Throughasimplecalculation,wecanobtain

T=

??

?1.00?1.000.00?1.00

??

,R=

??

1.000.000.000.000.001.000.000.000.000.001.000.000.000.000.001.00?1.00?1.000.00?1.00

??

.

SupposethattheweighteddiagonalmatrixisW=diag{0.12,0.24,0.44,0.43,0.09}.Bychoosingσ=1.64,wehave

?e=L

??

0.21

?0.120.000.00

0.090.24?0.24?0.24

0.000.000.440.00

0.090.000.000.43

??

?=,LO

??

0.12

?0.120.000.000.000.24?0.24?0.240.000.000.440.000.000.000.000.43?0.090.00?0.000.00

??

.

SolvingtheLyapunovequation(17)leadsto

H=

??

2.470.160.07?0.26

0.162.860.390.45

0.070.391.14?0.01

?0.260.45?0.011.22

??

.

Directedcalculationyieldsλmax(P)=8.098,λmin(P)=0.6157,??P??=8.098,????

??T??

??PLT1??=6.7121and??R??=2.5.1UniformQuantizer

First,weconsiderthequantizedprotocol(5)withthefollowinguniformquan-tizerastheoneusedin[15]

qu(x)=δu

????

x

2

??

.

Thesimulationresultswithδu=1areshowninFig.5,fromwhichwecanseethatxe(t)andve(t)indeedconvergetoasmallneighbourhoodneartheequilibriumpoints.Toshowthee?ectofδuontheagreementerrors|ess|,wefurthertakeδu=0.01,0.1,2and3torunthesimulation.TheresultsinTab.1showsthat|ess|trendstozerowhenδu→0,whichaccordstoTheorem11.

18

Fig.5.Edgeagreementunderuniformquantizerwithδu=1.

5.2LogarithmicQuantizer

Next,weapplythefollowinglogarithmicquantizertothequantizedprotocol(5):

???????

equ(lnx)

ifx>0ifx=0ifx<0

ql=

whereδl=1?e?δu[15].Tosatisfythestabilityconstraints(23),werequire

δl<0.0301.Thesimulationresultswithδu=0.01andδl=1?e?0.01=0.01areshowninFig.6,fromwhichwecanseethatedgeagreementisindeedachievedaswellasxi(t)andvi(t)reachthedesiredagreementvalues.The

Table1

Thee?ectofδuontheagreementerrors.

δu|ess|

0.10.05

21.98

0???????equ(ln(?x))

estimationoftheconvergencerateisgivenas

|ψ|=

λmax(P)

λmax(P)

t

|zT(0)|fort≥0

withπ=0.5387.From

Fig.7,onecanseethat|zT|exponentiallyconvergetotheorigin.Toillustratethee?ectsofthequantizedintervalontheconvergencerate,wetakeδl=0.01(δu=0.01),andthesimulationconsumes11.88timeunitsforthequantizedsystemtoconverge.Further,whenwetakeδl=0.02(δu=0.0202),ittakes12.18timeunitstoconvergeandwhilechoosingδl=0.03(δu=0.0305),ittakes12.45timeunitstoreachagreement.Obviously,theseresultskeepalignwithouranalysis.Finally,whilechoosingδl=0.9933(δu=5),whichbreaksthecondition(23),thecorrespondingresultsareshowninFig.8,whereagreementcannotbeachieved.

Fig.6.Edgeagreementunderlogarithmicquantizerwithδl=0.01.

6Conclusions

Inthispaper,weproposedageneralconceptofdirectededgeLaplacianwithitsalgebraicproperties.Baseduponthenewgraph-theoretictool,wederivedamodelreductionrepresentationoftheedgeagreementmodel,whichallowsaconvenientanalysisofmulti-agentsystem.Theedgeagreementofsecond-order

20

Fig.7.Theconvergenceestimationof|zT(t)|.

Fig.8.Thetrajectoriesofxeandvearedivergencewithδl=0.9933.

nonlinearmulti-agentsystemunderquantizedmeasurementswasstudied.Un-likepreviousworks,weprovidedexplicitresultsontheedgeagreementerrorsandstabilityconstraintswiththequantizationinterval.Speci?cally,wepro-videdtheexplicitupperboundoftheradiusoftheagreementneighbourhoodfortheuniformquantizers,whichindicatesthattheradiusincreaseswiththequantizationinterval.Whileforthelogarithmicquantizers,wepointedoutthattheagentsconvergeexponentiallytothedesiredagreementequilibrium.Besides,wealsoprovidedtheestimatesoftheconvergencerate.

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