x i( i k +lh) 2 λ2 e i( i k +lh) 2 +2 G 2, (9) where e i ( i k +lh)=x i ( i k ) x i ( i k +lh). (10) Remark 1. Each sensor broadcass is sae informaion o he neighbors and also receives sae informaion from is neighbors for even deecion a each sampling insan. The even deecor in (9) guaranees ha i reduces he sensor energy consumpion and nework bandwidh usage because i only checks he even condiion a discree sampling insans.

3 Discree Dynamics in Naure and Sociey 3 Nex, we consider he average consensus conrol proocol for he sysem (8) as follows: dδ () = L[δ () + (I J) e ()] d + (I J) GdW (), where δ() = x() Jx() and J = (1/n)1 n 1 T n. If we define f (δ, ) = L[δ () + (I J) e ()], hen (11)becomes g (δ, ) = (I J) G, (11) (12) dδ () =f(δ, ) d + g (δ, ) dw (). (13) In WSNs, each sensor node communicaes wih oher sensor nodes hrough he unreliable neworks. If he communicaion channel beween sensors i and j is (j, i) E f,and E f is he se of he communicaion channels which probably los he signal, hen he ime-varying opologies under link failure or creaion can be described by he Markov swiching opology. Le {r() 0 } be a righ-coninuous Markov chain on he probabiliy space aking values in a finie sae se S = {1,2,...,N}wih generaor Γ=(γ ij ) N N given by P{r(+Δ) =j r() =i}={ γ ijδ+o(δ), if i =j, 1+γ ii Δ+o(Δ), if i=j, (14) where Δ>0and γ ij 0is he ransiion rae from i o j if i =j, while γ ii = γ ij (15) j =i and r(0) = r 0. We denoe he undireced communicaion graph by G = {G(1), G(2),...,G(N)}, whereg(k) = {V, E G(k), A G(k) } is he undireced graph. Denoe he opology graph by G a momen ( 0),sor() = k if and only if G = G(k). Under Markovian swiching opology, we have dx () = [L (r ()) (x () +e())] d (16) +G(r ()) dw (), dδ () = L(r ()) [δ () + (I J) e ()] d (17) + (I J) G (r ()) dw (). So, under his condiion, he even condiion for agen i has he following form: λ n (L (r ())) x i( i k +lh) 2 λ 2 (L (r ())) e i( i k +lh) 2 +2 G 2, (18) 3. Main Resuls In his secion, we consider he average consensus for he WSNs wih fixed opology and Markovian swiching opology by even-based conrol WSNs wih he Fixed Topology Theorem 2. Consider he sysem (8) over a conneced communicaion graph driven by even condiion in (9). Then he sysem (11) is asympoically sable in mean square; i means ha he sysem (8) reaches average consensus in mean square if 0 h 1/2λ n. Proof. Define he funcion V: R n R + R + by V (δ, ) = 1 2 δt () δ (). (20) Compuing dv(δ, ) along he rajecory generaed by he sysem (11)forany [, (k + 1)h),wehave dv (δ, ) =[δ T () f (δ, ) + race [g T (δ, ) g (δ, )]] d +δ T ()(I J) GdW () =[ δ T () L [δ () + (I J) e ()] + (I J)G 2 ]d+δ T ()(I J) GdW () =[( ) [δ () + (I J) e ()] T L T L [δ () + (I J) e ()] δ T () L [δ () + (I J) e ()] + (I J)G 2 ]d (W T () W T ())G(I J) L[δ () + (I J) e ()] +δ T ()(I J) GdW () (hλ n [δ () + (I J) e ()] T L[δ () + (I J) e ()] δ T () L [δ () + (I J) e ()] + (I J)G 2 )d (W T () W T ())G(I J) where e i ( i k +lh)=x i ( i k ) x i ( i k +lh). (19) L[δ () + (I J) e ()] +δ T ()(I J) GdW ()

4 4 Discree Dynamics in Naure and Sociey = ((hλ n 1)δ T () Lδ () +(2hλ n 1) δ T () L (I J) e () +hλ n I J 2 e T () Le () + (I J) G 2 )d (W T () W T ())G(I J) L[δ () + (I J) e ()] +δ T ()(I J) GdW (). Applying he inequaliy δ T () L (I J) e () 1 2 L 2 δ T () δ () I J 2 e T () e (), i can be derived ha (21) (22) Inegraingbohsidesofhepreviousinequaliyfrom o ( [,(k + 1)h)), making use of (9), and aking he expecaion, one can obain ha e EV (δ (),) e EV (δ (),) d e δ T () δ () ( 1 2 δt () Lδ () 1 2 I J 2 e T () Le () (I J) G 2 )(e e ) e EV (δ (),) d e δ T () δ () +( 1 2 δt () Lδ () dv (δ (),) ( 1 2 δt () Lδ () 1 2 I J 2 e T () Le () (25) wih 2hλ n 1. We can see ha d[e V (δ (),)] I J 2 e T () Le () + (I J)G 2 )d (W T () W T ())G(I J) L[δ () + (I J) e ()] +δ T ()(I J) GdW () =e [ V (δ (),) d + dv (δ (),)] e [ V (δ (),) 1 2 δt () Lδ () I J 2 e T () Le () + (I J)G 2 ]d e (W T () W T ())G(I J) L[δ () + (I J) e ()] +e δ T ()(I J) GdW (). (23) (24) (I J) G 2 )(e e (k+1)h ) = e EV (δ (),) d e δ T () δ () +( 1 2 I J 2 x T () Lx () 1 2 I J 2 e T () Le () (I J)G 2 )(e e (k+1)h ) e EV (δ (),) d e δ T () δ (). By using Gronwall inequaliy, we have EV (δ (),) 1 2 e δ() 2 e ( ) = 1 2 δ() 2 e. (26) Leing (k ),weobain lim E δ() 2 =0. (27) So he sysem (8) reaches average consensus in mean square.

5 Discree Dynamics in Naure and Sociey WSNs wih Markovian Swiching Topology Theorem 3. Consider he sysem (16) over a number of conneced swiching communicaion graphs driven by even condiion in (18).Thenhesysem(17) is asympoically sable in mean square; i means ha he sysem (16) reaches average consensus in mean square if 0 h 1/2λ n (r()). Proof. Define he funcion V: R n R + S R + by V (δ,, r ()) = 1 2 δt () δ (). (28) Compuing dv(δ,, r()) along he rajecory generaed by he sysem (11)forany [, (k + 1)h),wehave dv (δ,, r ()) =[δ T () f (δ,, r ()) + race [g T (δ,, r ()) g (δ,, r ())]] d +δ T ()(I J) G (r ()) dw () For γ>0, γ=max{ γ ii :i S },compue d[e 2γ V (δ (),,r())] =e 2γ [ 2γV (δ (),,r()) d + dv (δ (),,r())] e 2γ [ 2γV (δ (),,r()) 1 2 δt () L (r ()) δ () I J 2 e T () L (r ()) e () + (I J)G(r()) 2 + γv (δ,, r ())]d e 2γ (W T () W T ())G(I J) L(r ()) [δ () + (I J) e ()] = [ δ T () L (r ()) [δ () + (I J) e ()] [ +e 2γ δ T ()(I J) G (r ()) dw (). (30) + (I J) G 2 + N j=1 γ ij V (δ,, r ())] d ] +δ T ()(I J) G (r ()) dw () ((hλ n 1)δ T () L (r ()) δ () +(2hλ n 1)δ T () L (r ()) (I J) e () +hλ n I J 2 e T () L (r ()) e () + (I J) G (r ()) 2 + γv (δ,, r ()))d (W T () W T ())G(r ()) (I J) L(r ()) [δ () + (I J) e ()] +δ T ()(I J) G (r ()) dw () ( 1 2 δt () L (r ()) δ () I J 2 e T () L (r ()) e () + (I J)G(r()) 2 + γv (δ,, r ()) )d (W T () W T ())G(r ()) (I J) L(r ()) [δ () + (I J) e ()] Inegraing boh sides of he above inequaliy from o ( [, (k + 1)h)), making use of (18), and aking he expecaion, one can obain ha e 2γ EV (δ (),,r()) γe 2γ EV (δ (),,r()) d e 2γ δ T () δ () ( 1 2 δt () L (r ()) δ () 1 2 I J 2 e T () L (r ()) e () (I J) G (r ()) 2 )(e 2γ e 2γ ) γe 2γ EV (δ (),,r()) d e 2γ δ T () δ () +( 1 2 δt () L (r ()) δ () 1 2 I J 2 e T () L (r ()) e () wih 2hλ n 1. +δ T ()(I J) G (r ()) dw () (29) (I J)G(r()) 2 ) (e 2γ e 2γ(k+1)h )

6 6 Discree Dynamics in Naure and Sociey = γe 2γ EV (δ (),,r()) d e 2γ δ T () δ () +( 1 2 I J 2 x T () L (r ()) x () 1 2 I J 2 e T () L (r ()) e() (I J) G (r ()) 2 ) (e 2γ e 2γ(k+1)h ) γe 2γ EV (δ (),,r()) d e 2γ δ T () δ (). By using Gronwall inequaliy, we have EV (δ (),,r()) 1 2 e 2γ δ() 2 e γ( ) = 1 2 e γ(+) δ() 2. (31) (32) The relaed degree marix is The relaed Laplacian marix is D=( ). (35) L=( ). (36) The nonzero larges and smalles eigenvalues are λ n =4 and λ 2 =1,respecively. Le he noise inensiy marix G be G=( ). (37) The sampling period for all sensor nodes is chosen as h = which saisfies 0 h 1/2λ n.usinghe even condiion in (9), we can draw he dynamic curve of he saes of he sensors by Malab as Figure 2. Ishowsus ha he four sensor nodes reach he average consensus in meansquarewihfixedopology.theconrolsignalwhenhe evens occur for each sensor is shown in Figure 3.Wecansee hahenumberofsensorconrolupdaeshaonlyoccur a he sampling insan under he even rigger condiion is rapidly reduced o reach average consensus. Leing (k ),weobainha lim E δ() 2 =0. (33) So he sysem (16) reaches average consensus in mean square. 4. Numerical Examples In his secion, we give wo examples o examine he average consensus of he sysems (8)and(16). Example 1. Consider a WSN composed by four sensors in which each dynamic sae of he sensor is x i = u i,where i = 1, 2, 3, 4 (see he opology in Figure 1, whichisusedin Dimarogonas [16]) and he iniial sae is X(0) = (4 1, 5, 2) T. The adjacen marix is A=( ) (34) Example 2. The swiching opology of he four sensors is deermined by he Markov chain r() whose sae space is S = {1,2,3}. The relaed opology graph is G(i) = {V(i), E(i), A(i)} (see he opologies in Figure 4). The adjacen marices are A (1) =( ), A(2) =( ), (38) A (3) =( ) The relaed degree marices are D (1) =( ), D(2) =( ), (39) D (3) =( )

7 Discree Dynamics in Naure and Sociey Figure 1: The opology of he four sensors. x() 2 The relaed Laplacian marices are L (1) =( ), L (2) =( ), L (3) =( ) (40) The nonzero larges and smalles eigenvalue are λ n = and λ 2 = ,respecively. Le he noise inensiy marices G be G (1) =( ), (s) x 1 () x 2 () x 3 () x 4 () Figure 2: The rajecories of he sae vecors x 1 (), x 2 (), x 3 (),and x 4 (). u() G (2) =( ), G (3) =( ) (41) (s) u 1 () u 2 () u 3 () u 4 () The sampling period for all sensor nodes is chosen as h= which saisfies 0 h 1/2λ n (i) (i = 1, 2, 3).Usinghe even condiion in (18), we can draw he dynamic curve of he saes of he sensors by Malab as Figure 5. Ishowsusha he four sensor nodes reach he average consensus in mean square wih Markovian swiching opologies. Figure 3: Conrol inpus for he sensors Conclusions Inhispaper,wehavedealwihheproblemofaverage consensus in mean square of he WSNs. By using he evenbased conrol mechanism, we have obained several sufficien condiions o ensure he average consensus in mean square for WSNs wih fixed and Markovian swiching opologies. There are many oher opics worh invesigaing, such as he G(1) (a) G(2) (b) G(3) (c) Figure 4: The opologies of he four sensors in saes 1, 2, and 3.

8 8 Discree Dynamics in Naure and Sociey x() (s) x 1 () x 2 () x 3 () x 4 () Figure 5: The rajecories of he sae vecors x 1 (), x 2 (), x 3 (),and x 4 (). packe loss and ime-delay cases of he WSN s consensus problem under even condiion. Acknowledgmens This work is suppored by he Naional Naural Science Foundaion of China ( , ), he Innovaion Program of Shanghai Municipal Educaion Commission (12zz064), he Specialized Research Fund for he DocoralProgramofHigherEducaionunderGranno , and he Naural Science Foundaion of Shanghai (12ZR ). [7] Q. Zhang and J. Zhang, Disribued consensus of coninuousime muli-agen sysems wih markovian swiching opologies and sochasic communicaion noises, Sysems Science and Mahemaical Sciences,vol.31,no.9,pp ,2011. [8]Z.Wang,D.Ding,H.Dong,andH.Shu, H consensus conrol for muli-agen sysems wih missing measuremens: he finie-horizon case, Sysems and Conrol Leers,vol.62, no. 10, pp , [9]D.Ding,Z.Wang,H.Dong,andH.Shu, DisribuedH sae esimaion wih sochasic parameers and nonlineariies hrough sensor neworks: he finie-horizon case, Auomaica, vol. 48, no. 8, pp , [10]H.Dong,Z.Wang,andH.Gao, DisribuedH filering foraclassofmarkovianjumpnonlinearime-delaysysems over lossy sensor neworks, IEEE Transacions on Indusrial Elecronics,vol.60,no.10,pp ,2013. [11] H. Dong, Z. Wang, and H. Gao, Disribued filering for a class of ime-varying sysems over sensor neworks wih quanizaion errors and successive packe dropous, IEEE Transacions on Signal Processing,vol.60,no.6,pp ,2012. [12] X. Wang and M. D. Lemmon, Even-riggering in disribued neworked conrol sysems, IEEE Transacions on Auomaic Conrol, vol. 56, no. 3, pp , [13] M. Mazo and P. Tabuada, Decenralized even-riggered conrol over wireless sensor/acuaor neworks, IEEE Transacions on Auomaic Conrol,vol.56,no.10,pp ,2011. [14] G. Seyboh, D. Dimarogonas, and K. Johansson, Even-based broadcasing for muli-agen average consensus, Auomaica, vol. 49, no. 1, pp , [15] X. Meng and T. Chen, Even based agreemen proocols for muli-agen neworks, Auomaica,vol.49,no.7,pp , [16] D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, Disribued even-riggered conrol for muli-agen sysems, IEEE Transacions on Auomaic Conrol,vol.57,no.5,pp , References [1] J. A. Fax and R. M. Murray, Informaion flow and cooperaive conrol of vehicle formaions, IEEE Transacions on Auomaic Conrol,vol.49,no.9,pp ,2004. [2] R. Olfai-Saber, Disribued Kalman filer wih embedded consensus filers, in Proceedings of he 44h IEEE Conference on Decision and Conrol, and he European Conrol Conference (CDC-ECC 05), pp , Seville, Spain, December [3] N. Lynch, Disribued Algorihms, Morgan Kaufmann, San Maero,Calif,USA,1996. [4] R. Olfai-Saber and R. M. Murray, Consensus problems in neworks of agens wih swiching opologyandime-delays, IEEE Transacions on Auomaic Conrol,vol.49,no.9,pp , [5] D. B. Kingson and R. W. Beard, Discree-ime averageconsensus under swiching nework opologies, in Proceedings of he American Conrol Conference, pp , Minneapolis, Minn, USA, June [6] L. Xiao and S. Boyd, Fas linear ieraions for disribued averaging, Sysems and Conrol Leers,vol.53,no.1,pp.65 78, 2004.

9 Advances in Operaions Research Advances in Decision Sciences Applied Mahemaics Algebra Probabiliy and Saisics The Scienific World Journal Inernaional Differenial Equaions Submi your manuscrips a Inernaional Advances in Combinaorics Mahemaical Physics Complex Analysis Inernaional Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Mahemaics Discree Mahemaics Discree Dynamics in Naure and Sociey Funcion Spaces Absrac and Applied Analysis Inernaional Sochasic Analysis Opimizaion