(t)=?LTe(G)?InxT(t)+yC(t)HT:?y(t)=?E(G)TEw(G)?Inx(t)⊙TTCTT??x˙(12)

whichisshowninFigure

2.(t)=?LCe(G)?InxC(t)+yT(t)HC:?y(t)=?E(G)TEw(G)?Inx(t)⊙CCTCC??x˙(13)

H,

Fig.2.EdgedynamicsasanoutputfeedbackinterconnectionstructurebetweenHT-subsystemandHC-subsystem.

Remark8Thedecompositionofthespanningtreeandco-spanningtreesub-graphactuallyhasbeenwildlyappliedtosolvethemagnetostaticproblems,suchastree-cotreegauging[25],?niteelementanalysis[26],inwhichthede-compositionisreferredasTree-CotreeSplitting(TCS)technique.

Asknownthatthespanningtreestructureplaysavitalroleintheanalysisofnetworkedmulti-agentsystem,butscarcelyanyliteraturescouldo?eradetailedinterpretationforthatfromthecontrolpro?les.Next,wearegoingtohighlighttheroleofthespanningtreesubgraphbyprovidingamodelreductionrepresentationintermsofthecorrespondingdynamicsonit.NoticethatET(G)T(G)=EC(G)asmentionedin(1),thereforetheco-spanningtree

statescanbereconstructedthroughthematrixT(G)as

xC(t)=T(G)T?InxT(t)(14)

whichshowstheco-spanningtreestatescanserveasaninternalfeedbackontheedgesofthespanningtreesubgraphshowninFigure3.Inthemeantime,wealsohave

xe(t)=R(G)T?InxT(t).

10(15)

?Taking

(14)into(12)leadtoareducedmodelHT

TwT

x˙T(t)=(?LT

e(G)?ET(G)E⊙C(G)T(G))?InxT(t)

ww

=?ET(G)T(E⊙(G)+E⊙(G)T(G)T)?InxT(t)TC

w=?ET(G)TE⊙(G)R(G)T?InxT(t)

?e(G)?Inx(t)=?L

T

(16)

whichcapturesthedynamicalbehaviourofthewholesystem.Inthispaper,

?e(G)=E(G)TEw(G)R(G)TastheessentialedgeLaplacian.wereferL⊙T

Fig.3.HC(t)-subsystemservesasaninternalfeedbackstate.

Inthesubsequentanalysis,thereducedmodelassociatedwiththeessentialedgeLaplacianwillplayanimportantrole.

?e(G)containsexactlyallthenonzeroLemma9TheessentialedgeLaplacianL

eigenvaluesofLe(G).Additionally,wecanconstructthefollowingLyapunovequationas

?e(G)+L?e(G)TH=IN?1HL

whereHisapositivede?nitematrix.

(17)

PROOF.Beforemovingon,weintroducethefollowingtransformationma-trix:

Se(G)=R(G)Tθe(G)

????

?

??

??

??

Se(G)

?1

=

R(G)R(G)

θe(G)

??T?1

T

R(G)?

whereθe(G)denotetheorthonormalbasisofthe?owspace,i.e.,E(G)θe(G)=0.Sincerank(E(G))=N?1,onecanobtainthatdim(θe(G))=N(E(G))andθe(G)Tθe(G)=IL?N+1.ThematrixR(G)isde?nedvia(2).Applying

11

thesimilartransformationleadto

Se(G)Le(G)Se(G)=?1????e(G)L

0L?N+1×N?1TwET(G)E⊙(G)θe(G)?0L?N+1?

Clearly,theeigenvaluesoftheblockmatrixarethesolutionof

λ(L?N+1)?.?e(G)=0detλI?L????

?e(G)hasexactlyallthenonzeroeigenvaluesofLe(G).AswhichshowsthatL

thus,wecanconstructthefollowingLyapunovequationas

?e(G)+L?e(G)TH=IN?1HL

whereHispositivede?nite.

Remark10In[3],byusingthesimilartransformationmentionedabove,theedge-descriptionsystemcouldbeseparatedintocontrollableandobservableparts.Italsopointsoutthattheminimalrealizationofthesystemcontainsonlythestatesacrosstheedgesofspanningtree.Additionally,byusingtheessentialedgeLaplacian,wecouldextremelysimplifythecomplexityoftheanalysisofmulti-agentsystems,sinceitonlypreservesthenonzeroeigenval-uesoftheedge(graph)Laplacian.Infact,similarrepresentationshavebeenimplicitlyrealizedinrecentworks[18][27][28][29].However,theseliteraturesdidnotrevealtheexplicitconnectionofthealgebraicpropertiesfromsystem-aticandstructuralview.

4QuantizedEdgeAgreementwithSecond-orderNonlinearDy-

namicsunderDirectedGraph

Whilethequantizatione?ectsonmulti-agentsystemassociatedwithundi-rectedgraphhasbeenwidelystudied,thescenarioconsideringthedirectedgraphisstillverychallenging,sincethequantizationmaycauseundesirableoscillatingbehaviourunderdirectedtopology[15].Inthissection,theedgeagreementofsecond-ordernonlinearmulti-agentsystemunderquantizedmea-

wsurementsisstudied.Toeasethenotation,wesimplyuseE,E⊙andLeinstead

wofE(G),E⊙(G)andLe(G)inthefollowingparts.

Consideringthedynamicsofthenetworkedagentsasdescribingin(3)and

12

(4),bydirectlyapplyingthequantizedprotocol(5),weobtain

??x˙i(t)?????????????=vi(t)N??j∈NiN??v˙i(t)=f(xi(t),vi(t),t)+α

+βj∈NiaijQ(xj(t)?xi(t))aijQ(vj(t)?vi(t))

Toeasethedi?cultyoftheanalysis,wetechnicallychoseα=σ2andβ=σ3(σ>0)asin[30].Thenthesystemcanbecollectedas

???x˙(t)??????=v(t)????2wTv˙(t)=F(x(t),v(t),t)?σE⊙?InQE?Inx(t)

w?σ3E⊙?InQET?Inv(t)????(18)

withx(t),v(t)andF(x(t),v(t),t)denotingthecolumnstackvectorofxi(t),vi(t)andf(xi(t),vi(t),t),respectively.

Byleft-multiplyingET?Inofbothsidesof(18),weobtain

??x˙

withxe=ET?Inx,ve=ET?Inv.=ve?v˙e=ET?InF?σ2Le?InQ(xe)?σ3Le?InQ(ve)e(19)

De?neexe=Q(xe)?xeandeve=Q(ve)?veasin[12],then(19)canbewrittenasthefollowingform:

??˙e(t)??x=ve(t)

T23v˙(t)=E?IF?σL?Ix?σLe?Inveenene????σ2Le?Inexe?σ3Le?Ineve.

TLetz=xTandω=exeTeveTvee

acompactmatrixformasfollows:(20)????T????T,thensystem(20)canberecastin

withL=???z˙=F+L?Inz+L1?Inω0LIL???,?σ2Le?σ3Le

ConsideringthenonlineartermFaswellasthezeroeigenvaluesthatLecontains,intuitiveanalysisisnotsimple.However,thereducededgeagreement

13L1=???0L?σ2Le?σ3Le0L???andF=???0ET?InF???.

model(16)willbeofgreathelp.Tobeginwith,wemakeuseofthefollowingtransformation

?1Se

?Inxe=

?

?xT???

?

?1Se

?Inve=

?

?vT???

?

?1Se

?Inexe=

?????

RRT

θe?Inexe

???1

T

???1

R?Inexe?

??

?

?1Se

?Ineve=

??T

?????

RRT

θe?Ineve

T

R?Ineve?

TThenwede?nezT=xTvTT

writteninto

??

?.

w?=ETE⊙andL.Finally,system(20)canbeOT

withLT=?

??

z˙T=FT+LT?InzT+LT1?Inω

0N?1

?

?

?

(21)

?

?

Tofurtherlookattherelationbetweenthequantizationintervalandtheedge

agreement,weproposethefollowingtheorem.

0IN?1????0N?1×L0N?1×L?

andF=,L=???.?T1?T

T2?3?2?3?ET?InF?σLe?σLe?σLO?σLO

Theorem11Consideringthequasi-stronglyconnecteddirectedgraph??Gas-??

sociatedwiththeedgeLaplacianLe,supposeQ=?PLT+LTPwithTHσH

λmin(Q)?2max(ξ1,ξ2)??P??>0.Then,underthequantizedprotocol(5),system(21)hasthefollowingconvergenceproperties:

(1):Withuniformquantizers,theagentsconvergetoaballofradius

√

????2??????zT??≤

λmin(Q)?2max(ξ1,ξ2)??P??P=

??σH?

H?

?

?,whereHisobtainedby(17).Ifσ>

??

2

+1and

(22)

whichiscentredattheagreementequilibrium;

(2):Withlogarithmicquantizers,theagentsconvergeexponentiallytothede-siredagreementequilibrium,providedthatδlsatis?es

14

δl<λmin(Q)?2max(ξ1,ξ2)??P??

λmin(P)e?π

λmax(H)

Inthemeanwhile,wenoticethat

|FT|=????T??ET?????InF??≤??

2nLδu.

Bycombining(25)and(26),onecanobtain

??22˙(z)≤?λmin(Q)????+2max(ξ,ξ)??P??|z|??zV12TTT√+2????(26)

2nLδu??PLT1??.

Clearly,theedgeagreementcanbereachedandtheradiusoftheagreementneighbourhoodisas(22).

Consideringthatz=?????RT

0(N×L?N+1)R

quantizer|Qu(a)?a|≤δl|a|,wehave

|ω|≤δl|z|≤

Combining(25)and(27),wehave

????0(N×L?N+1)?T??zT,thenforthelogarithmic??????T??δl??R??|zT|.(27)??22˙(z)≤?λmin(Q)????+2max(ξ,ξ)??P??|z|??zV12TTT??????T??+2δl??PLT1????R??|zT|2????2????=?π??zT??.

Obviously,while(23)issatis?ed,theedgeLaplaciandynamics(21)convergesexponentiallytothedesiredagreementequilibrium.

Moreover,onecanobtainthat

??2˙(z(t))≤?π??V??z??≤?TT????π

ByapplyingtheComparisonLemma[31],wehave

V(zT(t))≤e?π

λmin(P)e?π

λmin(P)r

πln

inherentnonlineardynamicsf(xi(t),vi(t),t):R×R3→R3isdescribedbyChua’scircuit

f(xi(t),vi(t),t)=(ζ(?vi1(t)+vi2(t)?l(vi1(t))),

τ(vi1(t)?vi2(t)+vi3(t)),?χvi2(t))T

wherel(vi1(t))=bvi1(t)+0.5(a?b)(|vi1(t)+1|?|vi1(t)?1|).Thesystemischaoticwhenζ=0.01,τ=0.001,χ=0.018,a=?4/3andb=?3/4.InviewofAssumption2,bycalculationonecanobtainξ1=0andξ2=4.3871×10?3[23].