accordingtocertain??permutations,??theincidencematrixE(G)canalwaysberewrittenasE(G)=ET(G)EC(G)aswell.Sincetheco-spanningtreeedges

canbeconstructedfromthespanningtreeedgesviaalineartransformation

[3],suchthat

ET(G)T(G)=EC(G)

withT(G)=ET(G)ET(G)

[21].Wede?ne??T(1)???1ET(G)TEC(G)andrank(E(G))=N?1from

????R(G)=IT(G)(2)

andthenobtainE(G)=ET(G)R(G).ThecolumnspaceofE(G)Tisknown

asthecutspaceofGandthenullspaceofE(G)iscalledthe?owspace,whichistheorthogonalcomplementofthecutspace.Interestingly,the????rowsofR(G)formabasisofthecutspaceofandtherowsof?T(G)TIformabasisofthe?owspacerespectively[21].

Lemma1([22]):ThegraphLaplacianLG(G)ofadirectedgraphGhasatleastonezeroeigenvalueandallofthenonzeroeigenvaluesareintheopenright-halfplane.Inaddition,LG(G)hasexactlyonezeroeigenvalueifandonlyifGisquasi-stronglyconnected.

2.2ProblemFormulationandQuantizedInformation

WeconsideragroupofNnetworkedagentsandthedynamicsofthei-thagentisrepresentedby

x˙i(t)=vi(t)

v˙i(t)=f(xi(t),vi(t),t)+ui(t)(3)(4)

5

wherexi(t)∈Rnistheposition,vi(t)∈Rnisthevelocityandμi(t)∈Rnisthecontrolinput.Thenonlineartermf(xi(t),vi(t),t):Rnandsatisfythefollowingassumption.

×Rn→RnisunknownAssumption2Foranonlinearfunctionf,thereexistsnonnegativecon-stantsξ1andξ2suchthat

|f(x,v,t)?f(y,z,t)|≤ξ1?x,|xv,?y,yz|+∈ξ2Rn|v;??tz≥|,

0.

Thegoalfordesigningdistributedcontrollawμi(t)istosynchronizetheve-locitiesandpositionsoftheNnetworkedagents.

Thegenerallystudiedsecond-orderconsensusprotocolproposedin[23]isde-scribedasfollows:ui(t)=α

j∈Nj(t)?vi(t)),

fori=1,2···,N,whereα??

Naij(xj(t)?xi(t))+β

>ij0andβ>0arethecoupling??

N∈Naij(vi

strengths.Asin[12],weassumethateachagentihasonlyquantizedmeasurementsoftherelativepositionQ(xi):Rn→Rnisthequantization?xj)andvelocityinformationQ(vifunction.Asthus,theprotocol?isvj),whereQ(.modi?edas

ui(t)=α

??

NaijQ(xj(t)?xi(t))+β

(vj(t)?vi(t))(5)

j∈Nij??

NaijQ∈Ni

fori=1,2···,N.Speci?cally,thequantizercanberepresentedbyQ(ν)=

[q(ν1),q(ν2),···,q(νn)]Tforavectorν=[ν1,ν2,···,νn]T?Rn.Twotypi-calquantizationoperatorsareconsidered:uniformandlogarithmicquantizer.Foragivenδu>0,auniformquantizerqu:R→Rsatis?es|qu(a)?a|≤δu,?a∈R;foragivenδl>0,alogarithmicquantizerql:R→Rsatis-?es|ql(a)?a|≤δlquantizationinterval.|a|,?aWe∈Rde?ne.TheQpositiveconstants[qδuandδlareknown

asu(ν)=?

u(ν1),qu(ν2),···,qu(νn)]T

andQl(ν)=?[ql(ν1),ql(ν2),···,ql(T

|Qu(ν)?ν|≤

√νn)],thenthefollowingboundshold:

3.1DirectedEdgeLaplacian

TheedgeLaplacianin[3]stillremainstoanundirectednotionandisthusinadequatetohandleourproblem.ExtendingtheconceptoftheedgeLapla-ciantodirectedgraphwillbeofgreathelptounderstandmulti-agentsystemsfromthestructuralperspective.

Beforemovingon,wegivethede?nitionofthein-incidencematrixandout-incidencematrixat?rst.

De?nition3(In-incidence/Out-incidenceMatrix)TheN×Lin-incidencematrixE⊙(G)foradirectedgraphGisa{0,?1}matrixwithrowsandcolumnsindexedbynodesandedgesofG,respectively,suchthatforanedgeek=(j,i)∈E,[E⊙(G)]mk=?1form=i,[E⊙(G)]mk=0otherwise.Theout-incidencematrixisa{0,+1}matrixwith[E?(G)]nk=+1forn=j,

[E?(G)]nk=0otherwise.

Incomparisonwiththede?nitionoftheincidencematrix,wecanrewriteE(G)inthefollowingway:

E(G)=E⊙(G)+E?(G).(6)

w(G)canbede?nedasOntheotherhand,theweightedin-incidencematrixE⊙wE⊙(G)=E⊙(G)W(G),whereW(G)isadiagonalmatrixofwk.Thiswilllead

usto?ndoutanovelfactorizationofthegraphLaplacianLn(G).

Lemma4ConsideringadirectedgraphGwiththeincidencematrixE(G)

wandweightedin-incidencematrixE⊙(G),thegraphLaplacianofGhavethe

followingexpressionwLn(G)=E⊙(G)E(G)T.(7)

TwTwTPROOF.Byusing(6),wehaveEw⊙(G)E(G)=E⊙(G)E⊙(G)+E⊙(G)E?(G).

wLetEw⊙i(G),E?i(G)bethei-throwofE⊙(G)andE?(G).Accordingto

Ttheprecedingde?nition,it’sclearthat,Ew⊙i(G)E⊙j(G)=?i(G)fori=j,

TwTEw⊙i(G)E⊙j(G)=0otherwise.ThenwecancollecttermsasE⊙(G)E⊙(G)=

TD(G).Besides,wealsohaveEw⊙i(G)E?j(G)=?wkforj∈Ni,ek=(j,i)and

TwTEw⊙i(G)E?j(G)=0otherwise,whichimpliesthatE⊙(G)E?(G)=?A(G).

wwwThen,wehaveE⊙(G)E(G)T=E⊙(G)E⊙(G)T+E⊙(G)E?(G)T=D(G)?

A(G)=Ln(G).

De?nition5(DirectedEdgeLaplacian)TheedgeLaplacianofadirectedgraphGisde?nedas

wLe(G):=E(G)TE⊙(G).(8)

7

ToprovideadeeperinsightsintowhattheedgeLaplacianLe(G)o?ersintheanalysisandsynthesisofmulti-agentsystems,weproposethefollowinglemma.

Lemma6ForanydirectedgraphG,thegraphLaplacianLGandtheedgeLaplacianLe(G)havethesamenonzeroeigenvalues.IfGisquasi-stronglyconnected,thentheedgeLaplacianLe(G)containsexactlyN?1nonzeroeigen-valueswhichareallintheopenright-halfplane.

PROOF.TheprooffortheweightedversionofLe(G)canbeeasilyextendedfromlemma5ofourpreviouswork[19],thusthedetailisomittedhere.Obviously,ifG=GT,thenGhasL=N?1edgesandalltheeigenvaluesof

Le(G)arenonzero.Inthefollowingpaper,whenwedealwithaquasi-stronglyconnectedgraph,itreferstoageneraldirectedgraphG=GT∪GCunlessnoted

otherwise.

Lemma7Consideringaquasi-stronglyconnectedgraphGoforderN,theedgeLaplacianLe(G)hasL?N+1zeroeigenvaluesandzeroisasimplerootoftheminimalpolynomialofLe(G).

PROOF.Theresultcanbelightlyextendedfromlemma6ofourpreviouswork[19].

Lemma7impliesthatthelinearsystemassociatedwith?Le(G)ismarginallystable.Asthus,theweighteddirectededgeLaplacianholdsthesimilarfunc-tionsasthegraphLaplacianforanalyzingtheinteractingmulti-agentsystem.TheexplicitconnectionbetweentheedgeandgraphLaplacianhasbeenhigh-lightedbyasimilaritytransformationin[3].Actually,byusingthistransfor-mation,wecanderiveareducedmodelrepresentationfortheedgeagreementdynamics.

3.2EdgeAgreementandModelReduction

AlthoughthegraphLaplacianisaconvenientmethodtodescribethegeomet-ricinterconnectionofnetworkedagents,anotherattractivenotion,theedgeagreement,whichhasnotbeenextensivelyexplored,deservesadditionalat-tentionbecausetheedgesareadoptedtobenaturalinterpretationsoftheinformation?ow.

8

Consideringthequasi-stronglyconnectedgraphGandthemostcommonlyusedconsensusdynamics[24]describedas:

x˙=?Ln(G)?Inx

where?denotestheKroneckerproduct.Contrarytothemostexistingworks,

westudythesynchronizationproblemfromtheedgeperspectivebyusingLe.Inthisavenue,wede?netheedgestatevectoras

xe(t)=E(G)T?Inx(t)

(9)

whichrepresentsthedi?erencebetweenthestatecomponentsoftwoneigh-bouringnodes.Takingthederivativeof(9)leadto

x˙e(t)=?Le(G)?Inxe(t)

(10)

whichisreferredasedgeagreementdynamicsinthispaper.Incomparisonto

thenodeagreement(consensus),theedgeagreement,ratherthanrequiringtheconvergencetotheagreementsubspace,desirestheedgedynamics(10)convergetotheorigin,i.e.,limt→∞|xe(t)|=0.Essentially,theevolutionofanedgestatedependsonitscurrentstateandthestatesofitsadjacentedges.Besides,theedgeagreementimpliesconsensusifthedirectedgraphGhasaspanningtree[3].

Asthequasi-stronglyconnectedgraphGcanbewrittenasG=??GT∪GC,then??

wwwtheweightedin-incidencematrixcanberepresentedasE⊙(G)=E⊙(G)E⊙(G)TCviasomecertainpermutationsinlinewithE(G)=ET(G)EC(G).According

tothepartition,wecanrepresenttheedgeLaplacianintotheblockformas

T

wE⊙(G)

????

Le(G)=E(G)=

??

?

LT

e(G)EC(G)

TwE⊙

T

ET(G)(G)

T

Le(G)

C

wE⊙(G)?C

??

TwTwC

withLTe(G)=ET(G)E⊙T(G)andLe(G)=EC(G)E⊙C(G).

(11)

TheedgeLaplaciandynamics(10)canbetranslatedintoaoutputfeedbackinterconnectionofthespanningtreesubsystemHTandtheco-spanningtreesubsystemHC.Actually,byusing(11),theedgeLaplaciandynamicsxe(t)describedby(10)canberewrittenas

?

˙T?x?

x˙C(t)

(t)?

?

?

=

?

??

?

LT

e(G)

TwE⊙

T

ET(G)(G)

T

EC(G)Le(G)

C

wE⊙(G)?C

?

?xT

??In?

?

xC(t)

(t)?

?

?.

9

Asthus,onecanobtainthefollowingoutputfeedbackinterconnectionsystem: