Stochastic Positive Real Lemma and Synchronization Over Uncertain Network

Transcription

1 Sochasic Posiive Real Lemma and Synchronizaion Over Uncerain ework Ami Diwadkar, Sambara Dasgupa and Umesh Vaidya Absrac n his paper, we prove he sochasic version of Posiive Real Lemma PRL) o sudy he sabiliy problem of nonlinear sysem in Lur e form wih sochasic uncerainy. We consider he mean square sabiliy problem of sysem in Lur e form wih sochasic parameric uncerainy affecing he linear par of he sysem dynamics. The resuls from sochasic PRL are used o sudy he problem of synchronizaion in a sysem of couple Lur e sysem wih sochasic uncerainy in he ineracion. We provide sufficiency condiion for synchronizaion expressed in erms of he second smalles eigenvalue of he coupling nominal mean) Laplacian and he saisics of link uncerainy in he form of coefficien of dispersion CoD). Under he assumpion ha he individual subsysem has idenical dynamics, we show ha he sufficiency condiion is only he funcion of dynamics of he individual subsysem and mean nework characerisics. This makes he sufficiency condiion aracive from he poin of view of compuaion for a large size nework sysem. The sufficiency condiion specialized for he case of large scale nework wih firs order agen dynamics clearly bring ou he inerplay of inernal agen dynamics, nework opology, and uncerainy saisics for he synchronizaion of he nework. Simulaion resuls for nework of coupled oscillaors wih sochasic link uncerainy are presened o verify he developed heoreical framework.. TRODUCTO The sudy of nework conrol sysems is a opic ha has received los of aenion among he research communiy laely. There is exensive lieraure on his opic involving boh deerminisic and sochasic nework sysems. Among various problems, he problem of characerizing he sabiliy of esimaor and conroller design for linear ime invarian LT) nework sysems in he presence of channel uncerainy is sudied in ], ]. The similar problem involving nonlinear and linear ime varying dynamics is sudied in 3], 4], 5], 6]. The resuls in hese papers discover fundamenal limiaions ha arise in he design of sabilizing conroller and esimaor in he presence of channel uncerainy. Passiviybased ools are used o sudy he sabiliy problem for deerminisic nework sysems in 7], 8]. Synchronizaion of inerconneced sysems from inpu-oupu approach has been sudied in 9] and shown o have applicaions in biological neworks. These ools provide for sysemaic procedure for he analysis and synhesis of deerminisic nework sysems. n his paper, we combine echniques from passiviy heory and sochasic sysems o provide sufficien condiion for he synchronizaion of uncerain nework sysems. This is A. Diwadkar, S. Dasgupa and U. Vaidya are wih he Deparmen of Elecrical & Compuer Engineering, owa Sae Universiy, Ames, A 5 dasgupa@iasae.edu, diwadkar.ami@gmail.com, ugvaidya@iasae.edu achieved by proving guaraneeing sochasic sabiliy of he error dynamics of hese sysems. We prove sochasic version of Posiive Real Lemma and provide L-based verifiable sufficien condiion for he mean square exponenial sabiliy of sochasic nework. One of he imporan feaure of he sochasic exension of Posiive Real Lemma is ha he uncerainy eners muliplicaively in he sysem dynamics. Exiing lieraure on he use of passiviy based ools for analysis of sochasic sysems assume addiive uncerainy models ], ]. The sufficien condiion are applied o sudy he synchronizaion problem in nework of Lur e sysems wih uncerain linear ineracions among he nework componens. The sufficiency condiion for mean square synchronizaion of he nework is posed in erms of a sufficiency condiion for mean square sabiliy for a single subsysem of he nework wih parameric uncerainy and second smalles eigenvalue of he mean nework, also known as he Fiedler eigenvalue. The Fiedler eigenvalue is conneced in graph heory lieraure wih conneciviy of a graph. The sufficiency condiion derived here can be solved using sandard L echniques o sudy synchronizaion of he nework, analyze effec of uncerainy in links or design nework coupling. As he condiion is posed in erms of a single subsysem i significanly reduces he compuaional complexiy associaed wih he verifying he sufficiency condiion. This makes our proposed sufficiency condiion very aracive for he sabiliy analysis of large scale uncerain nework sysem. Anoher ineresing resuls proved in his paper is he dependence of he sufficiency condiion on coefficien of dispersion of he nework links. The coefficien of dispersion CoD), defined as a raio of variance o mean of a random variable indicaes he amoun of clusering behavior in he random variable. A CoD less han uniy indicaes paerns of occurences ha are more regular. A CoD greaer han uniy indicaes clusers of occurences. Thus if he link in he nework urns off hen i ends o say off if he uncerainy has high CoD. Some real life neworks display his behavior due o heavy ail disribuions of uncerainies ], 3]. The sufficiency condiion derived shows ha he synchronizaion of he nework can be characerized by he mean square sabiliy of a single subsysem wih parameric uncerainy having CoD wice ha of he maximum CoD for he uncerain links in he nework. Undersanding he inerplay and dependence of inernal dynamics, uncerainy, and inerconnecion opology for he nework synchronizaion is a challenging problem. os of he curren lieraure focus on eiher one of hese hree elemens. The resuls derived

2 in his paper specialize for he case of firs order agen dynamics provide a sysemaic approach for undersanding he inerplay of all hree facors simulaneously. We discuss hese resuls in secion -D. The res of he paper is srucured as follows : n secion - A we formulae he general problem of sabilizaion of Lur e sysems wih parameric uncerainy and prove he main resuls on he sochasic varian of Posiive Real Lemma. The problem of synchronizaion is formulaed and solved using he sochasic varian of PRL in secion -A. The resuls on inerplay beween inernal dynamics, uncerainy, and inerconnecion opology is discussed in secion -D. Simulaion resuls are presened in secion V followed by conclusions in secion V.. STABLZATO OF UCERTA LUR E SYSTES n his secion, we firs presen he problem of sochasic sabiliy of a Lur e sysem wih parameric uncerainy. The uncerainy is modeled as an independen idenically disribued i.i.d.) random processes. The main resul of his secion proves he sochasic version of he Posiive Real Lemma. A. Problem Formulaion We consider a Lur e sysem, which has parameric uncerainy in he linear sysem dynamics. The uncerain sysem dynamics are described as follows: x + = AΞ))x Bφy,) ) y = Cx where, x R n, and y R m and φy,) R m is a nonlinear funcion. The sae marix AΞ)) R n n is uncerain. B R n m andc R m n are he inpu and oupu marices. The uncerainy is characerized by Ξ) = ξ ),...ξ )] T, where ξ i ) s for i {,...,} are i.i.d. random processes wih zero mean and σi variance, i.e., Eξ i )] = and Eξ i ) ] = σi. The schemaic of he sysem is depiced in Fig.. We make he following assumpions on he Fig.. u x A )) x Bu y Cx, ) y y Schemaic of he sysem wih parameric uncerainy. nonlineariy φy,) Assumpion : The nonlineariy φy,) is a monoonic non-decreasing funcion of y such ha, φ y,)y Dφy,)) >. The sysem, described by ), encompasses a broad class of problems like sabilizaion under parameric uncerainy, conrol and observaion of Lur e sysem over uncerain channel 4], and nework synchronizaion of Lur e sysems over uncerain links. ex, we sae and prove a sochasic version of he Posiive Real Lemma and successively use he resul for nework synchronizaion. The sochasic noion of sabiliy ha we use is he mean square exponenial sabiliy and is defined as follows: Definiion : The sysem described by Eq. ) is mean square exponenially sable if K >, and < β < such ha E Ξ x Kβ x, x R n. ) where, x evolves according o ). B. ain Resuls The following heorem is he sochasic version of he Posiive Real Lemma providing sufficien condiion for he mean square sabiliy of he sochasic Lur e sysem, described by ). Theorem 3: Le Σ = D + D and A T Ξ)) = AΞ)) BΣ C. Then he uncerain Lur e sysem in ) is mean square sable if - ) here exis symmeric posiive definie marices P and R P such ha Σ B PB > and, P =E Ξ) A T Ξ))PA T Ξ)) ] + R P +C Σ C + E Ξ) A T Ξ))PBΣ B PB) B PA T Ξ)) ] ) here exis symmeric posiive definie marices Q and R Q such ha Σ CQC > and, Q =E Ξ) AT Ξ))QA T Ξ)) ] + R Q + B Σ B + E Ξ) AT Ξ))QC Σ CQC ) CQA T Ξ)) ] 4) Proof: Please refer o he Appendix secion for he proof. The generalized version of sochasic Posiive Real Lemma, as given by Theorem 3, is now specailaized o he case of srucured uncerainies. n paricular, he srucured uncerainies are assumed o be of he form AΞ) = A+ ξ ia i, where {ξ i } are zero mean i.i.d. random variables, he mean value having been incorporaed in he deerminisic par of he marix given by A. The sae and oupu equaion for uncerain sysem becomes, x n+ = 3) y n = Cx n A + ξ i A i )x n Bφy n,n) 5) The marices A i, adjoining o he uncerainies, could be predeermined or could be designed depending on he problem. For insance, he resuls developed in 4] are for he scenario, where he marix A i is conroller gain. The following Lemma simplifies he generalized sochasic PRL o sudy he mean square sabiliy of sysem described by 5).

3 Lemma 4: The sysem, described in 5), would be mean square exponenially sable if here exiss a symmeric marix P >, such ha Σ B PB > and, P =A PA + σ i A ipa i + A PBΣ B PB) B PA σ i A ipbσ B PB) B PA i + R +C Σ C 6) A. Formulaion of Synchronizaion Problem We consider a nework of iner-conneced sysems in Lur e form. The individual subsysems could be described as follows:, { x k S k := + = Ax k Bφy k,) y k = Cx k ), k =,..., where, x k R n, and y k R m are he saes and he oupu of for some symmeric marix R > and A := A BΣ C. Proof: We subsiue A Ξ) = A + ξ ia i in he 3) and uilize he fac ξ i s are zero mean i.i.d. random variables wih variance σi. We also A T Ξ) = A+ ξ ia i BΣ C := A + ξ ia i. Hence we ge, E Ξ) A T Ξ))PA T Ξ)) ] = A PA + Also we ge, σ i A ipa i 7) S! S!! 4 ) S 4! E Ξ) AT Ξ)) PBΣ B PB) B PAΞ)) ] 8) = A PBΣ B PB) B PA 9) σ i A ipb B PB) B PA i ) Combining equaions 7) and 8) and subsiuing in 3) we ge he desired resul. Corollary 5: The sysem, described in 5), would be mean square exponenially sable if here exiss a symmeric marix Q >, such ha Σ CQC > and, Q =A QA + σ i A i QA i + A QC Σ CQC ) CQA σ i A i QC Σ CQC ) CQA i + R + B Σ B ) for some symmeric marix R > and A := A BΣ C. Proof: Corollary 5 follows from Theorem 3, Lemma 4 and dualiy.. SYCHROZATO OF LUR E SYSTES WTH UCERTA LKS n his secion, we apply he resuls developed in previous secion in analyzing sabiliy of nework of Lur e sysems, coupled hrough uncerain links. We consider a se of linearly coupled sysems in Lur e form. The links, which connec hese sysems, are uncerain in naure. n he subsequen secion we derive sufficiency condiion for he sabiliy of nework expressed in erms of he saisics of uncerainy and he mean propery of he nework, in paricular he second larges eigenvalue of he inerconnecion Laplacian. The condiion could be used o judge wheher he coupled sysem wih uncerainy could reain is sabiliy if he links binding he individual subsysems sar o fail. The sabiliy could be achieved if he uncerainies saisfy prescribed bounds. Fig.. S 3!! 34 ) Schemaic of he inerconneced sysem wih uncerain links. k h subsysem. The φy n,n) R l is a nonlinear funcion. The sae marix A R n n is he sae marix for k h subsysem. B R n m and C R m n are he inpu and oupu marices of he k h subsysem. The iner-conneced sysem is depiced in Fig.. The non-lineariy saisfies he following assumpion, Assumpion 6: The nonlineariy φ k y k,) R is globally Lipschiz monoonically nondecreasing funcion and C funcion of y s n R ha saisfies Assumpion. Furhermore, i also saisfies he following condiion, φ y k ) φ y j )) y k y j ) D φy k ) φy j )) >, for any wo sysems S k and S j and some Σ = D + D >. The aforemenioned assumpion is essenial for he synchronizaion of he nework. ex, we considered he coupled subsysems, whose sabiliy is o be analyzed. We consider he subsysems described by equaion ) are linearly coupled. The coupled sysem saisfies he following equaion, ) x+ k = Axk Bφ y k p + y k = Cx k, k =,..., j= a k j Gy j y k ) 3) where, a k j R represen he coupling link beween subsysems S k and S j, a kk = and G R n m. Remark 7: The coupled sysem as described by 3) is he mos general form of ineracion possible beween subsysems. The coupling beween subsysems could be eiher in form of oupu feedback or sae feedback. As he oupu and saes of individual subsysems are relaed linearly so he form of coupling, as described by 3) includes boh he oupu feedback and sae feedback.

4 ex, we define he graph laplacian L g := l i j ] R as following, l i j := a i j, i j, l ii := a i j, i =,... 4) j,i j ex, all he saes of he subsysems are combined o creae he saes of he coupled sysem. Finally he coupled sysem can be rewrien as, x + = Ã x B φ ỹ ) L g GC) x, ỹ = C x, 5) where, is he Kronecker produc, n is an n n deniy marix and, A... A... Ã := A = A We similarly define B := B, C := C, D := D and Σ := D + D >. We also define x = x )...x ) ], ỹ = y )...y ) ], φ = φ )...φ ) ]. B. odeling Uncerain Links We are now ready o sudy he problem of synchronizaion where he links of he graph are uncerain i.e. enries of he Laplacian marix are uncerain). Le S = {i, j) he link i, j) is uncerain, i > j} be he collecion of uncerain links in he nework. Hence, for links i, j) S, we have a i j = µ i j + ξ i j, where ξ i j are zero mean i.i.d. random variables wih variance σi j. f i, j) / S when we have a i j = µ i j o be purely deerminisic. This framework allows us o sudy synchronizaion for Lur e ype sysems wih a deerminisic weighed Laplacian as a special case. Le Ξ = {ξ i j } i, j) S. Then, he uncerain Laplacian L g Ξ) will be given as, L g Ξ) = L d + i, j) S ξ i j L i j 6) where L d is he deerminisic par of he Laplacian replacing a i j in L g wih µ i j, i.e. L d = µ i j ]. We may also wrie L i j = l i j l i j where l i j := l i j ),...,l i j )] R is a column vecor given by l i j k) = if k i j if k = i if k = j We are ineresed in finding a sufficiency condiion involving σi j for i, j) S, which would guaranee he mean square exponenial synchronizaion. The coupled nework of Lur e sysem can be wrien as, x + = Ã L g Ξ) GC) ) x n B φ ỹ ), ỹ = C x 7) We would analyze he sochasic synchronizaion of sysem, described by 7). We sar wih following definiion of mean square exponenial synchronizaion. Definiion 8: The sysem, described by 7) is mean square exponenially synchronizing if here exiss a β < and Kẽ ) > such ha, E Ξ x k x j Kẽ )β x k x j, k, j,]8) where, ẽ is funcion of difference x i x l for i,l,] and K) = K for some consan K. We now apply change of coordinaes o decompose he sysem dynamics on and off he synchronizaion manifold. The synchronizaion manifold is given by =,...,]. We show ha he dynamics on he synchronizaion manifold is decoupled from he dynamics off he manifold and is essenially described by he dynamics of he individual sysem. The dynamics on he synchronizaion manifold iself could be sable, oscillaory, or complex. Le L d = V d Λ d V d where V d is an orhonormal se of vecors given by V d = U d ], = ] and U d is orhonormal se of vecors also orhonormal o. Le z = V d n) x. uliplying 7) from he lef by V d n we ge z + = Ã V d L gξ)v d GC )) z B ψ w ) 9) where w = C z, and ψ = V d n) φ ỹ ). We can now wrie z = x ẑ ], ψ = φ ˆψ ] ) where x := x = k=x k, ẑ := U d n) x φ := φ ỹ ) = k=φy k ), ˆψ := U d n) φ ỹ ) Subsiuing ) in 9) we ge x + = A x B φ ȳ ) ẑ + = Â U d L gξ)u d GC )) ẑ ˆB ˆψ ŵ ) ) where ŵ = Ĉẑ, Â := A, ˆB := B, Ĉ := C, and ˆD := D. We now show ha for he synchronizaion of sysem 7), we only need o sabilize ẑ dynamics. The sabiliy of he sysem wih sae ẑ, implies he synchronizaion of he acual coupled sysem. This feaure is exploied o derive sufficiency condiion for sochasic synchronizaion of he coupled sysem. n he following Lemma we show he connecion beween he sabiliy of he described by ) o he synchronizaion of he sysem described by 7). Lemma 9: ean square exponenial sabiliy of sysem described by ) implies mean square exponenial synchronizaion of he sysem 7) as given by Definiion 8. Proof: To prove his resul, we show ha second momen of ẑ dynamics is equivalen o he mean square error dynamics of each pair of sysems. We hen apply sabiliy resuls o he error dynamics o complee he proof. For he complee proof please refer o he Appendix secion of his paper. n he following subsecion we will provide sufficiency condiions for he mean square exponenial synchronizaion

5 of 7) by provifing sufficiency condiions for mean square exponenial sabiliy of ). Bu firs, we rewrie he equaion ) in a more suiable forma. We noe ha L g Ξ) = L d + S ξ i j L i j, and L d = V d Λ d V d where V d = Hence we have U d L gξ)u d = U d L du d + S U d ]. ξ i j U d L i ju d := ˆΛ d +ξ i j ˆl i j ˆl i j S where L i j = l i j l i j, ˆl i j = U d l i j and ˆΛ d := U d L du d such ha ] ] Λ d = V d L dv d = U d L = du d ˆΛ d Le = {α k } k=, = S be an indexing on uncerain edges in S. f index α k corresponds o edge i, j) S hen le A αk := U d L i ju d GC = ˆl i j ˆl i j GC. Thus we can wrie equaion ) as ẑ + =  ˆΛ d GC ξ αk A αk )ẑ ˆB ˆψ ) α k C. Sufficiency Condiion for Synchronizaion wih Uncerain Links n previous subsecion, we have shown ha mean square exponenial sabiliy of ) guaranees he mean square exponenial synchronizaion of he coupled nework of Lur e sysem as given by 7). n he preceeding secion, we have derived sufficiency condiion for mean square sabiliy of Lur e sysem. n his subsecion, we combine hese wo resuls o obain sufficiency condiion for mean square exponenial synchronizaion of he nework of Lur e sysems. The following Lemma provides he sufficiency condiion for mean square synchronizaion. Lemma : The sysem described by 7) is mean square exponenial synchronizing if here exiss a symmeric posiive definie marix P R )n )n such ha, P = Â Λ d GC) PÂ Λ d GC) +σα k A α k PA αk + Â Λ d GC) P ˆB ˆΣ ˆB P ˆB ) ˆB PÂ Λ d GC) σi ja α k P B ˆΣ ˆB P ˆB ) ˆB PA αk + R 3) S and ˆΣ ˆB P ˆB > for some symmeric marix R > and  :=  ˆB ˆΣ Ĉ = A, A = A BΣ C. Proof: The proof follows from 7), ), Lemma 9 and Theorem 3. The above sufficiency condiion is very difficul o verify for large size neworks due o compuaional complexiy associaed wih solving he Riccai equaion. n paricular he marix P is of size ) n ) n having ) n + )n variables o be deermined. The number of variables increases quadraically wih change in sysem dimension or size of nework. n he following resuls, we exploi he idenical naure of sysem dynamics o provide more conservaive sufficien condiion bu wih subsanially reduced compuaional effors. The sufficiency condiion is based upon a single represenaive dynamical sysem modified using nework characerisics, reducing number of variables o nn+). The new sufficien condiion is also very insighful as i highlighs he role played by he nework propery, in paricular he second larges eigenvalue of he inerconnecion Laplacian, and he saisics of uncerainy in he sufficiency condiion. The saisics of uncerainy is capured using he following definiion of coefficien of dispersion. Definiion Coefficien of Dispersion): Le ξ R be a random variable wih mean µ > and variance σ >. Then, he coefficien of dispersion γ is defined as γ := σ µ To uilize he above definiion in subsequen resuls we make an assumpion on he sysem Assumpion : For all edges i, j) in he nework, he mean weighs assigned are posiive, i.e. µ i j > for all i, j). Furhermore, he coefficien of dispersion of each link is given by γ i j = σ i j µ i j, and γ = max i, j) {γ i j }. This assumpion simply saes ha he nework connecions are posiively enforcing he coupling. The following heorem provides a sufficiency condiion for synchronizaion of he coupled sysems based on he sabiliy of a signle modified sysem. Theorem 3: The coupled sysem 7) is mean square exponenially synchronized if here exiss a symmeric posiive definie marix P > such ha Σ B PB > and P = A λ GC) PA + λ GC) +C Σ C + R + A λ GC) PBΣ B PB) B PA + λ GC) + γλ C G PGC +C G PBΣ B PB) B PGC ) 4) for R >, A = A BΣ C and λ is second smalles eigenvalue of Λ Fiedler eigenvalue, λ > implies conneced graph) Proof: To prove his resul, we assume a block diagonal Lyapunov marix P = P and simplify he problem using maximum coefficien of dispersion. We hen pose i as an L using Schur s complemen mehod and prove he resul. Please refer o he Appendix secion of his paper for he complee proof. Remark 4: We have derived he sufficien condiion for mean square exponenial synchronizaion of coupled n- dimensional Lur e sysems by providing a sufficien condiion for a single n-dimensional Lur e sysem. This significanly reduces he compuaional load in deermining he sufficien condiion for synchronizaion of he coupled dynamics as he nework size increases. should be noed ha he sufficien condiion as provided in 4) can be wrien as a Riccai equaion obained for he sochasic Posiive Real Lemma condiion as derived in Theorem 3. Wriing µ c := λ and σ c := γλ we can wrie 4) as P > A µ c GC) PA µ c GC) + σ c C G PGC + A µ c GC)PB Σ B PB ) B PA µ c GC) + σ c C G PB Σ B PB ) B PGC 5)

6 Comparing wih condiion in Theorem 3, equaion 5) is he sufficien condiion for mean square sabiliy of x + = A ξ GC)x Bφy ), y = Cx 6) where ξ is an i.i.d. random variable wih mean µ c and variance σc. Thus he coefficien of dispersion of ξ is given by γ c = σ c µ c = γ. Thus he synchronizaion of he coupled dynamics is guaraneed by he mean square exponenial sabilizaion of an individual sysem, wih parameric uncerainy in he sae marix muliplying he coupling marix, having coefficien of dispersion wice ha of he maximum coefficien of dispersion of he uncerain links of he nework. D. nerplay of inernal dynamics, nework opology, and uncerainy characerisics The objecive of his secion is o undersand he inerplay of inernal dynamics, nework opology, and he uncerainy characerisics for he synchronizaion of large scale nework sysem. To achieve his goal we consider a firs order agen dynamics. The firs order agen dynamics allows us o simply he condiion of main Theorem 3 and bring ou explicily he connecion beween he inernal dynamics, nework opology and uncerainy. We consider firs order agen dynamics x + = ax φx ) 7) where x R is a scalar and φ ) : R R saisfies φx) φy)) x y δ φx) φy)) > for all x,y R. The above firs order dynamics is special case of Eq. ) wih B = C =. The goal is o synchronize firs order sysems over a nework wih mean Laplacian graph Laplacian L d having Fiedler eigenvalue λ, and maximum link uncerainy dispersion coefficien γ. The oher parameers for he problem are inernal dynamics of he agens capured by parameer a, secor for nonlineariy δ, and he inerconnecion gain g refer o Eq. 3)) of he Laplacian. We have following heorem. Theorem 5: Consider he problem of synchronizaion of sysems of he form 7) over an uncerain nework wih mean Laplacian L d and nework inerconnecion gain g. The sufficien condiion for synchronizaion is given by ) δ >, and ) κ < δ ) where κ = a λ g) + γλ g and a = a δ. Proof: We subsiue he parameers for he D sysem from 7) in equaion 4) o obain he sufficiency condiion for synchronizaion of D sysems. f we suppose he Lyapunov funcion P in 4) is denoed by q R, simplifying he equaion we ge q > a λ g) + γλ g ) qδ δ q ) + δ 8) where δ > q and a = a δ. This may be simplified as > q + δκ ) ) q + 9) δ where κ = a λ g) + γλ g. Equaion 9) is quadraic in he Lyapunov funcion q. We check for feasibiliy of a soluion for his equaion when δ > q. For he value of he quadraic in 9)o be negaive we mus have wo real roos for he equaion given by q and q. The quadraic funcion of q is negaive if and only if q < q < q. Furhermore, we also impose he condiion ha a leas δ > q so ha a leas some roos saisfy he condiion δ > q. Using hese condiions we ge he sysem has posiive roos only if δκ ) ) 4 > 3) δ Simplifying his condiion along wih he given condiion δ > and requiremen δ > q we ge he desired resul. To undersand he inerplay of various parameers involved in he problem, we perform simulaion resuls. We idenify he feasibiliy region in λ γ space i.e., range of parameer values where he sufficiency condiions from Theorem 5 are saisfied. We vary γ beween,5] and λ beween,]. Examining Fig. 3a, which is obained for fixed values of a =.5,g =., and δ = 4, we noice some ineresing inerplay beween parameer γ and λ which will be noed in he following remark. Remark 6: We observe ha, here exiss a criical value of λ, say λl, below which no synchronizaion occurs Region yellow) in Fig. 3a). Recall ha λ is he measure of conneciviy of mean nework wih larger value of λ implies more conneced nework. Hence, we require minimum degree of conneciviy for he synchronizaion o occur. The parameer values where synchronizaion is possible, is marked as Region blue) in Fig 3. For fixed value of λ, an increase in he value of γ i.e., he uncerainy), will cause he sysem o cease o be in synchronized sae. So as expeced, high noise level requires higher degree of conneciviy of he nework. However, ineresing inerplay beween λ and γ is obained for large values of λ Region in Fig 3). There exiss a criical value of λ, say λu, above which no synchronizaion is possible. Thus if here is oo much communicaion beween he sysems he uncerainy doesn remain localized and spreads wihin he nework faser. Furhermore, we see ha wha ever he graph here is a maximum amoun of noise γ in Fig. 3a) he nework can handle for a given sysem, and here exiss an opimal nework conneciviy for i. Similar simulaion resuls are also performed by varying parameers a,g, and δ o observe he effecs of hese parameers. Following conclusions are drawn from hese parameer variaions. The increase in inernal insabiliy for higher value of a wih require improved nework conneciviy for synchronizaion and hence increase in criical value λ l. λ u is also proporional o a indicaing high insabiliy in sysems inerferes wih synchronizaion if here is high level of communicaion. n Fig. 3b, we show he region of synchronizaion/desynchronizaion in λ γ space for parameer values of a =.5,δ = 4, and g =.. ncrease in gain g leads o increase in region where synchronizaion occurs in λ γ parameer space. λ u

7 and λl are inversely proporional o he value of g indicaing ha high gain is derimenal o synchronizaion of highly conneced nework bu migh aid synchronizaion of sparsely conneced neworks. n Fig. 3c, we show he region of synchronizaion/desynchronizaion in λ γ space for parameer values of a =.5, δ = 4, and g =.. The parameer δ is inversely proporional o secor of nonlineariy i.e., increase in δ leads o smaller secor of nonlineariy. λi is independen of δ while λu is direcly proporional o δ. Thus we conclude ha high level of communicaion is harmful for synchronizaion of highly nonlinear sysems wih large secors. n Fig. 3d, we show he region of synchronizaion/desynchronizaion in λ γ space for parameer values of a =.5, δ = 8, and g =.. wih periodic moion in each of he well. The oscillaors are iniialized so ha wo of he oscillaors sars in one poenial well and he oher wo in he second well. We sudy he synchronizaion of four coupled oscillaors conneced over he nework as shown in Fig. 4 wih oupu error coupling. We make he links connecing sysems S o S3, S o S3 and S o S4 uncerain which are shown as dashed red lines in Fig. 4 Fig. 4. a) ework conneciviy of he sysems wih uncerain edges in red The mean Laplacian for his nework is given by + µ3 µ3 + µ3 + µ4 µ3 µ4 Ld = µ3 µ3 µ3 + µ3 µ4 µ4 b). The uncerainies are modelled as i.i.d. Bernoulli uncerainies where he link connecs wih probabiliy p and disconnecs wih probabiliy p for each uncerainy. The mean value for each connecion is µ3 = µ3 = µ4 = p while he coefficien of variaion γ = p. We choose he coupling marix G as G = g ] where we se g =.. 6 d) 4.5 θ Fig. 3. Region of synchronizaionblue)/desynchronizaionyellow) in λ γ parameer space a) a =.5, g =., and δ = 4; b) a =.5, g =., and δ = 4; c) a =.5, g =.5, and δ = 4; d) a =.5, g =., and δ = 8.5 do θ c) V. S ULATO R ESULTS We consider nework of coupled oscillaor sysem wih linear coupling and sochasic uncerainy in heir ineracions. The dynamics of he individual oscillaor is given by second order differenial equaion θ k = κ sin θk. We wrie he individual oscillaor sysem in Lur e form as follows x φ y), y = x x = κπ κ where we se κ =. The above sysem is hen discreized using a zero order hold. We assume ha he nonlineariy and he nework ineracion change only a discree inervals and are consan during an inerval. We choose he sampling ime o be T =. seconds. The phase space dynamics of he discree ime uncoupled oscillaor sysem in shown in Fig. 5. The phase space dynamics consiss of wo poenial wells 6 4 θ a) Time b) Fig. 5. a) Sae space dynamics of uncoupled oscillaor; b) θ dynamics of four oscillaors. n Fig. 6, we show he simulaion resuls for wo differen value of non-erasure probabiliy p. We noice ha for p =.5 he oscillaors canno synchronize and some are reained wihin heir iniial condiion well. The minimum non-erasure probabiliy required for synchronizaion, as prediced by solving he sufficiency condiion is p =.6. This is obained by solving he Riccai equaion using L s and semidefinie programming SDP) echniques. A p =.6 we see he sysems synchronize and are able o pull he oscillaors ino a common well.

8 x dynamics error y dynamics y dynamics error x dynamics error S S x dynamics error S S3 x dynamics error S S Time x a) Time x y dynamics error S S y dynamics error S S3 y dynamics error S S4 c) 4 x dynamics e) x dynamics error y dynamics error y dynamics 6 4 x dynamics error S S x dynamics error S S3 x dynamics error S S Time x 5 3 b) y dynamics error S S y dynamics error S S3 y dynamics error S S Time x d) x dynamics Fig. 6. a) X-dynamics error for p =.5, b) X-dynamics error for p =.6, c) Y-dynamics error for p =.5, d) Y-dynamics error for p =.6, e) Phase space plo for p =.5, f) Phase space plo for p =.6 V. COCLUSOS n his paper we sudy he problem of synchronizaion of Lur e sysems over an uncerain nework. This problem is presened as a special case of he problem of sabilizaion of Lur e sysem wih parameric uncerainy. Oher special case of his problem include conrol of Lur e sysem over an uncerain nework which have been previously sudied by he auhors. These resuls are used o obain some insighful resuls for he problem of synchronizaion over uncerain neworks. We conclude ha mean square exponenial syncronizaion of he coupled dynamics is governed by mean square exponenial sabiliy of a specific sysem wih parameric uncerainy in he sae marix muliplying he coupling marix, and having coefficien of dispersion wice ha of he maximum dispersion in he nework links. This sufficien condiion may be solved as an L using he L decomposiion similar o he Posiive Real Lemma. This can also be used o deermine he maximum amoun of dispersion olerable wihin he nework links. f he randomness in he nework links is highly clusered hen i will be more difficul o synchronize he sysem. We provide a sufficiency condiion for synchronizaion expressed in erms of he second smalles eigenvalue of he Laplacian for he mean inerconnecion and he wors case saisics of link uncerainy f) in he form of coefficien of dispersion CoD). Uilizing idenical dynamics of individual subsysems, we show ha he sufficiency condiion is only a funcion of individual subsysem dynamics and mean nework characerisics. This makes he sufficiency condiion aracive from he poin of view of compuaional complexiy for large scale neworks. Furhermore, sudying he sufficiency condiion for special case of -D sysems, we derive imporan inferences as o he inerplay of variaous parameers, like sysem dynamics, coupling, mean nework conneciviy and randomness in inerconnecion, and heir effec on nework synchronizaion. is shown ha in general a very high amoun of communicaion is deerimenal o synchronizaion of he sysems as his allows uncerainy o spread faser hrough he nework raher han remain localized. Also, i is observed ha a high coupling gain is deerimenal for highly inerconneced graph. Finally we presen simulaion resuls which show he synchronizaion of oscillaors which is an imporan problem in various fields from power sysems o biology. REFERECES ] L. Schenao, B. Sinopoli,. Franceschii, K. Poolla and S. Sasry, Foundaions of conrol and esimaion over Lossy neworks, Proceedings of EEE, vol. 95, no., pp , 7. ]. Elia, Remoe sabilizaion over fading channels, Sysems and Conrol Leers, vol. 54, pp , 5. 3] A. Diwadkar and U. Vaidya, Limiaion on nonlinear observaion over erasure channel, EEE Transacions on Auomaic Conrol, vol. 58, no., pp , 3. 4] A. Diwadkar and U. Vaidya, Conrol of LTV sysems over uncerain channel, nernaional Journal of Robus and onlinear Conrol,. 5] U. Vaidya and. Elia, Limiaion on nonlinear sabilizaion over packe-drop channels: Scalar case, Sysems and Conrol Leers,. 6] U. Vaidya and. Elia, Limiaion on nonlinear sabilizaion over erasure channel, in Proceedings of EEE Conrol and Decision Conference, Alana, GA,, pp ] G. B. San, A. Hamadeh, R. Sepulchre, and J. Gonalves, Oupu synchronizaion in neworks of cyclic biochemical oscillaors, in American Conrol Conference, 7, pp ] J. Yao, H. O. Wang, Z. H. Guan, and W. Xu, Passive sabiliy and synchronizaion of complex spaio-emporal swiching neworks wih ime delays, Auomaica, vol. 45, no. 7, pp. 7 78, 9. 9] L. Scardovi,. Arcak, and E. Sonag, Synchronizaion of inerconneced sysems wih applicaions o biochemical neworks: An inpuoupu approach, Auomaic Conrol, EEE Transacions on, vol. 55, no. 6, pp ,. ] P. Florchinger, A passive sysem approach o feedback sabilizaion of nonlinear conrol sochasic sysems, in Proceedings of he 38h EEE Conference on Decision and Conrol, vol. 4, 999, pp ] W. Lin and T. Shen, Robus passiviy and feedback design for minimum-phase nonlinear sysems wih srucureal uncerainy, Auomaica, vol. 35, no., pp , 999. ] R. ondragn, D. Arrowsmih, and J. Pis, Chaoic maps for raffic modelling and queueing performance analysis, Performance Evaluaion, vol. 43, no. 4, pp. 3 4,. 3] K. Park and W. Willinger, Self-Similar ework Traffic and Performance Evaluaion. ew York: Wiley,. 4] A. Diwadkar, S. Dasgupa, and U. Vaidya, Sabilizaion of sysem in lure form over uncerain channels, in American Conrol Conference,, pp ] P. Lancaer and L. Rodman, Algebraic Riccai Equaions. Oxford: Oxford Science Publicaions, ] W. Haddad and D. Bernsein, Explici consrucion of quadraic Lyapunov funcions for he small gain heorem, posiiviy, circle and Popov heorems and heir applicaion o robus sabiliy. Par : Discree-ime heory, nernaional Journal of Robus and onlinear Conrol, vol. 4, pp , 994.

9 V. APPEDX n he appendix we provide proofs for some of he imporan resuls we prove in he paper. Proof: Theorem 3] We show he condiions in Theorem 3 are indeed sufficien by consrucing an appropriae Lyapunov funcion ha guaranees mean square sabiliy. We will prove he resul in Theorem 3 for Case. We hen use ha resul o prove Case as he dual of Case. Firs, noe ha 3) holds if and only if P =E Ξ) A Ξ))PAΞ)) ] + R P + E Ξ) A Ξ))PB C )Σ B PB) C B PAΞ))) ] 3) The equivalence of he wo equaions 3) and 3) is observed based on 5] Proposiion.,). ow consider he Lyapunov funcion V x ) = x Px. Then, he condiion for he sysem o be mean square sable wih Lyapunov funcion V x ) is given by E Ξ) V x + ) V x )] =x EΞ A Ξ)PAΞ) ] P ) x + x E Ξ A Ξ)BP]φy,) Subsiuing from 3) in 3) we ge + φ y,)b PBφy,) 3) E Ξ) V x + ) V x )] = x R P x) + φ y,)b PBφy,) E Ξ x B PAΞ) C ) Σ B PB) B PAΞ) C ) ] x + Ex A Ξ)BPφy,)] + Eφ y,)b PAΞ)x ] 33) Le ζ Ξ)) be given by ζ = Σ B PB) B PAΞ)) C ) x Σ B PB) φy,) 34) Applying algebraic manipulaions as adoped in 6] o 33) and 34), we ge, E Ξ V n + )) V n) = x R ] P x E Ξ ζ ζ φ y,)y Dφy,)) From condiion of secor nonlineariy as given in Assumpion we ge φ y,)y Dφy,)) >, which gives us, E Ξ V x + )) V x )] < x Rx < This implies mean square exponenial sabiliy of x and hence Case is proved. Case is now he dual o Case by a simple argumen as shown in 4]. Proof: Lemma 9] Consider equaion ). We have We now have ẑ = ẑ ẑ = x U d n ) U d n) x x Ud U d n) x 35) U d U d = V dv d = 36) Subsiuing 36) in 35) we ge ẑ = x ) ) n x ) ) ) = x n n n x = x x x x = j i, j= x i x j ) x i x j ) 37) ow, mean square exponenial sabiliy of ) implies here exiss K > and < β < such ha E Ξ k= j k, j= k= j k, j= This gives us E Ξ ẑ Kβ ẑ, x k x j Kβ k= E Ξ x k x j Kβ k= j k, j= j k, j= x k x j, x k x j, E Ξ x k x l K +,i k j=, j i xi x j ) x k β x k xl xl. Wriing Kẽ ) := K +,i k j=, j i xi x j we ge for all sysems S k and S l, x k xl ) E Ξ x k x l Kẽ )β x k xl. Hence he proof. Proof: Theorem 3] We know mean square exponenial synchronizaion is guaraneed by condiions in Lemma. Consider P = P where P > is a symmeric posiive definie marix ha saisfies Σ B PB >. This gives us Σˆ ˆB P ˆB >. Using his we wrie he condiion in 3) as follows P > Â Λ d GC) P)Â Λ d GC) +σα k A α k P)A αk + Â Λ d GC) P) ˆB ˆΣ ˆB P) ˆB ) ˆB P)Â Λ d GC) +σα k A α k P) B ˆΣ ˆB P) ˆB ) ˆB P)A αk + C Σ C 38) Since A αk = ˆl i j ˆl i j GC we can wrie 38) as P > A λ j GC] P)A λ j GC] +σα k ˆl αk ˆl α k GC) P) ˆl αk ˆl α k GC) + A λ j GC] PB Σ B PB ) ) B P) A λ j GC] + σ α k ˆl αk ˆl α k GC) PB Σ B PB ) ) B P) ˆl αk ˆl α k GC) + C Σ C 39)

10 where A λ j GC] = Â Λ d GC). nequaliy 39) can furher be simplified using he kronecker produc muliplicaion rule as P > A λ j GC] P)A λ j GC] + σα k ˆl αk ˆl α k C G PGC + A λ j GC] PB Σ B PB ) ) B P) A λ j GC] C G PB Σ B PB ) ) B PGC + σα k ˆl αk ˆl α k + C Σ C 4) We know ha σ α k ˆl αk ˆl α k = γ αk µ αk ˆl αk ˆl α k γ µ αk ˆl αk ˆl α k γ ˆΛ d 4) conclude ha 44) is mos vulnerable o he smalles nonzero eigenvalue of L d given by λ. λ is known as he Fiedler eigenvalue and is associaed wih graph conneciviy. We hus prove he desired resul given in 4). ow, subsiuing 4) in 4) a sufficien condiion for inequaliy 4) o hold is given by P > A λ j GC] P)A λ j GC] + A λ j GC] PB Σ B PB ) ) B P) A λ j GC] + γ ˆΛ d C G P + PB Σ B PB ) ) ) B P GC + C Σ C 4) Equaion 4) is essenially a block diagonal equaion which gives he sufficien condiion for mean square synchronizaion o be P > A λ j GC) PA λ j GC) + γλ j C G PGC +C Σ C + A λ j GC) PB Σ B PB ) B PA λ j GC) + γλ j C G PB Σ B PB ) B PGC 43) for all non-zero eigenvalues λ j of ˆΛ d. Using Schur complemen we can equivalenly wrie 43) for a given λ j and Ḡ = G, as an L given by where = + λ j > 44) P C Σ C A P A PB PA P B PA Σ B PB, 45) C Ḡ P C Ḡ PB γc Ḡ P γc Ḡ PB PḠC = B PḠC γpḡc P. γb PḠC Σ B PB arix given in 45) is posiive only if individual sysem dynamics is sable. We do no consider his case as his makes he synchronizaion problem rivial. Thus is indeerminae. Since we require 44) o hold for all nonzero eigenvalues λ j of he deerminisic Laplacian L d, we