Multi-Agent Consensus With Relative-State-Dependent Measurement Noises

Transcription

1 I TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 9, SPTMBR Mul-Agen Consensus Wh Relave-Sae-Dependen Measuremen Noses Tao L, Fuke Wu, and J-Feng Zhang Absrac In hs noe, he dsrbued consensus corruped by relavesae-dependen measuremen noses s consdered. ach agen can measure or receve s neghbors sae normaon wh random noses, whose nensy s a vecor uncon o agens relave saes. By nvesgang he srucure o hs neracon and he ools o sochasc derenal equaons, we develop several small consensus gan heorems o gve sucen condons n erms o he conrol gan, he number o agens and he nose nensy uncon o ensure mean square (m.s.) and almos sure (a.s.) consensus and quany he convergence rae and he seady-sae error. specally, or he case wh homogeneous communcaon and conrol channels, a necessary and sucen condon o ensure m.s. consensus on he conrol gan s gven and s shown ha he conrol gan s ndependen o he specc nework opology, bu only depends on he number o nodes and he nose coecen consan. For symmerc measuremen models, he almos sure convergence rae s esmaed by he Ieraed Logarhm Law o Brownan moons. Index Terms Dsrbued consensus, dsrbued coordnaon, adng channel, measuremen noses, mul-agen sysem. I. INTRODUCTION In recen years, he dsrbued coordnaon o mul-agen sysems wh envronmenal unceranes has been pad much aenon o by he sysems and conrol communy. There are varous knds o unceranes n mul-agen neworks, whch have sgncan nluence on he success o coordnaon algorhms and perormances o he whole nework. For dsrbued neworks, he unceranes o a sngle node and lnk may propagae over he whole nework along wh he normaon exchange among agens. Compared wh sngle-agen sysems, he eec o unceranes o mul-agen sysems on he overall perormances s closely relaed o he paern o normaon neracon. Fruul resuls have been acheved or dsrbued consensus wh sochasc dsurbances. For dscree-me models, he dsrbued sochasc approxmaon mehod s nroduced n [1] [3] o aenuae he mpac o communcaon/measuremen noses and condons are gven o ensure m.s. and a.s. consensus. For connuous-me models, L and Zhang [4] gave a necessary and sucen condon on he conrol gan o ensure m.s. consensus. Wang and la [5] made a sysemac sudy o unsable nework dynamc behavors wh whe Manuscrp receved February 17, 13; revsed July 19, 13, Ocober 3, 13, and January 1, 14; acceped January 7, 14. Dae o publcaon February 3, 14; dae o curren verson Augus, 14. The work o T. L was suppored by he Naonal Naural Scence Foundaon o Chna under Gran and he Program or Proessor o Specal Apponmen (asern Scholar) a Shangha Insuons o Hgher Learnng. The work o F. Wu was suppored by he Program or New Cenury xcellen Talens n Unversy. The work o J.-F. Zhang was suppored by he Naonal Naural Scence Foundaon o Chna under Grans and Recommended by Assocae dor S. Zamper. (Correspondng auhor: Tao L.) T. L s wh he School o Mecharonc ngneerng and Auomaon, Shangha Unversy, Shangha 7, Chna (e-mal: sxumuz@qq.com). F. Wu s wh he School o Mahemacs and Sascs, Huazhong Unversy o Scence and Technology, Wuhan 4374, Chna (e-mal: wuuke@mal. hus.edu.cn). J.-F. Zhang s wh he Key Laboraory o Sysems and Conrol, Academy o Mahemacs and Sysems Scence, Chnese Academy o Scences, Bejng 119, Chna (e-mal: j@ss.ac.cn). Dgal Objec Idener 1.119/TAC Gaussan npu noses, channel adng and me-delay. Furhermore, compuaonal expressons or checkng m.s. sably under crculan graphs are developed n [6]. Aysal and Barner [7] and Medvedev [8] suded he dsrbued consensus wh addve random noses or dscree-me and connuous-me models, respecvely. In a general ramework, Aysal and Barner [7] gave a sucen condon o ensure a.s. consensus, and Medvedev [8] gave a sucen condon o ensure closed-loop saes o be bounded n m.s. Mos o he above leraure assume ha he nensy o noses s me-nvaran and ndependen o agens saes. However, hs assumpon does no always hold or some mporan measuremen or communcaon schemes. For consensus wh quanzed measuremens ([9]), he logarhmc quanzer ([1]) s used, hen he unceranes nroduced by he quanzaon are modeled by relave-saedependen whe noses n a sochasc ramework ([1]). I he relave saes are measured by analog adng channels, he unceranes o he measuremen are also relave-sae-dependen noses ([5], [6], [11]). I s a promnen eaure o mul-agen neworks wh relavesae-dependen noses ha he dynamc evoluon o unceranes o he whole nework neracs wh he dynamc evoluon o he agens saes n a dsrbued normaon archecure, whch resuls n essenal dcules or he conrol desgn and closed-loop analyss o hs knd o unceran mul-agen neworks. In hs noe, we consder he dsrbued consensus o hghdmensonal rs-order agens wh relave-sae-dependen measuremen noses. The normaon neracon o agens s descrbed by an undreced graph. ach agen can measure or receve s neghbors sae normaon wh random noses. Deren rom our prevous work or he case wh whe Gaussan measuremen noses ([4]), here, he nose nensy s a vecor uncon o he relave saes o he agens. So deren rom mos o he exsng leraure, he sascal properes o he mpac o he noses on he nework are me-varyng and coupled by he dynamc evoluons o he agens saes. Snce he nose nensy depends on relave saes, our model can no be covered by he case wh me-varyng bu ndependen-o-sae nose nensy uncons consdered n [7], [8]. Typcal examples or our model are he logarhmc quanzaon model n he sochasc ramework ([1]) and he dsrbued averagng sysem wh Gaussan adng communcaon channels ([6]). We show ha he closed-loop consensus error equaon becomes a sochasc derenal equaon wh mulplcave noses, whch presens us an neresng propery ha he couplng s que well-organzed beween he nose process over he whole nework and he dynamcs o he agens. Ths equaon quanavely shows how he nensy coecen marx assocaed wh he nework noses relaes o he nework opology. For he case wh ndependen and homogeneous conrol channels and lnear nose nensy uncons, he quadrac sum o coecen marces over all measuremen channels s exacly he dagonal marx composed o non-zero egenvalues o he Laplacan marx L mulpled by a consan dependen on he conrol gan, he nose nensy coecen o a sngle lnk and he number N o nework nodes. We develop several small consensus gan heorems and show ha he nose nensy uncon lnearly grows wh rae bounded by σ, hen a conrol gan k whch sases <k<n/[(n 1)σ ] can ensure asympocally unbased m.s. and a.s. average-consensus, and he m.s. seady-sae error and convergence rae can be gven n quanave relaon o he conrol gan, he nose and nework opology parameers. specally, or he case wh ndependen and homogeneous channels, he nose nensy grows wh he rae σ, hen <k<n/[(n 1)σ ] s also a necessary and sucen condon o ensure m.s. consensus. We show ha hough a small conrol gan can decrease he mean-square seady-sae I. Personal use s permed, bu republcaon/redsrbuon requres I permsson. See hp:// or more normaon.

2 464 I TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 9, SPTMBR 14 error or achevng average-consensus, may slow down he m.s. convergence rae as well. For opmzng he m.s. convergence rae, he opmal conrol gan s N/[(N 1)σ ] n some sense. We prove ha or mul-agen neworks wh relave-sae-dependen measuremen noses, he condon or a.s. consensus s weaker han ha or m.s. consensus. specally, or neworks wh homogeneous lnear growh nose nensy uncons and conrol channels, consensus can be acheved wh probably one provded ha he conrol gan sases k + k σ / >. Ths s a promnen derence compared wh he case wh non-sae-dependen measuremen noses ([4]). For he case wh symmerc nose nensy uncons, by he Ieraed Logarhm Law o Brownan moons, s shown ha he convergence rae wh probably 1 s beween O(exp{( (k + k σ /)λ (L)+ɛ)}) and O(exp{( (k + k σ /)λ N (L) ɛ)}), ɛ>. The ollowng noaons wll be used. 1 denoes a column vecor wh all ones. η N, denoes he N-dmensonal column vecor wh he h elemen beng 1 and ohers beng zero. J N denoes he marx (1/N )11 T. I N denoes he N-dmensonal deny marx. For a gven marx or vecor A, A T denoes s ranspose, and A denoes s -norm. For wo marces A and B, A B denoes her Kronecker produc. For any gven real symmerc marx L, s mnmum real egenvalues s denoed by λ mn (L) and he maxmum by λ max (L).For any gven square marx A, deneˆλ mn (A) =mn 1 N { λ (A) }. [ ] denoes he mahemacal expecaon. II. PROBLM FORMULATION We consder he consensus conrol or a nework o he agens wh he ollowng dynamcs: ẋ () =u (), =1,,...,N, (1) where x () R n and u () R n. Here, each agen has n conrol channels, and each componen o x () s conrolled by a conrol channel. Denoe x() =[x T 1 (),...,x T N ()]T and u() =[u T 1 (),..., u T N ()]T. The normaon low srucure among deren agens s modeled as an undreced graph G = {V, A}, wherev = {1,,..., N} s he se o nodes wh represenng he h agen, and A = [a j ] R N N s he adjacency marx o G wh elemen a j =1or ndcang wheher or no here s an normaon low rom agen j o agen drecly. 1 Also, deg = N a j s called he degree o, The Laplacan marx o G s dened as L = D A,where D =dag(deg 1,...,deg N ).Theh agen can receve normaon rom s neghbors wh random perurbaon as he orm: y j () =x j ()+ j (x j () x ())ξ j (), j N, () where N = {j V a j =1} denoes he se o neghbors o agen, y j () denoes he measuremen o x j () by agen, andξ j () R denoes he measuremen nose. Assumpon.1: The nose nensy uncon j ( ) s a mappng rom R n o R n. There exss a consan σ> such ha j (x) σ x, =1,...,N, j N,oranyx R n. Assumpon.: The nose processes {ξ j (),,j =1,...,N} sasy ξ j(s)ds = w j (),, where {w j (),,j =1,...,N} are ndependen Brownan moons. Remark 1: Consensus problems wh quanzed measuremens o relave saes were suded n [9]. I he logarhmc quanzaon s used, hen by properes o logarhmc quanzers, he quanzed 1 Here, or concseness, we consder undreced graphs wh 1 weghs. I s no dcul o exend our resuls o he case wh general dgraphs wh nonnegave weghs. measuremen by agen o x j () x () s gven by z j () =x j () x ()+(x j () x ())Δ j (), whch can be vewed as a specal case o (), where he quanzaon uncerany Δ j () s regarded as whe noses ([1]) n he sochasc ramework. Remark : Dsrbued averagng wh Gaussan adng channels were suded n [6], where he measuremen o x j () x () s gven by z j (k) =ξ j (k)(x j (k) x (k)), where {ξ j (k)} are ndependen Gaussan noses wh mean value μ j. Followng he mehod n [11], Wang and la ([6]) ransormed he above equaon no z j (k) =μ j (x j (k) x (k)) + Δ j (k)(x j (k) x (k)), where Δ j (k) =ξ(k) μ j are ndependen zero-mean Gaussan noses. Ths can be vewed as a dscree-me verson o (), where μ j can be merged no he wegh o he weghed adjacency marx o he nework opology graph. We consder he ollowng dsrbued proocol: u () =K a j (y j () x ()),,=1,...,N, (3) where K R n n s he conrol gan marx o be desgned. For he dynamc nework (1) and () and he dsrbued proocol (3), we should consder he ollowng quesons. () Under wha condons s he closed-loop sysem can acheve m.s. or a.s. consensus? () Wha s he relaonshp beween he closed-loop perormances (.e., he convergence rae, he seady-sae error e al.) and he conrol gan marx K, he measuremen nose nensy uncon and he parameers o he nework opology graph? How o desgn he conrol gan marx o opmze he closed-loop perormances? III. MAN SQUAR AND ALMOST SUR CONSNSUS Denoe δ() =[(I N J N ) I n ]x(). Denoe φ =[φ,...,φ N ], where φ s he un egenvecor o L assocaed wh λ (L). Le δ() =(T L I n ) δ() and δ() =[ δt 1 (),..., δt N ()] T wh δ1 (). Denoe δ() =[ δt (),..., δt N ()] T. Denoe Λ L =dag(λ (L),, λ N (L)) and Ψ L (K) =Λ L [(K + KT )/] [(N 1)/N ] K σ (Λ L I n), whch s a symmerc marx. We have he ollowng heorem. Theorem 3.1: Suppose ha Assumpons.1. hold. I Ψ L (K) s posve dene, hen he dsrbued proocol (3) s an asympocally unbased m.s. and a.s. average-consensus proocol ([4]). Precsely, he closed-loop sysem o (1) and () under (3) sases: or any gven x() R Nn, here s a random vecor x R n wh (x )= (1/N ) N x j(), such ha [ x () x ]=, and x () =x,a.s. =1,...,N, and he m.s. seady-sae error s gven by x 1 N x j () K σ λ N (L) δ() (. (4) N λ mn Ψ L (K)) Moreover, he m.s. convergence raes o δ() s gven by [ δ() ] δ() exp { λ mn ( Ψ L (K)) }, (5) and he a.s. convergence rae s gven by log δ() λ mn ( Ψ L (K)),a.s. (6) Remark 3: Generally speakng, he momen exponenal sably and he a.s. exponenal sably do no mply each oher. Bu under he

3 I TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 9, SPTMBR lnear growh condons on he dr and duson erms, he momen exponenal sably mples he a.s. exponenal sably ([1]). In Secon IV, or he case wh lnear nose nensy uncons, one may see ha a.s. consensus requres weaker condon han m.s. consensus. Theorem 3.. Small Consensus Gan Theorem: Suppose ha Assumpons.1. hold. Le he conrol gan marx K = ki n, where k R. Then he dsrbued proocol (3) s an asympocally unbased m.s. and a.s. average-consensus proocol he graph G s conneced and <k<n/[(n 1)σ ]. Proo: From he condon o he heorem, we know ha Ψ L (K) =(kλ L (k σ (N 1)/N )Λ L ) I n,andsoψ L (K) s posve dene and only (k k σ (N 1)/N )λ (L) >, =,...,N. The above nequales hold and only he graph G s conneced and <k<n/[(n 1)σ ]. Then, by Theorem 3.1, we have he concluson o he heorem. Remark 4: Theorem 3. ells us ha or he case wh muually ndependen and homogeneous conrol channels, he graph s conneced and he produc o conrol gan k and he square upper bound o he nose nensy σ s less han N/(N 1), hen boh m.s. and a.s. consensus can be acheved. I s obvous ha <kσ < 1 suces or <k<n/[(n 1)σ ], so he selecon o conrol gan can be ndependen o N and he nework opology, and nuvely speakng, n nverse proporon o he growh rae o he nose nensy uncon. IV. LINAR NOIS INTNSITY FUNCTION In hs secon, we wll consder he case where he nosy nensy j ( ) s a lnear uncon o he relave sae x j () x (). Theorem 4.1: Suppose ha Assumpons.1. hold wh j (x) =Σ j x, j,, j =1,...,N,oranyx R n,whereσ j R n n.leb j =[b kl ] N N be an N N marx wh b = a j, b j = a j and all oher elemens beng zero,, j =1,,...,N. Le Φ K = N, (φt Bj T φφt B j φ) (Σ T j KT KΣ j ) and Ψ K = Λ L (K + KT ) Φ K. Apply he proocol (3) o he sysem (1) and (). Then he closed-loop sysem sases [ δ() ] δ() e λmax(ψ K ), [ δ() ] δ() e λ mn(ψ K ). (7) I he symmerc marx Ψ K s posve dene, hen he proocol (3) s an asympocally unbased m.s. and a.s. average-consensus proocol. And [ x 1 ] λ max (Φ K ) x j () N N(N 1)λ mn (Ψ K ) δ(), (8) where x s he lm o x (), =1,...,N, boh n m.s. and probably 1. Remark 5: For consensus problems wh precse communcaon, s always assumed ha he saes and conrol npus o agens are scalars. Ths assumpon wll no loose any generaly or he case wh precse communcaon and wh non-sae-dependen measuremen noses, snce he sae componens o he agens are decoupled. However, or he case wh relave-sae-dependen measuremen noses, rom model (), one may see ha he nose nensy o deren sae componens wll be generally coupled ogeher. For he case wh lnear nose nensy uncons, he couplng among communcaon channels o deren sae componens means ha Σ j, j,, j = 1,...,N, are no dagonal marces. From Theorem 4.1, one may see ha he non-dagonal elemens o Σ j ndeed have mpacs on he consensus condons and perormances. For he case wh decoupled communcaon channels, we have he ollowng resuls. Theorem 4.: Suppose ha Assumpons.1. hold wh j (x) =σ j x, σ j >, j,, j =1,...,N, or any x R n. Then he proocol (3) wh K = ki n, k R, s an asympocally unbased m.s. average-consensus proocol he nework opology graph G s conneced and <k<n/[σ (N 1)], and only he nework opology graph G s conneced and <k<n/[σ (N 1)], where σ =max{σ j,=1,...,n,j N } and σ =mn{σ j,= 1,...,N,j N }. Corollary 4.1: Suppose ha Assumpons.1. hold wh j (x) =σx j,, j =1,...,N,oranyx R n,whereσ>. Then he proocol (3) wh K = ki n, k R, s an asympocally unbased mean-square average-consensus proocol and only he nework opology graph G s conneced and <k<n/[σ (N 1)]. Remark 6: Theorems 4. and Corollary 4.1 are concerned wh he case where he communcaon and conrol channels or deren componens o he saes o agens are compleely decoupled. specally, n Corollary 4.1, when he nose nensy uncons are homogeneous or deren agens and sae componens, we gve a necessary and sucen condon on he conrol gan, he nose nensy and nework parameers o ensure m.s. consensus. Theorem 3. shows ha he nose nensy uncon grows lnearly wh rae bounded by σ, hena posve conrol gan k<1/σ s sucen or m.s. consensus. For he case o Corollary 4.1, we can see ha s necessary or m.s. consensus ha he upper bound o he conrol gan s nversely proporonal o he square o he growh rae o he nose nensy uncon. Remark 7: From (7), we can see ha or he case wh lnear nose nensy uncons, he m.s. convergence rae s conrolled by he maxmal and mnmal egenvalues o Ψ(K). A queson s wheher we can choose K o maxmze he m.s. convergence rae. Generally speakng, a gven conrol gan K ha maxmze λ mn (Ψ(K)) may no maxmze λ max (Ψ(K)) n he meanwhle. However, Corollary 4.1 ells us ha or he case wh ndependen and homogeneous communcaon and conrol channels, we can ndeed ge some opmal soluon o he conrol gan. Nong ha Σ j = σi n,, j =1,...,N, we have Φ K =((N 1)σ k /N )(Λ L I n), and Ψ K =(k ((N 1)σ k /N ))(Λ L I n). For hs case, he egenvalues o Ψ K are jus he nonzero egenvalues o he Laplacan marx mulpled by k (N 1)σ k /N.LeK =(N/((N 1)σ ))I n, hen Ψ K =max K=kIn,<k<(N/(σ (N 1))) Ψ K. Ths mples ha he conrol gan o opmze he m.s. convergence rae can be seleced as k = N/((N 1)σ ). Remark 8: From (8), we can see ha he m.s. seady-sae error or average-consensus s bounded by (λ max (Φ K )/(N(N 1)λ mn (Ψ K ))) δ(). The coecen o he bound depends on he conrol gan and he nework opology. For he case o Corollary 4.1, by Remark 7, can be compued ha λ max (Φ K )/(N(N 1)λ mn (Ψ K )) =σ kλ N (L)/(N(N (N 1)σ k)λ (L)), whch vanshes as kσ. To reduce he seady-sae error or average-consensus, one way s o decrease he conrol gan k, however,rom(7),we can see ha as k, he convergence wll become very slow; he oher way s o desgn he nework opology o maxmze he synchronzably o he nework λ (L)/λ N (L). Remark 9: For he asympoc analyss, we consder a sequence {G N,N 1} o conneced graphs. Nong ha λ N (L) d(g N ) and λ (L) 4/dam(G N ) ([13]), we have x (1/N ) N x j () (σ kd(g N )(N 1)/[N (1 ((N 1)/N )σ k)], where d(g N ) s he degree o G N and dam(g N ) s he dameer o G N. The dsance beween wo verces n a graph s he lengh o (.e., number o edges n) he shores pah beween hem. The dameer o a graph G s maxmum dsance beween any wo verces o G.

4 466 I TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 9, SPTMBR 14 Smlar o Theorem 4.1, he condons o Theorem 4. and Corollary 4.1 suce or a.s. consensus. I was shown ha or he case wh non-sae-dependen measuremen noses, he condons or a.s. consensus also suce or m.s. consensus ([4], [14]). From he ollowng heorems, we can see ha or he case wh relavesae-dependen measuremen noses, a.s. consensus requres weaker condon on he conrol gan han m.s. consensus. Theorem 4.3: Le μ =n. x R n(n 1),x {(1/ x )x T Ψ K x +(/ x 4 ) N, [xt ((φ T B j φ) (KΣ j ))x] }. Under he assumpons o Theorem 4.1, he proocol (3) s appled o he sysem (1) and (), hen he closed-loop sysem sases (log δ() /) (μ/) a.s. Parcularly, he proocol (3) s an asympocally unbased a.s. average-consensus proocol μ>. Remark 1: Theorem 4.1 showes ha Ψ K s posve dene, hen he proocol (3) can drve he dynamc nework o consensus boh n m.s. and probably 1. We know ha μ> s weaker han he posve deneness o Ψ K snce N, [xt ((φ T B j φ) (KΣ j ))x] / x 4 > or any x.. Ths mples ha μ λ mn (Ψ K ). Acually, le λ K = λ mn (Ψ K )+ (1/) N, ˆλ mn ((φt B j φ) (KΣ j )+(φ T B T j φ) (ΣT j KT )). I ollows ha μ λ K and λ K > Ψ K s posve dene. So, λ K > can be used as a sucen condon, whch s easer o be vered han μ>, o ensure a.s. consensus. I λ K >, hen he closedloop sysem sases (log δ() /) λ K / < a.s. I he measuremen model s symmerc and K s a symmerc marx, hen more precse esmaes o he convergence rae or a.s. consensus can be obaned. Assumpon 4.1: The nose processes {ξ j (),,j =1,...,N} sasy ξ j(s)ds=w j (), w j () w j (),,where{w j (),= 1,...,N 1,j=+1,...,N} are ndependen Brownan moons. Theorem 4.4: Suppose ha Assumpons.1 and 4.1 hold wh j (x) =σ j x j,, j =1,...,N,oranyx R n,whereσ j = σ j >. Apply he proocol (3) o he sysem (1) and (). I K s symmerc, hen he closed-loop sysem sases log δ() + λ mn (A L (K)) and log δ() + λ max (A L (K)) K a j σ j, j a.s., (9) K a j σ j, j a.s., (1) where A L (K) =[Λ L K +(1/)(φT ( N, B j σ j )φ) K ]. Corollary 4.: Suppose ha he nework opology graph G s conneced and Assumpons.1 and 4.1 hold wh j (x) =σx j,, j =1,...,N,oranyx R n,whereσ>. Then he proocol (3) wh K = ki n, k R, s an asympocally unbased a.s. averageconsensus proocol k + k σ / > and he convergence rae s gven by and log δ() +(k+ k σ )λ (L) log δ() +(k+ k σ )λ N (L) k σ a j, a.s., j k σ a j, a.s. j Proo: From Bj = B j and N B, j = L, we have A L (K) =(k + k σ /)(Λ L I n). Then he concluson ollows rom Theorem 4.4. Remark 11: Corollary 4. ells us ha provded ha he nework s conneced, any gven posve conrol gan or negave conrol gan sasyng kσ / < 1 can ensure a.s. consensus. Corollary 4.1 ell us ha o ensure m.s. consensus, he conrol gan has o be posve and small enough such ha kσ (N 1)/N < 1. Ths mples ha or he case wh homogeneous communcaon and conrol channels, a.s. consensus requre weaker condon han m.s. consensus, whch s conssen wh Theorems 4.1 and 4.3. Remark 1: For he consensus sysem wh precse communcaon: ẋ () =k (x j N j () x ()), was shown n [15] ha a necessary and sucen condon on he conrol gan k or consensus o be acheved s k>. In [4], or he consensus sysem wh non-saedependen addve nose: ẋ () =k (x j N j () x ()+ξ j ()), was shown ha a consan conrol gan k, no maer how small s, can no ensure he closed-loop sably. For he consensus sysem wh non-sae-dependen measuremen noses and he sochasc approxmaon ype conrol proocol: ẋ () =k() (x j N j () x ()+ξ j ()), was shown n [4] and [14] ha he necessary and sucen condon on he nonnegave conrol gan k() or consensus o be acheved almos surely s k() = and k () <. Corollary 4. ells us ha or he consensus sysem wh relave-saedependen measuremen noses: ẋ () =k (x j N j () x ()+ (x j () x ())ξ j ()), a sucen condon on he conrol gan k or consensus o be acheved almos surely s k + k σ / >, whch means ha even a negave conrol gan may ensure consensus as well. Ths ells us ha derenly rom he non-sae-dependen measuremen noses ([4], [14]), he relave-sae-dependen measuremen noses wll somemes be helpul or he a.s. consensus o he nework. Wheher or no nework noses need o be aenuaed depends on he paern ha noses mpac on he nework. Remark 13: For he consensus sysem wh non-sae-dependen measuremen noses and he sochasc approxmaon ype conrol proocol, was shown n [4] and [14] ha he vanshng conrol gan k() wh a proper vanshng speed s necessary and sucen o ensure he m.s. and a.s. consensus, however, he vanshng conrol gan may resul n a slower convergence o he closed-loop sysem, whch s no longer exponenally as. From he resuls o hs paper, we can see ha or he case wh relave-sae-dependen noses, he vanshng o he conrol gan s no necessary and he convergence speed o he closedloop sysem can be exponenally as. Remark 14: I s well known ha mulplcave noses can be used o sablze an unsable sysem n he sense o probably 1 ([16]), and p-momens wh p (, 1) ([17]). In Corollary 4., he condon k + k σ / > shows ha he noses play posve roles or a.s. consensus. However, For he m.s. consensus (p =), he condon <kσ (N 1)/N < 1 n Corollary 4.1 shows ha he noses play negave roles, whch means ha or a gven xed updae gan, he nose level σ could no be larger han he hreshold value N/(k(N 1)). Ths mples ha here exs undamenal derences beween he a.s. and he m.s. consensus or he consensus sysem wh relave-sae-dependen noses. Remark 15: In [6], he dscree-me dsrbued averagng s consdered wh adng channels and me-delays. By converng he adng channel model no a precse measuremen model wh relavesae-dependen noses and he assumpon ha he closed-loop sysem s npu-oupu sable, some necessary and sucen condons were gven under crculan graphs. I was shown ha as he number o agens or me-delay ncreases o nny, small conrol gans can lead o m.s. sably bu may slow down he convergence. Here, he closed-loop

5 I TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 9, SPTMBR sysem s no assumed o be npu-oupu sable n pror. The nework opology s no lmed o crculan graphs and he convergence and perormance are also consdered or a.s. convergence. Corollary 4.1 shows ha o ensure m.s. consensus, he conrol gan has o be small enough. Ths resul and hose o [6] boh reveal ha here s a naural rade-o beween he m.s. convergence speed and he robusness o nose or he choce o he conrol gan. Our mehod can be urher exended o he dscree-me case wh he noses modeled by marngale derence sequences, whch can cover he case wh Bernoull adng channels ([5]) and he sochasc logarhmc quanzaon n [1]. Remark 16: Consder a conneced wo-agen undreced nework. 1 (x) =σ 1 x, 1 (x) =σ 1 x, σ 1 >, σ 1 >,oranyx R n, hen he proocol (3) wh K = ki n, k R s an a.s. averageconsensus proocol and only k + k (σ 1 + σ 1)/ >, and s a m.s. average-consensus proocol and only 4k k (σ 1 + σ 1) >. The m.s. seady-sae error s gven by x (x 1 () + x ()/ = k(σ 1 + σ 1) x 1 () x () /(4[4 k(σ 1+σ 1)]). So he ac ha a.s. consensus requres weaker condon han m.s. consensus can be urher vered by he wo agen case even he channel s no symmerc. I can be vered ha or he wo agen case, (λ max (Φ K )/(N(N 1)λ mn (Ψ K ))) δ() = k(σ 1+ σ 1) x 1 () x () /(4[4 k(σ 1 + σ 1)]), whch mples ha he upper bound o he m.s. seady-sae error n Theorem 4.1 s gh or he wo agen case. V. C ONCLUDING RMARKS In hs noe, he dsrbued consensus o hgh-dmensonal rsorder agens wh relave-sae-dependen measuremen noses has been consdered. The normaon exchange among agens s descrbed by an undreced graph. ach agen can measure or receve s neghbors sae normaon wh random noses, whose nensy s a vecor uncon o agens relave saes. By nvesgang he srucure o he neracon beween nework noses and he agens saes and he ools o sochasc derenal equaons, we have developed several small consensus gan heorems o gve sucen condons o ensure m.s. and a.s. consensus and quany he convergence rae and he seady-sae error. specally, or he case wh lnear nose nensy uncons and homogeneous communcaon and conrol channels, a necessary and sucen condon o ensure m.s. consensus on he conrol gan k s <k<n/[(n 1)σ ],whereσ s he growh rae o he nose nensy uncon. I s shown ha or hs knd o mulagen neworks, a.s. consensus requres weaker condons han m.s. consensus. specally, or neworks wh homogeneous lnear nose nensy uncons and conrol channels, consensus can be acheved wh probably one provded k + k σ / >, whch means ha even a negave conrol gan can also ensure almos consensus. For uure research on he dsrbued coordnaon o mul-agen sysems wh relave-sae-dependen measuremen noses, here are many neresng opcs, such as he dscree-me case wh he noses modeled by marngale derence sequences, he case wh random lnk alures, he me-delay and dsrbued rackng problems. APPNDIX Lemma A.1: The N (N 1) dmensonal marx φ sases φφ T = I N J N,andφ T φ = I N 1. Lemma A.: Le B j =[b kl ] N N,, j =1,,...,N be marces dened n Theorem 4.1. Then Bj T 11T B j =(N/N 1)Bj T φφt B j. Lemma A.3: Suppose ha he assumpons o Theorem 4.1 hold. Applyng he proocol (3) o he sysem (1) and (), he closed-loop sysem sases dδ() = (Λ L K)δ()d + N, [(φt B j φ) (KΣ j )]δ()dw j (). Lemma A.4: Suppose ha he assumpons o Theorem 4.1 hold. Apply he proocol (3) o he sysem (1) and (), hen or all δ(), he closed-loop sysem sases P{δ() on all } =1. Lemma A.5: Suppose ha he assumpons o Theorem 4.4 hold. Apply he proocol (3) o he sysem (1) and (), hen he closed-loop sysem sases δ()= exp{ A L (K)+M L,K ()}δ(),wherea L (K)= [Λ L K +(1/) N, [(φt B j φ) (Kσ j) ]], andm L,K () = N, [(φt B j φ) (Kσ j )]w j (). The proos o Lemmas are omed here. Proo o Theorem 3.1: Subsung he proocol (3) no he sysem (1) gves ẋ () =K N a j(x j () x ())+K N a j j (δ j () δ ())ξ j (). By Assumpon., we have dx()= (L K)x() d + N N a =1 j[η N, (K j (δ j () δ ()))] dw j (), whch ogeher wh he denon o δ() gves dδ() = (L K)δ()d + a j [(I N J N )η N,, Then by he denon o δ(), wehave (K j (δ j () δ ()))] dw j (). dδ() = ( Λ L K ) δ()d [ + a j φ T (I N J N )η N,, (K j (δ j () δ ()))] dw j (). By he denons o η N, and J N,wehaveη T N (I N J N )η N, = (N 1)/N. By Lemma A.1, nong ha (I N J N ) = I N J N, applyng he Iô ormula o δ(),wege d δ() = δ T () ( Λ L (K + K T ) ) δ()d + dm 1 () + N 1 a j N =1 ( T j (δ j () δ ()) K T K j (δ j () δ ()) ) d, where dm 1 ()= N, δt ()a j [φ T (I N J N )η N, (K j (δ j () δ ()))]dw j (). By Assumpon.1, we have d δ() λ mn (Ψ L (K)) δ() + dm 1 (), Then by he comparson heorem ([18]), we ge (5), whch ogeher wh he posve deneness o Ψ L (K) leads o [ δ() ]=. By he properes o he marx L, we have (1 T I n )x() =(1 T I n )x() + a j M j (),, where M j () = [1T η N, (K j (δ j (s) δ (s)))]dw j (s). By Assumpon.1, nong ha 1 T η N, =1, s esmaed ha 1 T η N, (K j (δ j (s) δ (s))) ds K σ δ() ( λ mn Ψ (K)), L whch mples ha M j () s a square-negrable connuous marngale. Then we know ha as, (1/N )(1 T I n )x() converges o a random varable wh ne second-order momen boh n mean square and almos surely. Denoe he lm random varable by x =(1/N )(1 T I n )x() + (1/N ) N a, j K j (δ j () δ ())dw j () wh (x )=(1/N ) N x j(). Ths ogeher wh he convergence o [ δ() ] means ha (3) s an asympocally

6 468 I TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 9, SPTMBR 14 unbased m.s. average-consensus proocol. By he denon o x, we have [ x 1 ] x j () N K σ N = K σ N =1 a j δ j (s) δ (s) ds δ T (s) ( Λ L I n ) δ(s) ds K σ λ N (L) δ() ( N λ mn Ψ (K)), L whch gves he seady-sae error (4). I s known ha here exss a posve consan α 1, such ha (Λ L K)δ() α 1 δ(). By Assumpon.1 and he C r nequaly, we know ha here exss a posve consan α, such ha N N a =1 j[φ T (I N J N ) η N, (K j (δ j () δ ()))] α δ(). Then by [1, Th. 4.], we know ha (3) s an asympocally unbased a.s. average-consensus proocol. Proo o Theorem 4.1: Applyng Lemma A.3 and he Io ormula gves d δ() λ mn (Ψ K ) δ() + N, δt ()[(φ T B j φ) (KΣ j )]δ()dw j (), and d δ() λ max (Ψ K ) δ() + N, δt ()[(φ T B j φ) (KΣ j )]δ()dw j (). Ths ogeher wh he comparson heorem gves (7). I Ψ K s posve dene, hen [ δ() ]. Also, smlar o Theorem 3.1, we have [ x () x ]=,wherex =(1/N )1 T x() + (1/N ) N, (1T B j KΣ j )δ()dw j (). By Lemma A. and he denons o δ() and δ(), applyng (7) gves ha x 1 N = x j () 1 N(N 1) =1 δ T () ( ) φ T Bjφφ T T B j φ Σ T jk T KΣ j δ() d λ max(φ K ) δ() e λ mn (Ψ K ) d, N(N 1) whch mples (8). Then Smlar o Theorem 3.1, we know ha (3) s a m.s. and a.s. average-consensus proocol. Proo o Theorem 4.: The par ollows drecly rom Theorem 3.1. By he denon o B j and φ, we have N, φ T Bj T φφt B j φ =((N 1)/N )φ T Lφ =((N 1)/N )Λ L. Ths ogeher wh K = ki n leads o Ψ K (k((kσ (N 1)/N ) 1)Λ L ) I n. Then smlarly o Theorem 4.1, we have d δ() k((kσ (N 1)/N ) 1)λ (L) δ() + N, δt ()[(φ T B j φ) (KΣ j )]δ()dw j (), whch mply he only par. Proo o Theorem 4.3: By Lemmas A.3 and A.4, applyng he Iô ormula o log δ() gves d log δ() μd + (/ δ() ) N, δt ()[(φ T B j φ) (KΣ j )]δ()dw j (). Thereore, ollows rom he denon o μ ha log δ() log δ() μ + M(), (A.1) where M()= δ T () [ (φ T B δ() j φ) (KΣ j ) ] δ() dw j (), s a local marngale wh M() = and he quadrac varaons ([1]) M,M / = N ˆλ, max((φ T B j φ) (KΣ j )+(φ T Bj T φ) (ΣT j K T )) <. Applyng he law o large number gves (M()/ ) =, a.s., whch ogeher wh (A.1) gves (log δ() /) (μ/) < a.s. Then smlar o Theorems 4.1 and 3.1, we know ha he proocol (3) s an a.s. average-consensus proocol. Proo o Theorem 4.4: From Lemma A.5, nong ha A L (K), M L,K () and exp{ A L (K) + M L,K ()} are all symmerc marx and he egenvalues o exp{ A L (K) + M L,K ()} are all nonnegave, we know ha δ() exp{ λ mn (A L (K)) + λ max (M L,K ())} δ(), whch gves log δ() + λ mn (A L (K)) λ max(m L,K ()) log δ() +. log log (A.) Thus, by he Law o he Ieraed Logarhm o Brownan moons, nong ha B j a j,wehave(9).smlarly,wehave δ() exp{ λ max (A L (K)) + λ mn (M L,K ())} δ(). From above and he Law o he Ieraed Logarhm o Brownan moons, smlar o (A.), we have (1). RFRNCS [1] M. Huang and J. H. Manon, Coordnaon and consensus o neworked agens wh nosy measuremen: Sochasc algorhms and asympoc behavor, SIAM J. Conrol & Opmz.: Specal Issue Conrol Opmzaon Cooperave New., vol. 48, no. 1, pp , 9. [] S. Kar and J. M. F. Moura, Dsrbued consensus algorhms n sensor neworks wh mperec communcaon: Lnk alures and channel nose, I Trans. Sgnal Process., vol. 57, no. 1, pp , 9. [3] T. L and J. F. Zhang, Consensus condons o mulagen sysems wh me-varyng opologes and sochasc communcaon noses, I Trans. Auom. Conrol, vol. 55, no. 9, pp , 1. [4] T. L and J. F. Zhang, Mean square average consensus under measuremen noses and xed opologes: Necessary and sucen condons, Auomaca, vol. 45, no. 8, pp , 9. [5] J. Wang and N. la, Dsrbued averagng under consrans on normaon exchange: mergence o Levy lghs, I Trans. Auom. Conrol, vol. 57, no. 1, pp , 1. [6] J. Wang and N. la, Mgaon o complex behavor over neworked sysems: Analyss o spaally nvaran srucures, Auomaca, vol. 49, no. 6, pp , 13. [7] T. C. Aysal and K.. Barner, Convergence o consensus models wh sochasc dsrubances, I Trans. Sgnal Process., vol. 56, no. 8, pp , 1. [8] G. S. Medvedev, Sochasc sably o connuous me consensus proocols, SIAM J. Conrol & Opmz., vol.5,no.4,pp ,1. [9] D. V. Dmarogonas and K. H. Johansson, Sably analyss or mulagen sysems usng he ncdence marx: Quanzed communcaon and ormaon conrol, Auomaca, vol. 46, no. 4, p. 695-, 1. [1] R. Carl, F. Fagnan, A. Speranzon, and S. Zamper, Communcaon consrans n he average consensus problem, Auomaca, vol. 44, no.3, pp , 8. [11] N. la, Remoe sablzaon over adng channels, Sys. and Conrol Le., vol. 54, no. 3, pp , 5. [1] X. Mao, Sochac Derenal quaons and Applcaons. Chcheser, U.K.: Horwood Publshng, [13] N. M. M. de Abreu, Old and new resuls on algebrac connecvy o graphs, Lnear Algebra Applc., vol. 43, no. 1, pp , 7. [14] B. C. Wang and J. F. Zhang, Consensus condons o mul-agen sysems wh unbalanced opology and sochasc dsurbances, J. Sys. Sc. Mahma. Sc., vol. 9, no. 1, pp , 9. [15] R. Ola-Saber and R. M. Murray, Consensus problem n neworks o agens wh swchng opology and me-delays, I Trans. Auom. Conrol, vol. 49, no. 9, pp , 4. [16] F. Wu and S. Hu, Suppresson and sablzaon o nose, In. J. Conrol, vol. 8, no. 11, pp , 9. [17] X. Zong, F. Wu, and T. Tan, Sably and sochasc sablzaon o numercal soluons o regme-swchng jump duson sysems, J. Derence quaons Applc., vol. 19, no. 11, pp , 13. [18] A. N. Mchel and R. K. Mller, Qualave Analyss o Large Scale Dynamcal Sysems. New York, NY, USA: Academc.