2011Consensus of linear multi-agent systems with reduced-order observer-based protocols

linear consensus

Systems&ControlLetters60(2011)

510–516

ContentslistsavailableatScienceDirect

Systems&ControlLetters

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Consensusoflinearmulti-agentsystemswithreduced-orderobserver-basedprotocols

ZhongkuiLia,XiangdongLiua,PengLinb,WeiRenc,

abc

SchoolofAutomation,BeijingInstituteofTechnology,Beijing100081,China

InstituteofAstronauticsandAeronautics,UniversityofElectronicsScienceandTechnologyofChina,Chengdu610054,ChinaDepartmentofElectricalandComputerEngineering,UtahStateUniversity,Logan,UT84322-4120,USA

articleinfoabstract

Thispaperconsiderstheconsensusproblemsforbothcontinuous-anddiscrete-timelinearmulti-agentsystemswithdirectedcommunicationtopologies.Distributedreduced-orderobserver-basedconsensusprotocolsareproposed,basedontherelativeoutputsofneighboringagents.Amulti-stepalgorithmispresentedtoconstructareduced-orderprotocol,underwhichacontinuous-timemulti-agentsystemwhosecommunicationtopologycontainsadirectedspanningtreecanreachconsensus.Thisalgorithmisfurthermodifiedtoachieveconsensuswithaprescribedconvergencerate.Thesetwoalgorithmshaveafavorabledecouplingproperty.InlightofthemodifiedalgebraicRiccatiequation,analgorithmisthengiventoconstructareduced-orderprotocolforthediscrete-timecase.

©2011ElsevierB.V.Allrightsreserved.

Articlehistory:

Received15November2010Receivedinrevisedform18February2011

Accepted4April2011

Availableonline5May2011Keywords:

Multi-agentsystemConsensus

Reduced-orderobserverConvergencerate

1.Introduction

Cooperativecontrolofagroupofagentshasreceivedcom-pellingattentionfromvariousscientificcommunities.Agroupofautonomousagentscancoordinatewitheachotherviacommuni-cationorsensingnetworkstoperformcertainchallengingtasks,whichcannotbewellaccomplishedbyasingleagent.Itspotentialapplicationsincludespacecraftformationflying,sensornetworks,andcooperativesurveillance[1,2].Intheareaofcooperativecon-trolofmulti-agentsystems,consensusisanimportantandfunda-mentalproblem,whichiscloselyrelatedtoformationcontrol[3]andflockingproblems[4,5].Themainideaofconsensusistode-velopdistributedcontrolpoliciesthatenableagroupofagentstoreachanagreementoncertainquantitiesofinterest.

Consensusproblemshavebeenextensivelystudiedbynumer-ousresearchersfromvariousperspectives.Atheoreticalexplana-tionisprovidedin[6]forthealignmentbehaviorobservedin[7]byusinggraphtheory.In[8],ageneralframeworkoftheconsensusproblemfornetworksofdynamicagentswithfixedorswitchingtopologiesisaddressed.Theconditionsgivenby[6,8]arefurtherrelaxedin[9].Thecontrolledagreementproblemformulti-agentnetworksisconsideredfromagraph-theoreticperspectivein[10].Trackingcontrolformulti-agentconsensuswithanactiveleaderis

Correspondingauthor.Tel.:+14357972831;fax:+14357973054.

E-mailaddresses:wei.ren@usu.edu,wren@u.edu(W.Ren).

consideredin[11]byusinganeighbor-basedstate-estimationrule.Adistributedalgorithmisproposedin[12]toachieveconsensusinfinitetime.ThedistributedH∞controlandconsensusproblemsareinvestigatedin[13,14]fornetworksofagentssubjecttoexternaldisturbances.Theconsensusproblemofnetworksofdouble-andhigh-orderintegratorsisstudiedin[15–18].Sampled-datacontrolprotocolsareproposedin[19,20]toachieveconsensusforfixedandswitchingagentnetworks.Onelimitationintheaforemen-tionedworksisthattheagentdynamicsareassumedtobefirst-,second-,orhigh-orderintegrators,whichmightberestrictiveinmanycases.

Thispaperextendstoconsiderthedistributedconsensusprob-lemsformulti-agentsystemswithcontinuous-anddiscrete-timegenerallineardynamicsanddirectedcommunicationtopologiesbyexpandingonourpreliminarywork[21].Distributedreduced-orderobserver-baseddynamicconsensusprotocols,relyingontherelativeoutputsofneighboringagents,areproposedforboththecontinuous-anddiscrete-timecases.Thedynamicprotocolsherecanberegardedasextensionsofthetraditionalreduced-orderobserver-basedcontrollerforasinglesystemtothoseformulti-agentsystems.Itisshownthattheseparationprincipleoftra-ditionalobserver-basedcontrollersstillholdsinthemulti-agentsetting.Previousworksrelatedtothispaperinclude[22–26].Incontrasttothestaticconsensusprotocolbasedontherelativestatesin[22],theprotocolsinthecurrentpaperrelyontherelativeoutputs.Incontrasttothedynamicprotocolsin[23–26],whosedi-mensionsareequaltoorevenhigherthanthatofasingleagent,the

0167-6911/$–seefrontmatter©2011ElsevierB.V.Allrightsreserved.doi:10.1016/j.sysconle.2011.04.008

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Z.Lietal./Systems&ControlLetters60(2011)510–516511

protocolsinthecurrentpaperarereduced-orderandhencehavelowerdimensions.Inparticular,thefull-orderobserver-basedpro-tocolin[25]possessesacertaindegreeofredundancy,whichstemsfromthefactthatwhiletheobserverconstructsanestimateoftheentirestate,partofthestateinformationisalreadyreflectedinthesystemoutputs.Thereduced-orderprotocolproposedhereelim-inatesthisredundancyandtherebycanconsiderablyreducethedimensionoftheprotocolespeciallyforthecasewheretheagentsareMIMOsystems.

Forthecontinuous-timecase,amulti-stepalgorithmispresentedtoconstructareduced-orderobserver-basedconsensusprotocolforamulti-agentsystemwhosecommunicationtopologycontainsadirectedspanningtree.Itisshownthatasufficientconditionfortheexistenceofsuchaprotocolisthateachagentisstabilizableanddetectable.Anotheralgorithmisfurtherproposedtoconstructaprotocol,underwhichtheagentscanreachconsensuswithaprescribedconvergencerate.Thesetwoalgorithmshaveafavorabledecouplingfeature.Specifically,thefirstthreestepsinthesealgorithmsdealwithonlytheagentdynamics,whilethelaststeptacklesthecommunicationtopology.Thecasewithdiscrete-timeagentdynamicsisalsoconsidered,wheretherow-stochasticmatrix,ratherthantheLaplacianmatrixasinthecontinuous-timecase,isutilizedtocharacterizethecommunicationtopology.InlightofthemodifiedalgebraicRiccatiequation,analgorithmisgiventoconstructareduced-orderprotocoltosolvetheconsensusproblemforadiscrete-timemulti-agentsystemwhosecommunicationtopologycontainsadirectedspanningtree.ItisobservedthatthenonzeroeigenvaluewiththesmallestrealpartoftheLaplacianmatrixplaysakeyroleinthecontinuous-timecase,whilethenon-oneeigenvalueofthestochasticmatrixwiththelargestmagnitudeiscriticalinthediscrete-timecase.

Therestofthispaperisorganizedasfollows.Somebasicno-tationandusefulresultsofthegraphtheoryarereviewedinSec-tion2.Theconsensusproblemsofcontinuous-anddiscrete-timemulti-agentsystemsareinvestigatedin,respectively,Sections3and4.Section5concludesthepaper.2.Conceptsandnotation

LetRn×n

andCn×n

bethesetofn×nrealmatricesandcomplexmatrices,respectively.ThesuperscriptTmeanstransposeforrealmatricesandHmeansconjugatetransposeforcomplexmatrices.INrepresentstheidentitymatrixofdimensionN.Matrices,ifnotexplicitlystated,areassumedtohavecompatibledimensions.Denoteby1thecolumnvectorwithallentriesequaltoone.Re(ζ)denotestherealpartofζ∈C.A BdenotestheKroneckerproductofmatricesAandB.ThematrixinequalityA>(≥)BmeansthatAandBaresquareHermitianmatricesandthatA Bispositive(semi-)definite.AmatrixisHurwitz(inthecontinuous-timesense)ifallofitseigenvalueshavenegativerealparts,whileitisSchurstable(inthediscrete-timesense)ifallofitseigenvalueshavemagnitudelessthan1.

AdirectedgraphGisapair(V,E),whereVisanonemptyfinitesetofnodesandE V×Visasetofedges,inwhichanedgeisrepresentedbyanorderedpairofdistinctnodes.Foranedge(i,j),nodeiiscalledtheparentnode,nodejthechildnode,andiisaneighborofj.Agraphwiththepropertythat(i,j)∈Eimplies(j,i)∈Eissaidtobeundirected.ApathonGfromnodei1tonodeilisasequenceoforderededgesoftheform(ik,ik+1),k=1,...,l 1.Adirectedgraphhasorcontainsadirectedspanningtreeifthereexistsanodecalledtheroot,whichhasnoparentnode,suchthatthereexistsadirectedpathfromthisnodetoeveryothernodeinthegraph.

Suppose)∈thattherearemnodesinagraph.TheadjacencymatrixA=(aijRm×misdefinedbyaii=0,aij=1if(j,i)∈Eandaij=

0∑otherwise.TheLaplacianmatrixL∈Rm×misdefinedasLm×m

ii=matrixj=iaij,Lij= aijfori=j.LetD∈R

bearow-stochasticwiththeadditionalassumptionthatd(ii>0,dij>0ifj,i)∈Eanddij=0otherwise.

Lemma2.1([8,9,27]).ZeroisaneigenvalueofLwith1andanonnegativevectorrT∈R1×N,respectively,asthecorrespondingrightandlefteigenvectors,andallnonzeroeigenvalueshavepositiverealparts.Furthermore,zeroisasimpleeigenvalueofLifandonlyifthegraphGhasadirectedspanningtree.

Lemma2.2([9]).OneisaneigenvalueofDwith1anda

nonnegativevector r

T∈R1×N,respectively,asthecorrespondingrightandlefteigenvectors,andallothereigenvaluesofDareintheopenunitdisk.Furthermore,oneisasimpleeigenvalueofDifandonlyifGcontainsadirectedspanningtree.3.Continuous-timemulti-agentsystems

ConsideragroupofNidenticalagentswithgeneralcontinuous-timelineardynamics.Thedynamicsofthei-thagentaredescribedby

˙x

i=Axi+Bui,yi=Cxi,

i=1,...,N,

(1)

wherexi∈Rnisthestate,ui∈Rpthecontrolinput,andyi∈Rqthemeasuredoutput.A,B,C,areconstantmatriceswithcompatibledimensions,whereCisassumedtohavefullrowrank.

Itisassumedthateachagenthasaccesstotherelativeoutputmeasurementswithrespecttoitsneighbors.Differingfromthedynamicprotocolsin[23–26],whosedimensionsareequaltoorevenhigherthanthatofasingleagent,weintroducehereareduced-orderobserver-basedconsensusprotocolas

v˙i=Fvi+Gyi+TBui,

u Ni=cKQ1

aij(yi yj)

j=1

(2)

+cKQ N2

aij(vi vj),

i=1,···,N,

j=1

wherevi∈Rn qistheprotocolstate,c>0isthecouplingstrength,aijisthe(i,j)-thentryoftheadjacencymatrixAofadirectedgraphG,F∈R(n q)×(n q)isHurwitzandhasnoeigenvaluesincommonwiththoseofA,G∈R(n q)×q,T∈R(n q)×nistheuniquesolutiontothefollowingSylvesterequation:TA FT=GC,

whichfurthersatisfiesthat (3)

C

T

isnonsingular,Q1∈Rn×qQ and

2∈R

n×(n q)

aregivenby[QC 1

1Q2]=

T

,andK∈Rp×nis

thefeedbackgainmatrixtobedesigned.Notethatprotocol(2)isdistributed,sinceitisbasedonlyontherelativeinformationofneighboringagents.

Letzi=[xTT]T

andz=[zTTTi,vi1,...,zN].Then,theclosed-loopnetworkdynamicsresultingfrom(1)and(2)canbewrittenas

˙z

=(IN M+cL R)z,(4)

whereL∈RN×N[

]

istheLaplacianmatrixofG,andM=

A0GC

,R=

[

BKQ1C

BKQ2

]

F

TBKQ1C

TBKQ2

.

Wesaythattheprotocol(2)solvestheconsensusproblemfor(1),ifthestatesof(4)satisfylimt→∞‖xi(t) xj(t)‖=0, i,j=1,...,N.

Next,analgorithmispresentedtoselectthecontrolparametersin(2).

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512Z.Lietal./Systems&ControlLetters60(2011)510–516

Algorithm3.1.Giventhat(A,B,C)isstabilizableanddetectable,theprotocol(2)canbeconstructedasfollows:

(1)ChooseaHurwitzmatrixFhavingnoeigenvaluesincommon

withthoseofA.SelectGsuchthat(F,G)isstabilizable.(2)Solve(3)togetasolutionT,whichsatisfiesthat

C T

nonsingular. Then,computematricesQ1andQ2by

Q1

Q2

is

=

C 1

T

.

(3)Solvethefollowinglinearmatrixinequality(LMI):

AP+PAT

2BBT

<0,

(5)

togetonesolutionP>0.Then,choosethematrixK= BTP 1.

(4)Selectthecouplingstrengthc≥1

λminRe,whereλiisthei-thi=0

i

eigenvalueofL.

Remark3.2.ByTheorem8.M6in[28],anecessaryconditionforthematrix Ttotheuniquesolutionto(3)andfurthertosatisfythat

C

T

isnonsingularisthat(F,G)isstabilizable,(A,C)is

detectable,andFandAhavenocommoneigenvalues.Inthecasewheretheagentin(1)issingle-inputsingle-output(SISO),thisconditionisalsosufficient.Undersuchacondition,itisshownforthegeneral multi-input multi-output(MIMO) casethattheprobabilityfor

C

T

tobenonsingularis1[28].If

CT

issingularin

step(2),weneedtogobacktostep(1)andrepeattheprocess.Asshownin[25],anecessaryandsufficientconditionfortheexistenceofapositive-definitesolutiontotheLMI(5)isthat(A,B)isstabilizable.Therefore,asufficientconditionforAlgorithm3.1tosuccessfullyconstructaprotocol(2)isthat(A,B,C)isstabilizableanddetectable.

Theorem3.3.Forthemulti-agentnetwork(4)whosecommunica-tiontopologyGcontainsadirectedspanningtree,thedynamicpro-tocol(2)constructedbyAlgorithm3.1solvestheconsensusproblem.Specifically,

xi(t)→ (t) (rT eAt) x1( .0).. ,(6)

xN(0)

vi(t)→GC (t),i=1,...,N,ast→∞,

wherer∈RNisanonnegativevectorsuchthatrTL=0and

rT1=1.

Proof.ξLetξ=((IsatisfiesthefollowingN 1rTdynamics:

) I2n q)z.Then,itfollowsfrom(4)thatξ

˙=(IN M+cL R)ξ.(7)

Clearly,0isasimpleeigenvalueofIN 1rTwith1astherighteigenvector,and1istheothereigenvaluewithmultiplicityN 1.Thus,bythedefinitionofξ,ξ=0ifandonlyifz1=···=zN,i.e.,theconsensusproblemissolvedifsystem(7)isasymptoticallystable.

BecauseGcontainsadirectedspanningtree,itfollowsfromLemma2.1thatzeroisasimpleeigenvalueofLandallother

eigenvalueshavepositiverealparts.thatUT

LU=Λ=

LetUN×N

besucha00

∈Runitarymatrix0

,wherethediagonal

entriesof arethenonzeroeigenvaluesofL.Sincetherightandlefteigenvectorscorrespondingtothezeroeigenvalue[ofLare,respectively,1andrT,wecanchooseU=1

]

Y1 N,UT=rT

Y2

,withY1∈RN×(N 1),Y2∈R(N 1)×N.Letζ [ζTTT

1,...,ζN]=

(UT I2n q)ξ.Then,(7)canberewrittenasζ

˙=(IN M+cΛ R)ζ.(8)

Bythedefinitionofξ,itiseasytoseethatζ1=(rT I2n q)ξ=0.Noteζthatthestatematrixof(8)isblockuppertriangular.Hence,i,i=2,...,N,convergeasymptoticallytozero,ifandonlyiftheN 1subsystems

ζ

˙i=(M+cλiR)ζi,i=2,...,N,

(9)

areasymptoticallystable.Multiplying sidesofthematrixM+cλI

theleftandrightiRbyQ= TI

andQ 1,respectively,andin

virtueof(3),weget[

Q(M+cλiR)Q

1

=

A+cλiBK

cλiBKQ2

]

F

.(10)

Bysteps(3)and(4)inAlgorithm3.1,wecanobtainthatthereexistsaP>0satisfying

(A+cλiBK)P+P(A+cλiBK)H=AP+PAT 2cRe(λi)BBT

≤AP+PAT 2BBT<0,i=2,...,N.

Thatis,A+cλiBK,i=2,...,N,areHurwitz.Therefore,theN 1systemsin(9)areasymptoticallystable,implyingthatsystem(7)isasymptoticallystable,i.e.,theconsensusproblemissolved.Next,thesolutionof(4)canbeobtainedasz(t)=e(IN M+cL R)tz(0)

=(U I)e(IN M+cΛ R)t[(UT I)z(0)

Mt=(U I)e0

e

(0]I)t(UT

N 1 M+c R I)z(0).

(11)

IthasbeenshownabovethatIN 1 M+c RisHurwitz.Thus,z(t)→(1 I)eMt(rT I)z(0)

=(1rT) eMtz(0),

ast→∞,

implyingthat

zi(t)→rT eMtz(0),

ast→∞.

(12)

SinceFisHurwitz,(12)directlyleadsto(6).

Remark3.4.Theconsensusprotocol(2)canberegardedas

anextensionofthetraditionalreduced-orderobserver-basedcontrollerforasinglesystemtotheoneformulti-agentsystems.Theseparationprincipleofthetraditionalobserver-basedcontrollersstillholdsinthemulti-agentsetting,asshownin(10).Someobservationsonthefinalconsensusvaluein(6)canbeconcludedasfollows:IfAin(1)haseigenvalueslocatedintheopenright-halfplane,thentheconsensusvalue (t)reachedbythe (agentswilltendtoinfinityexponentially.IfAisHurwitz,thent)→0,ast→∞.Ontheotherhand,ifAhaseigenvaluesintheclosedleft-halfplane,thentheagentsin(1)mayreachconsensusnontrivially.Thatis,somestatesofeachagentmightapproachacommonnonzerovalue.Typicalexamplesbelongingtothelastcaseincludethecommonly-studiedfirst-,second-,andhigh-orderintegrators.

Remark3.5.Algorithm3.1hasafavorabledecouplingfeature.Specifically,thefirstthreestepsdealwithonlytheagentdynamicsandthefeedbackgainmatricesof(2),whilethelaststeptacklesthecommunicationtopology.Therefore,theconsensusprotocol(2)constructedviaAlgorithm3.1foragivencommunicationgraphcanbedirectlyusedforanyothercommunicationgraphcontainingadirectedspanningtree,withtheonlyadditionaltaskofappropriatelyadjustingthecouplingstrengthc.

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Z.Lietal./Systems&ControlLetters60(2011)510–516

513

Fig.1.Thecommunicationtopology.

Algorithm3.1constructsaprotocoltoachieveconsensus.Inthefollowing,theprotocol(2)willberedesignedtoachieveconsensuswithagivenconvergencerate.FromtheproofofTheorem3.3,itiseasytoseethattheconvergencerateoftheNagentsin(1)reachingconsensusundertheprotocol(2)isequaltotheminimaldecayrate

oftheN 1systemsin(9).Thedecayrateofthesystem˙x

=AxisdefinedasthemaximumofnegativerealpartsoftheeigenvaluesofA[29].Thus,bynoticing(10),theconvergencerateofagents(1)reachingconsensuscanbemanipulatedbyproperlyassigningtheeigenvaluesofA+cλiBK,i=2,...,N,andF.

Algorithm3.6.Giventhat(A,B,C)isstabilizableanddetectable,theprotocol(2)canbeconstructedasfollows:

(1)ChoosethematrixFwhoseeigenvalueslieintheleft-halfplane

ofx= α.SelectGsuchthat(F,G)isstablizable.(2)Step2inAlgorithm3.1.(3)SolvethefollowingLMI:

AQ+QAT 2BBT+2αQ<0,

(13)

togetonesolutionQ>0.Then,choosethematrixK= BTQ 1.

(4)Step4inAlgorithm3.1.

Theorem3.7.Forthemulti-agentnetwork(4)withGcontain-ingadirectedspanningtree,theprotocol(2)constructedbyAlgo-rithm3.6solvestheconsensusproblemwithaconvergenceratelargerthanα.Thefinalconsensusvaluesarethesameasin(6).

Proof.ItcanbeshownbyfollowingsimilarstepstothoseinTheorem3.3,andbyfurthernotingthefact:Thedecayrateofthe

system˙x

=Axislargerthanα>0,ifandonlyifthereexistsamatrixQ>0suchthatAQ+QAT+2αQ<0[29].

Example3.8.Consideranetworkofsecond-orderintegrators,i.e.,theagentdynamicsin(1)aregivenby[

]

[A=

010

]

00

,B=

1

,C=

1

0

.

Afirst-orderdynamicprotocolbasedonlyontherelativepositionsisintheformof(2).

TakeF= 2andG= ingthefunctiontosolvetheSylvesterEq.(3)givesT= lyapin 0.50.25 Matlab,whichobviouslysatisfiesthat

C

T

isnonsingular.Then,thematricesQ1

andQ2canbeobtainedasQ0

1=

4

andQ1

2=

2

.Solving

the LMI(5)byusing theSedumitoolbox[30],wehaveK= 0.8543 2.5628.Assumethatthecommunicationgraphis

givenbyFig.1.ThecorrespondingLaplacianmatrixis

300 1

1

1 10000L= 1 1 12000

1

00100 ,000 110

00

1

1

whosenonzeroeigenvaluesare1,1.3376±0.5623j,2,3.3247.ByAlgorithm3.1andTheorem3.3,theprotocol(2)withfeedbackgainmatricesgivenasabovesolvestheconsensusproblemforthecommunicationgraphinFig.1,ifthecouplingstrengthc≥1.Algorithm3.6canbeutilizedtoconstructaprotocolachievingconsensuswithaprescribedconvergencerate,e.g.,largerthan1.Thematricesin(2)exceptKremain(13)withα=1givesK= thesame.Solving 5.0141 3.7372 theLMI.ForthecommunicationgraphinFig.1,selectc=1forsimplicity.Thestatesofthenetwork(4)withtheprotocol(2)givenbyAlgorithm3.6asabovearedepictedinFig.2.Theconvergencerateoftheagentsreachingconsensuscanbeobtainedas1.5301.4.Discrete-timemulti-agentsystems

Thissectionfocusesonthediscrete-timecounterpartofthelastsection.ConsideranetworkofNidenticaldiscrete-timelinearagents,withthedynamicsofthei-thagentdescribedbyx+i=Axi+Bui,yi=Cxi,

i=1,...,N,

(14)

wherexi=xi(k)∈Rn×nisthestate,x+i=xi(k+1)isthestateat

thenexttimeinstant,ui∈Rp

isthecontrolinput,andyi∈Rqisthemeasuredoutput.ItisassumedthatCisoffullrowrank.

Similartothecontinuous-timecase,thefollowingreduced-orderobserver-basedconsensusprotocolisproposed

v +i=Fv i+Gyi+TBui,

u Ni=KQ1

dij(yi yj)+KQ N2

dij(v i v j),

(15)

j=1

j=1

i=1,...,N,

wherev i∈Rn qistheprotocolstate,F∈R(n q)×(n q)isSchurstableandhasnoeigenvaluesincommonwiththoseofA,G∈R (n q)×q,T∈R(n q)×nistheuniquesolutiontoC

(3),satisfyingthat

T

isnonsingular,Q1

QC 1

2

=

T

,K∈Rp×nisthefeedback

gainmatrixtobedesigned,anddijisthe(i,j)-thentryoftherow-stochasticmatrixDassociatedwiththegraphG.

Let zi=[xTi,v

Ti]Tand z=[ zT

TTnetworkdynamicscanbewritten1,...,zN].Then,thecollectiveas

z

+=(IN M+(IN D) R)z ,(16)

wherematricesMandRaredefinedin(4).

Wesaythattheprotocol(15)solvestheconsensusproblemfor(1)ifthestatesof(16)satisfylimk→∞‖xi(k) xj(k)‖=0, i,j=1,...,N.

Beforemovingforward,weintroducethefollowingmodifiedalgebraicRiccatiequation(MARE)[31,32]:P=ATPA δATPB(BTPB+I) 1BTPA+Q.

(17)

Forδ=1,theMARE(17)isreducedtothecommonly-useddiscrete-timeRiccatiequationdiscussedin,e.g.,[33].

ThefollowinglemmashowstheexistenceofsolutionsfortheMARE.

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514Z.Lietal./Systems&ControlLetters60(2011)

510–516

Fig.2.Thestatesofthenetwork(4)undertheprotocol(2)constructedviaAlgorithm3.6.Herexi(k)denotesthek-thcomponentofxi.

Lemma4.1([31,32]).For0<δ<1,theMARE(17)hasauniquepositive-definitesolutionP,ifthematrixAhasnoeigenvalueswithmagnitudelargerthan1,(A,Q1/2)isstablizable,and(A,C)isdetectable.Furthermore,P=limk→∞PkforanyinitialconditionP0≥0,wherePksatisfies

P(k+1)=ATP(k)A δATP(k)B(BTP(k)B+I) 1BTP(k)A+Q.Next,analgorithmfortheprotocol(15)ispresented,whichwillbeusedlater.

Algorithm4.1.Giventhat(A,B,C)isstabilizableanddetectable,theprotocol(15)canbeconstructedasfollows:

(1)SelectaSchurstablematrixFhavingnoeigenvaluesin

commonwiththoseofA,andGsuchthat(F ,G )isstablizable.(2)Solve(3)togetasolutionT,satisfyingthat

Then,computethematricesQ1andQ2byQ1

T

Theorem4.3.AssumethatAhasnoeigenvalueswithmagnitude

largerthan1.Forthemulti-agentnetwork(16)withGcontain-ingadirectedspanningtree,theprotocol(15)constructedbyAlgo-rithm4.1solvestheconsensusproblem.Specifically,

T Ak) xi(k+1)→ψ(k+1) (rv i(k+1)→GCψ(k+1),

. .. ,

xN(0)

x1(0)

(19)

i=1,...,N,ask→∞,

∈RNisnonnegativesuchthatr T(IN D)=0andr T1=1.wherer

=((IN 1r T) I2n q)z .AsdemonstratedintheProof.Letξ

proofofTheorem3.3,theconsensusproblemcanbereducedto

,whichevolvesaccordingtothetheasymptoticalstabilityofξ

followingdynamics:

+=(IN M+(IN D) R)ξ. ξ

(20)

C

T

isnonsingular.Q2=

1

CT

.

(3)ChooseK= (BPB+I)BPA,whereP>0istheunique

solutionofthefollowingMARE:

1T

i|2)ATPB(BTPBP=ATPA (1 max|λ

i|<1|λ

+I) 1BTPA+Q,

ibeingthei-theigenvalueofD.withQ>0andλ

(18)

Foranygraphcontainingadirectedspanningtree,itfollowsfrom

Lemma2.2that0isasimpleeigenvalueofIN Dandallothereigenvaluesliewithinadiskofradius1+0j[1centered]atthepoint T 1 1,U = T=rYinthecomplexplane.LetU 2,withY

N

1∈RN×(N 1),Y 2∈R(N 1)×N,besuchunitarymatricesthatY

Remark4.2.Asufficientconditionfortheexistenceofthe

consensusprotocolbyusingAlgorithm4.1isthat(A,B,C)isstabilizableanddetectable,andAhasnoeigenvalueswithmagnitudelargerthan1whichisrequiredheretoensurethesolvabilityoftheMARE(18).

T(IN D)U =Λ =U

,wherethediagonalentriesof

TTT [ζ 1 NarethenonzeroeigenvaluesofIN D.Letζ,...,ζ]=

T I2n q)ξ .Then,(20)canberewrittenas(U

+=(IN M+(IN Λ ) R)ζ.ζ

(21)

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Z.Lietal./Systems&ControlLetters60(2011)510–516515

Clearly,ζ

1=(r T I2n q)ξ =0.of(21)isblockuppertriangular,ζ

Bynotingthatthestatematrixi,i=2,...,N,convergetozeroasymptotically,ifandonlyiftheN 1subsystemsζ

+i=(M+(1 λ i)R)ζ i,i=2,...,N,

(22)

are asymptoticallystable.Itis knownthatM+(1 λ i)Rissimilar

to

A+(1 λ

i)BK(1 λ

i)BKQ20

F

.Inlightofstep(3)inAlgorithm4.1,

wecanobtain

(A+(1 λ

i)BK)HP(A+(1 λ i)BK) P=ATPA 2Re(1 λ

i)ATPB(BTPB+I) 1BTPA P+|1 λ

i|2ATPB(BTPB+I) 1BTPB(BTPB+I) 1BTPA=ATPA+( 2Re(1 λ

i)+|1 λ i|2)AT×PB(BTPB+I) 1BTPA P+|1 λ

i|2ATPB(BTPB+I) 1×( I+BTPB(BTPB+I) 1)BTPA

=ATPA+(|λ

i|2 1)ATPB(BTPB+I) 1BTPA P |1 λ

i|2ATPB(BTPB+I) 2BTPA≤ATPA (1 max| i|2|λ

<λ

)ATPB(BTPB+I) 1BTPA Pi|1= Q<0,

(23)

wheretheidentity I+BTPB(BTPB+I) 1= (BTPB+I) 1

hasbeenapplied.Then,(23)impliesthatA+(1 ,...,λ

i)BK,i=2N,areSchurstable.Therefore,theN 1systemsin(22)areasymptoticallystable,implyingthattheconsensusproblemissolved.

BynotingthatIN 1 M+

RisSchurstable,thesolutionof(16)canbeobtainedas

z

(k+1)=(IN M+(IN D) R)kz (0)=(U

I)(IN M+Λ R)k(U T I)z (0)=(U

[ I)Mk0] T0(I R)k(U I) z(0)N 1 M+ →(1 r

T) Mkz (0),ask→∞.

(24)

Therefore,wehave

z

i(k+1)→r T Mkz (0),ask→∞,

whichdirectlyleadsto(19).

Remark4.4.Theorem4.3givesthediscrete-timecounterpartof

theresultsinTheorem3.3.TheLaplacianmatrixLisusedinthelastsectiontorepresentthecommunicationgraphforcontinuous-timemulti-agentsystems.Incontrast,therow-stochasticmatrixDisutilizedhereforthediscrete-timecase.ByobservingAlgorithms3.1,3.6and4.1,itcanbeconcludedthatthenonzeroeigenvaluewiththesmallestrealpartoftheLaplacianmatrixplaysakeyroleincontinuous-timemulti-agentsystems,whilethenon-oneeigenvalueofthestochasticmatrixwiththelargestmagnitudeiscriticalforthediscrete-timecase.Itcanbeobservedfrom(19)thattheconsensusvalueψ(k+1)reachedbytheagentswilltendtoinfinityexponentially,ifAin(14)hasaneigenvaluewithmagnitudelargerthan1.Therefore,theassumptiononAinTheorem4.3doesnotinvolvemuchconservatism.SimilartoTheorem3.3,Ain(14)witheigenvalueswithaunitmagnitudeiscriticalfortheagentstoreachconsensusnontrivially.5.Conclusion

Inthispaper,theconsensusproblemsformulti-agentsystemswithcontinuous-anddiscrete-timelineardynamicsanddirected

communicationtopologieshavebeenconsidered.Distributed

reduced-orderconsensusprotocols,basedontheinformationofrelativeoutputsofneighboringagents,havebeenproposed.Severalmulti-stepalgorithmshavebeenpresentedtoconstructtheconsensusprotocols,whichsolvetheconsensusproblemforboththecontinuous-anddiscrete-timecases.Inthispaper,wedidnotconsidertheissuesoftimedelays,switchingtopologies,orrandomgraphs.However,theseissuesareinterestingtopicsthatdeservefurtherinvestigationinfuturework.Acknowledgments

ThisworkwassupportedinpartbytheNationalNaturalScienceFoundationofChinaunderGrantNos.10832006,10872030,ChinaPostdoctoralScienceFoundationunderGrantNo.20100480211,andNationalScienceFoundationCAREERAwardECCS–0748287.References

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