<%BANNER%>

Distributed Time Synchronization from Relative Measurement in Mobile Wireless Sensor Networks

Permanent Link: http://ufdc.ufl.edu/UFE0044850/00001

Material Information

Title: Distributed Time Synchronization from Relative Measurement in Mobile Wireless Sensor Networks
Physical Description: 1 online resource (109 p.)
Language: english
Creator: Liao, Chenda
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: clock -- consensus -- distributed -- estimation -- mobile -- network -- sensor -- synchronization -- time -- wireless
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A wireless sensor network (WSN) consists of a set of devices (nodes) with sensing, data processing, and communicating components. They can monitor physical or environmental information around, and collaborate to process such information.They have been used in a variety of applications, such as habitat and environment monitoring, health care, military surveillance, industrial machinery surveillance, home automation and so on. In many of those applications, nodes in networks are mobile. Clock synchronization is critical in the effective use of sensor networks; particularly in applications such as range finding for target tracking and localization, intrusion detection, time correlation of telemetry data, sensor fusion, slot assignment in TDMA, duty cycling protocols, and so on. The problem of clock synchronization indeed has been widely investigated. Most algorithms are designed and tested in static networks, while little attention has been paid to that in mobile networks. In mobile networks, the communication links among networks varies frequently due to changes in inter-node distance and obstacles, which may affect the performance of current algorithms for static networks. At a given global time t, the local clock time at node u can be approximately written as Tau=alpha*t +beta, where alpha is the skew and beta is the offset. The global time to which all nodes need to be synchronized can be the local clock time at an arbitrarily chosen "reference" node. The time synchronization problem is effectively a problem of estimating the skews and offsets of every node, since the nodes can infer the global time from their local clock times once they know their own skew and offset estimates. It is not possible for a node to measure its skew and offset directly. However, it is possible for a pair of neighbors to measure the noisy relative difference between their offsets and logarithm of skews by exchanging a number of time stamped messages.We show the existing protocols to perform so-called pairwise synchronization can be used to obtain such relative measurements. The focus of this work is howto achieve network-wise synchronization, i.e. estimating skews and offsets of clocks in nodes from those relative measurements with respect to a global clock. Two different distributed algorithms have been proposed, with which each node can estimate its offset/skew from these noisy relative measurements by communicating only with its neighbors. The algorithms are simple and easy to implement. The convergence of the two algorithms is guaranteed under certain conditions and it is robust to measurement noise and time-varying network topology. The first algorithm was inspired by the existing Jacobi type of algorithms for skews and/or offsets estimation. The algorithm ensures that mean of estimation error converges to zero (if relative measurement is unbiased) and variance to a limiting value. The second algorithm was inspired by stochastic approximation type of consensus principles. It performs better than the first algorithm in that it ensures the variance of estimation error converges to zero.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chenda Liao.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Barooah, Prabir.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0044850:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044850/00001

Material Information

Title: Distributed Time Synchronization from Relative Measurement in Mobile Wireless Sensor Networks
Physical Description: 1 online resource (109 p.)
Language: english
Creator: Liao, Chenda
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: clock -- consensus -- distributed -- estimation -- mobile -- network -- sensor -- synchronization -- time -- wireless
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A wireless sensor network (WSN) consists of a set of devices (nodes) with sensing, data processing, and communicating components. They can monitor physical or environmental information around, and collaborate to process such information.They have been used in a variety of applications, such as habitat and environment monitoring, health care, military surveillance, industrial machinery surveillance, home automation and so on. In many of those applications, nodes in networks are mobile. Clock synchronization is critical in the effective use of sensor networks; particularly in applications such as range finding for target tracking and localization, intrusion detection, time correlation of telemetry data, sensor fusion, slot assignment in TDMA, duty cycling protocols, and so on. The problem of clock synchronization indeed has been widely investigated. Most algorithms are designed and tested in static networks, while little attention has been paid to that in mobile networks. In mobile networks, the communication links among networks varies frequently due to changes in inter-node distance and obstacles, which may affect the performance of current algorithms for static networks. At a given global time t, the local clock time at node u can be approximately written as Tau=alpha*t +beta, where alpha is the skew and beta is the offset. The global time to which all nodes need to be synchronized can be the local clock time at an arbitrarily chosen "reference" node. The time synchronization problem is effectively a problem of estimating the skews and offsets of every node, since the nodes can infer the global time from their local clock times once they know their own skew and offset estimates. It is not possible for a node to measure its skew and offset directly. However, it is possible for a pair of neighbors to measure the noisy relative difference between their offsets and logarithm of skews by exchanging a number of time stamped messages.We show the existing protocols to perform so-called pairwise synchronization can be used to obtain such relative measurements. The focus of this work is howto achieve network-wise synchronization, i.e. estimating skews and offsets of clocks in nodes from those relative measurements with respect to a global clock. Two different distributed algorithms have been proposed, with which each node can estimate its offset/skew from these noisy relative measurements by communicating only with its neighbors. The algorithms are simple and easy to implement. The convergence of the two algorithms is guaranteed under certain conditions and it is robust to measurement noise and time-varying network topology. The first algorithm was inspired by the existing Jacobi type of algorithms for skews and/or offsets estimation. The algorithm ensures that mean of estimation error converges to zero (if relative measurement is unbiased) and variance to a limiting value. The second algorithm was inspired by stochastic approximation type of consensus principles. It performs better than the first algorithm in that it ensures the variance of estimation error converges to zero.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chenda Liao.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Barooah, Prabir.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0044850:00001


This item has the following downloads:


Full Text

PAGE 1

DISTRIBUTEDTIMESYNCHRONIZATIONFROMRELATIVEMEASUREME NTIN MOBILEWIRELESSSENSORNETWORKS By CHENDALIAO ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

PAGE 2

c r 2013ChendaLiao 2

PAGE 3

Idedicatethistomyparentsandmywife. 3

PAGE 4

ACKNOWLEDGMENTS MyPhDeducationintheDepartmentofMechanicalandAerospaceE ngineeringat theUniversityofFloridahasbeenaextremelychallengingexperience .Plentyofpeople haveprovidedhelptotheaccomplishmentofthisdissertation.Iwou ldliketoexpressmy sincereappreciationtothem. Firstandforemost,Iwouldliketoexpressmydeepgratitudetomya dvisorDr. PrabirBarooahforguidingmeduringthewholecourseofmyPhDeduc ation.An outstandingacademicadvisorisnotonlyaself-motivatedexpertan dresearcherinhis/her areasofinterest,butalsowillingtospendenergyontheeducationo fhis/herstudents.Dr. PrabirBarooahissuchaperson,andwithouthissupport,guidance andencouragement,I wouldnevermakethisdissertationpossible.DuringmyrstyearPhD study,heprovided mestraightforwardandeectiveguidelineandsuggestiontothepa thofresearch.His supervisiononmyresearchisnotonlyonamacro-scalebutalsoonma thematicaldetails, whichcomesfromhisin-depthknowledge.Hisconstructivecriticisma ndcommentshelped mecorrectproblems,developnewthoughtsandreneideas.Ifee ltrulyfortunatetohave thispreciousopportunitytoworkwithhimduringmyPhDstudy,andI thankhimforhis dedication. IamindebtedtoDr.WarrenDixon,Dr.NormanFitz-CoyandDr.Joh nMarkShea forconstantsupportinmyPhDdissertation:beinginmycommitteea ndspendingtime andeortinprovidingsuggestionfortheimprovementofthisdisser tation. Ialsowanttoextendmythankstomycolleagues,HeHao,JosephKn uth,Yashen Lin,SiddharthGoyal,RahulSubramanyandJonathanBrooksinthe DistributedControl Systemlab.Iappreciatethehelpanddiscussionfromthemandalsoc herishthegreat momentswiththem. Finally,Iwouldliketothankmyparentsandmywife,fortheirlove,sup portand encouragement. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFFIGURES ....................................7 ABSTRACT ........................................9 CHAPTER 1INTRODUCTION ..................................11 1.1Motivation ....................................11 1.2ClockModelandSynchronization .......................13 1.3SkewandOsetEstimationfromNoisyRelativeMeasurements ......16 1.4RelatedWork ..................................17 1.5Contributions ..................................20 2RELATIVEMEASUREMENTSOFSKEWANDOFFSET ...........25 2.1PairwiseSynchronization ............................26 2.2FromPairwiseSynchronizationtoRelativeMeasurement ..........28 2.3Simulation ....................................30 3THEJATALGORITHMINMARKOVIANSWITCHINGNETWORKS ....34 3.1ProblemStatement ...............................35 3.2ProposedJATAlgorithm ............................36 3.2.1PriorWork:Jacobi-typeAlgorithms ..................36 3.2.2AlgorithmforDistributedEstimationfromRelativeMeasurement .37 3.3AsynchronousImplementationforTimeSynchronization ..........38 3.4MarkovianModelofTopologySwitch .....................39 3.5ConvergenceAnalysiswithMarkovianSwitching ...............43 3.5.1ProofofTheorem3.1 ..........................46 3.5.2VericationofTheorem3.1 .......................53 3.6NumericalEvaluationofTimeSynchronizationAccuracywithJAT ....54 3.7ConclusionandOpenProblems ........................58 3.8ProofofLemma3.2 ...............................60 4THESTOALGORITHMINTWOTYPESOFSWITCHINGNETWORKS ..66 4.1ProposedSTOAlgorithm ...........................67 4.2ConvergenceAnalysis ..............................71 4.2.1DeterministicTopologySwitching ...................73 4.2.2MarkovianTopologySwitching .....................75 4.2.3VericationoftheTheorems ......................77 4.2.3.1VericationofTheorem4.1 .................77 5

PAGE 6

4.2.3.2VericationofTheorem4.2 .................79 4.3NumericalEvaluationofTimeSynchronizationAccuracywithSTO ....79 4.4NumericalComparisonamongSTO,JATandATSAlgorithm .......83 4.4.1BriefDiscussionofATSAlgorithms ..................83 4.4.2Comparison ...............................84 4.5ImprovementofConvergenceRate .......................89 4.5.1UpdatewithAverageDistance .....................89 4.5.2CombinationofJATandSTOAlgorithms ..............91 4.5.3NumericalComparison .........................92 4.6ConclusionandOpenProblems ........................94 4.7TechnicalProofs ................................95 4.7.1ProofofLemma4.1 ...........................95 4.7.2ProofofLemma4.3 ...........................99 5CONCLUSIONANDFUTUREWORK ......................101 5.1Conclusion ....................................101 5.2FutureWork ...................................103 5.2.1AsymmetricUpdate ...........................103 5.2.2AsynchronousDecreaseoftheStepSize ................103 5.2.3ImplementationinRealSensorNetworks ...............104 REFERENCES .......................................105 BIOGRAPHICALSKETCH ................................109 6

PAGE 7

LISTOFFIGURES Figure page 1-1Wirelesssensornetworks ...............................12 1-2Anexampleofeventdetectioninunsynchronizednodes. .............13 1-3Thelinearclockmodel ................................15 2-1Two-waymessagesexchangeforpairwisesynchronization. ............26 2-2Therelativeclockmodel ...............................29 2-3Histogramsoferrorratios ..............................31 2-4Themeanandvarianceoferrorofrelativemeasurementofske w .........32 2-5Themeanandvarianceoferrorofrelativemeasurementofos et .........33 3-1Thestandardshapesofestimatedconditionalentropy ...............42 3-2Empiricallyestimatedconditionalentropyofthegraphprocess fG k g .......43 3-3Ameasurementgraph G andthecorrespondingweightgraph ~ G .........47 3-4Thethreegraphs G 1 ; G 2 ; G 3 thatcomprises G ....................53 3-5Thetimeseriesofthemeanandvarianceoftheestimateofnode 3'snodevariable 54 3-6Twographsthatoccurduringasimulationwith50nodes .............56 3-7ThetimeseriesoftheskewestimatesbyJATAlgorithm .............56 3-8ThemeanandvarianceoftheskewestimationerrorbyJATAlgor ithm .....57 3-9ThetimeseriesoftheosetestimatesbyJATAlgorithm .............57 3-10ThemeanandvarianceoftheosetestimationerrorbyJATAlg orithm .....58 3-11TheerrorofglobaltimeestimationbyJATAlgorithm ..............58 4-1Therelationshipbetweenlocalandglobalinterval .................70 4-2Allthegraphsthatoccurinsimulation#1. ....................78 4-3Meanandvarianceofestimationerrorinthedeterministicswitch ingnetwork ..78 4-4MeanandvarianceofestimationerrorintheMarkovianswitching network ...79 4-5Twographsthatoccurduringasimulationwith50nodes .............81 4-6TheskewestimatesbySTOAlgorithm .......................81 7

PAGE 8

4-7ThemeanandvarianceoftheskewestimationerrorbySTOAlgor ithm .....81 4-8TheosetestimatesbySTOAlgorithm .......................82 4-9ThemeanandvarianceoftheosetestimationerrorbySTOAlgo rithm ....82 4-10TheerrorofglobaltimeestimationbySTOAlgorithm ..............82 4-11Pairwisetime-stampexchangesinATSalgorithm. .................84 4-12Twographsthatoccurduringasimulationwith10nodes .............85 4-13Themeanandvarianceoftheskewestimationerrorincomparis on .......86 4-14Themeanandvarianceoftheosetestimationerrorincompar ison .......87 4-15Theerrorofglobaltimeestimationincomparison .................87 4-16Themeanandvarianceoftheglobaltimeestimationerror ............88 4-17Syncerroralongtimeincomparison ........................88 4-18Themeanandvarianceofthesyncerror ^ t ( t k )incomparison ..........89 4-19Themeanoftheglobaltimeestimationerror ...................92 4-20Thevarianceoftheglobaltimeestimationerror ..................93 4-21Thesynchronizationerror ^ t ( t k )incomparison ..................93 4-22Themeanofthesynchronizationerror ^ t ( t k )incomparison ...........94 4-23Thevarianceofthesynchronizationerror ^ t ( t k )incomparison .........94 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DISTRIBUTEDTIMESYNCHRONIZATIONFROMRELATIVEMEASUREME NTIN MOBILEWIRELESSSENSORNETWORKS By ChendaLiao May2013 Chair:PrabirBarooahMajor:MechanicalEngineering Awirelesssensornetwork(WSN)consistsofasetofdevices(node s)withsensing, dataprocessing,andcommunicatingcomponents.Theycanmonito rsurroundingphysical orenvironmentalinformation,andcollaboratetoprocesssuchinf ormation.Theyhave beenusedinavarietyofapplications,suchashabitatandenvironme ntmonitoring,health care,militarysurveillance,industrialmachinerysurveillance,homea utomationandsoon. Inmanyofthoseapplications,nodesinsensornetworksaremobile. Clocksynchronization iscriticalfortheeectiveuseofsensornetworks;particularlyina pplicationssuchas rangendingfortargettrackingandlocalization,intrusiondetect ion,timecorrelation oftelemetrydata,sensorfusion,slotassignmentinTDMA,dutycy clingprotocols,and soon.Theproblemofclocksynchronizationindeedhasbeenwidelyinv estigated.Most algorithmsaredesignedandtestedinstaticnetworks,whilelittleatt entionhasbeenpaid tothatinmobilenetworks.Inmobilenetworks,thecommunicationlink samongnetworks variesfrequentlyduetochangesininter-nodedistanceandobsta cles,whichmayaectthe performanceofalgorithmsdesignedforstaticnetworks. Atagivenglobaltime t ,thelocalclocktimeatnode u canbeapproximatelywritten as u ( t )= u t + u ,where u isthe skew and u isthe oset .Theglobaltimetowhich allnodesneedtobesynchronizedcanbethelocalclocktimeatanar bitrarilychosen \reference"node.The timesynchronizationproblem iseectivelyaproblemofestimating 9

PAGE 10

theskewsandosetsofeverynode,sincethenodescaninferthe globaltimefromtheir localclocktimesoncetheyknowtheirownskewandosetestimates Itisnotpossibleforanodetomeasureitsskewandosetdirectly.H owever,itis possibleforapairofneighborstomeasurethenoisyrelativedieren cebetweentheir osetsandlogarithmofskewsbyexchanginganumberoftimestamp edmessages.We showtheexistingprotocolstoperformso-called pairwisesynchronization [ 1 { 5 ]canbe usedtoobtainsuchrelativemeasurementsinChapter 2 .Thefocusofthisworkishowto achieve network-widesynchronization ,i.e.,estimateskewsandosetsofclocksinnodes fromtheserelativemeasurements.InChapter 3 andChapter 4 ,twodierentdistributed algorithmsareproposed,withwhicheachnodecanestimateitsose t/skewfromthese noisyrelativemeasurementsbycommunicatingonlywithitsneighbors .Thealgorithms aresimpleandeasytoimplement.Theconvergenceofthetwoalgorit hmsisguaranteed undercertainconditions.Theyarealsoshowntoberobusttomeas urementnoiseand time-varyingnetworktopologies.Therstalgorithm(JAT)wasinsp iredbytheexisting Jacobitypeofalgorithmsforskewsand/orosetsestimation[ 6 { 10 ].Thealgorithm ensuresthatthemeanofestimationerrorconvergestozero(ifr elativemeasurementsare unbiased)andvariancetoalimitingvalue.Thesecondalgorithm(STO) wasinspiredby stochasticapproximationtypeofconsensusprinciples[ 11 12 ].Itperformsbetterthan JATalgorithmintermsoftheestimationoftheglobaltimeasitensure sthevarianceof skewestimationerrorconvergestozero.Furthermore,wealsoc omparethetwoproposed algorithmwiththestate-of-the-artATSalgorithm[ 13 ]intermsofsynchronization errorthatisthemaximumabsolutedierenceoftimeestimatesofall pairsofnodesin networks.STOachievesbetteraccuracywhileitsconvergencera teisrelativelyslow. Wethenprovidemethodstoimprovetheconvergencerateandcor respondingnumerical validation. 10

PAGE 11

CHAPTER1 INTRODUCTION 1.1Motivation Awirelesssensornetwork(WSN)consistsofasetofdevices(node s)withsensing, dataprocessing,andcommunicatingcomponents.Theycanbedep loyedoverawidegeographicalregiontomonitorphysicalorenvironmentalinformation (suchastemperature, humidity,pressure,noise,vibration,CO2,moisture,speed,andim age)andcollaborateto processsuchinformation.Nodesareusuallylowcost,withlimitedonboardpowerand energy,andlimitedcommunicationrangeandability.Onenodecancom municatewith anotherifawirelesscommunicationlinkisavailable.WSNshavebeenuse dinavariety ofapplications,suchashabitatandenvironmentmonitoring,health care,militarysurveillance,industrialmachinerysurveillance,homeautomationandsoon .Inmanyofthose applications,nodesinnetworksaremobile.Forinstance,aproject calledZebraNet[ 14 ] waslaunchedtounderstandthemigrationpatternsofzebraswhe rethesensornodes weremountedonzebras.Asetofsmallroboticvehicleswithon-boa rdsensorsandradios isanotherexampleofmobilesensornetworks[ 15 ].Intherstexample,themobilityof thenetworkispassive,whileinthesecondexample,themobilityisactiv e.Ontheother hand,duetolimitedpower,communicationability,storageinnodes,w irelesssensor networksalsosuerfromproblemssuchasfrequentnodefailurea ndchangeofnetwork topology.Signicantamountofresearchhasbeenconductedonis suesrelatedtoWSNs, suchasenergyconservation,localization,clustering,routing,ne tworkconnectivity,clock synchronization,datafusion,etc. Inthiswork,weareinterestedinclocksynchronizationbecauseclo cksindistinct nodesusuallydonothavethesamereadingoftime.Clocksynchroniz ationinawireless sensornetworkisthereforecrucialforprovidingacommonnotion oftimeacrossthe networktoensurefunctionality.InTDMAbasedcommunicationsch emes,multiple senderscansharethesametransmissionchanneltosendpackag estoonereceiverusing 11

PAGE 12

Figure1-1.Wirelesssensornetworksdierenttimeslots.Inordertomakesureeachsenderonlyusesits owntimeslot,the clocksofsendersneedtobeaccuratelysynchronized.Operation onapre-scheduled sleep-wakecycleforenergyconservationandlifetimemaximizationr equiresaccurate knowledgeoftimeaswell.Basedonthistechnique,nodesaresuppos edtowakeupatthe sametimetoperformacooperativeoperationsuchasdataexchan geandthengoback tosleep.Ifclocksarenotsynchronized,onenodemaywakeuptoo latetoconductthe operation.Similarly,detectionofeventsalsorequiresclocksinnode saresynchronized. Figure 1-2 providesanexamplewhichshowshowpoorlyeventdetectionisperfo rmedin sensornetworksiflocalclocksarenotwellsynchronized.Inthe gure,twosensornodes candetectthetimeofeventthatanintruderenterstheirsensing ranges.Let'ssaya vehiclerstenterstherangeofnode1at2:29pmandthenintothat ofnode2at2:30pm. Supposetheclocksofthetwonodesarenotsynchronized.Theac tualreadingsinnodes are2:32pmand2:29pmrespectively.Therefore,theymakeanincor rectconclusionabout theorderofevents. Thediscussionaboveshowstheneedforclocksynchronizationinwir elesssensor networks.Clockmaybeun-synchronizedduetotworeasons:dist inctinitialtimes anddierentclockspeeds.Apairofclocksusuallydriftsawayfrome achotherdueto suchspeeddierencecausedbyimperfectnessofcrystaloscillat ors.Therefore,even iftwoclocksareinitiallysynchronized,theywillgetunsynchronizede ventually.The 12

PAGE 13

Figure1-2.Anexampleofeventdetectioninunsynchronizednodes .Clocktimesshown arelocaltimes. eectofclockdierenceswouldbeconsiderableoveralongtimeinter val.Traditional synchronizationmethodssuchasNetworkTimeProtocol(NTP)and GlobalPositioning System(GPS),whichisdirectlysynchronizedtotheCoordinatedUn iversalTime(UTC), arenotsuitableforsensornetworks.NTPisusedintraditionalcom putersthrough Internet.However,manyoftherequirementofNTP,whilemetinth eInternet,donot meetinsensornetworks[ 16 ].Forexample,NTPassumesCPUisalwaysavailable,and anodeisalwayslisteningtothenetwork.Thisisnotpossibleforsenso rnodeswith limitedpowerandenergy.Ontheotherhand,GPSistooexpensiveto bemountedon low-costsensornodes.Besides,GPSmaynotbeavailableinsomeplac es(underground, underwater,etc). Therefore,manynewtimesynchronizationprotocolshavebeenpr oposedthattarget wirelesssensornetworks[ 4 5 8 9 17 { 22 ].However,littleattentionhasbeenpaidtoclock synchronizationinmobileWSNs.Inmobilenetworks,thecommunicatio nlinksamong networksvariesfrequentlyduetochangesininter-nodedistance andobstacles.Some algorithmsaredesignedandtestedinstaticnetworksmaybeadopt edtomobilenetworks. However,littleisknownabouttheperformanceofsuchalgorithmsin mobilenetworks.We arespeciallyinterestedintheclocksynchronizationprobleminmobileW SNs. 1.2ClockModelandSynchronization Aclockinanelectronicdeviceisacomponentwhichcountsoscillationsin an accurately-machinedquartzcrystalataparticularfrequency. Afteracertainnumberof 13

PAGE 14

oscillations,aninterrupt(i.e.,aclocktick)occursandtheinterrupt handlerincrements thetimereadingoftheclockbyone.However,thefrequenciesofq uartzcrystalsarenot identical.Furthermore,thefrequencyistime-varyingduetoenvir onmentchanges,suchas temperature,atmospherepressure,batteryvoltage,etc.If aquartzcrystalclockiscooled to 55 o C ,itlosesabout20secondperday[ 23 ].Eveninamoderatecondition,anytwo clocksgovernedbysimplecrystaloscillatorsthatareusedinnodes ofsensornetworkswill driftapartupto100 sec=sec [ 23 ],whichendsupwithabout8 s dierenceperday.This isenoughtocauseseriousproblemsinthelongterm. Thereadingofaclock,basedonclockticks,canbemeasuredasafu nctionofthe frequencyofcrystaloscillator[ 24 ] T ( t )= a Z t 0 ( ) d + T (0) ; (1{1) where t isthetheglobalreferencetime, ( )istheangularfrequencyoftheoscillator, a isaconstantand T (0)istheinitialreadingoftheclockwhenglobalreferencetime t is denedas0.Inthiswork,wemodeltheclockinasimplefashionbecau seitcapturesthe twomainsourcesofun-synchronization:dierentclockspeedand initialdierence.Each node u onlyknowsitslocaltime u ( t ),whichisdierentfromtheglobaltime t .Thelocal time(refertoFigure 1-3 )ismodeledas: u ( t )= u t + u ; (1{2) wherethescalars u ; u arecalledits skew (relativespeedofclockwithrespectto globaltime t )and oset (localtimereading u (0)whenglobaltime t is0),respectively [ 2 5 25 26 ].Notethat u isingeneraltime-varyingduetoslowchangeoftheoscillation frequencycausedbyenvironmentalchangesoraging[ 23 ].Hereweassume u isconstant, evenifthismodeldoesnotholdinthelongterm.Themodelwithconst ant u isstill widelyadopted[ 2 5 25 26 ].Anodecandeterminetheglobaltime t fromitslocalclock 14

PAGE 15

Global clock tLocal clockt u v u v u vFigure1-3.Thelinearclockmodel.Twosolidlinesarerepresentingtwo dierentclocksin node u and v .Theskewsandosetsaremarkedingure,wheretheskewis theslopeofthelineandosetistheintersectiononyaxisoftheline.N ote thatnegativeosetiswelldenedasshowninnode v .Thedashedlineis representingaglobalclock,thereforetheskewis1andosetis0. timebyusingtherelationship ^ t = u ( t ) ^ u ^ u (1{3) aslongasitcanobtainestimates^ u ; ^ u ofitslocalclock.Hencetheproblemofclock synchronizationinanetworkcanbealternativelyposedasthatofn odesestimatingtheir skewsandosetswithrespecttoacommonglobalreferencetime t .Inthecontextof timesynchronization,theexistenceofareferencenodemeansth atatleastonenodehas accesstotheglobaltime t .Thisisthecasewhenatleastonenodeisequippedwitha GPSreceiverandhasaccesstotheUTC.IfnonodehasaGPSreceiv er,thenonenode hastobeelectedasthe\root"sothatit'slocalclocktimeisconsider edastheglobaltime thateveryonehastosynchronizeto.Tosynchronizetothegloba lclock,anode u canuse aso-called pairwisesynchronization method,suchasthosein[ 1 2 4 ],toestimate u and u ifitcandirectlycommunicatetothereferencenode.However,this isnotthecasefor mostofthenodesduetolimitedcommunicationrange.Itistherefor enotpossibleforall nodestomeasuretheirskewsandosetsdirectly. 15

PAGE 16

1.3SkewandOsetEstimationfromNoisyRelativeMeasureme nts Althoughmostofthenodescannotestimatetheirskewsandoset sdirectlyfromthe referencenode(s)duetolimitedrangeofcommunication,apairofn eighboringnodes u and v areabletomeasurethenoisyvaluesoflog( u ) log( v )and u v bypairwise communication.Suchmeasurementsarecalledrelativemeasuremen tofskewsandosets andwewillintroducetheprocedureforobtainingtheminChapter 2 .Thenextquestion iswhetheranodecanestimateitsskewfromrelativemeasurements ofskewsfromonlyits neighborsovertime(similarlyforosets).Toanswerthisquestion, wetakeskewsasan exampleandformulateanetwork-wideestimationproblem.Forconv enience,denethe nodevariable fornode u as x u =log u .Ifnodes u and v areneighborsatdiscretetime interval k k =0 ; 1 ;::: ,thentheycanobtainarelativemeasurement u;v ( k )= x u x v + u;v ( k ) ; (1{4) where u;v ( k )ismeasurementerror.Theestimationproblemisforeachnodetoe stimate itsnodevariablefromtherelativemeasurementsitcollectsovertime withoutrequiring anycentralizedinformationprocessingorcoordination.Onceanod e u obtains^ x u ( k ),it canestimatetheskewbyusing^ u ( k )=exp(^ x u ( k )).Itturnsoutthattheosetestimation problemissimilar,whichisalsoaspecialcaseofthe\estimationfromno isyrelative measurementproblem".Inviewofthis,wefocusontheproblemofe stimatingscalar valuednodevariablesfromnoisyrelativemeasurements.Twocopies ofthisproblemcan besolvedinaparallelfashionforbothskewandoset. InChapter 3 and 4 ,weproposetwodierentalgorithmstosolvethenodevariable estimationproblemintroducedabove.Indesigninganalgorithm,wee speciallycareabout thefollowingaspects.(i)Thealgorithmshouldbedistributed.(ii)The algorithmshould berobusttothenoiseintherelativemeasurement(seeChapter 2 foradiscussiononthe noise).(iii)Thealgorithmmustbeimplementedasynchronouslysincen odesdon'thavea 16

PAGE 17

commonclocktomaintainsynchronouscommunication.(iv)Thealgor ithmneedstohave theoreticalguaranteesonitsconvergencesformobilenetworks 1.4RelatedWork Timesynchronizationinsensornetworkshasbeenintenselystudied inrecentyears, andseveralclassicationsofexistingsynchronizationmethodsar epossible.Wecan classifythemaspairwisesynchronizationandglobalsynchronizatio nmethods.Inpairwise synchronization,apairofnodestrytosynchronizetheirclocksto eachother.Inpractice, thisisoftenachievedbyoneofthenodesestimatingitsrelativeose tand/orskew withrespecttotheothernode,sothatthelocaltimeofoneofthe nodesservesasa reference[ 1 { 5 ].Inglobaltimesynchronization,alsocallednetwork-widesynchron ization, allnodessynchronizethemselvestoacommontime,whichcouldbeth euniversaltime (UTC)thatisaccessedthroughGPSorthelocaltimeofoneofthen odeselectedasa referencenode.Theglobaltimesynchronizationapproachescan begroupedintothree categories:cluster-based,tree-basedandfullydistributedpro tocols. ReferenceBroadcastSynchronization(RBS)[ 17 ]isaprotocolthatperformsclusterbasedsynchronization.First,severalnodesareelectedasclust erleaders,andeachleader broadcastsabeacontonodesinitscluster.Apairofnodescompar ethearrivaltime ofthebeaconfromtheirclusterleaderinordertondtheclockos etsbetweenthem. Furthermore,twonodesindierentclusterscanusegatewaynod es(agatewaynode belongstoatleasttwoclusters)asabridgetocomputetheclocko setbetweenthem.In theend,nodescandiscoveratableofosetsforallpairsofnodes anduseittoachieve globaltimesynchronizationifanodeisselectedtobereferencenod e.Furthermore,skews estimationisalsoaddressedin[ 17 ]. Thetree-basedapproachistorstelectarootnodeandconstru ctaspanningtree ofthenetworkwiththerootnodebeingthe\level0"node.Then,e verynodethereafter synchronizesitselftoanodeoflowerlevel(higherupinthehierarch y).Examplesofsuch 17

PAGE 18

spanning-treebasedprotocolsincludethewidelyusedTiming-SyncP rotocolforSensor Networks(TPSN)[ 18 ]andFloodingTimeSynchronizationProtocol(FTSP)[ 19 ]. Bothtypesofalgorithmssuerfromhighoverheadofconstructin gapre-existing communicationstructure.Incluster-basedalgorithm,theoverh eadisfromelecting clusterleadersandgatewaynodes,whileintree-basedalgorithm,t heoverheadisfrom constructingaspanningtree.Inaddition,cluster-basedandtre e-basedapproachesare primarilytargetedtonetworksofstaticorquasi-staticnodes.Wh ennodesaremobile,the communicationlinksfrequentlychange.Therefore,theclusteror spanningtreeneedstobe re-computedfrequently,whichintroducesmoreoverhead. Recently,anumberoffullydistributedglobalsynchronizationalgor ithmshavebeen proposedthatdonotneedclusterorspanningtreecomputation. Distributedprotocols are,therefore,morereadilyapplicabletomobilenetworksthantre e-basedprotocols. Amongthedistributedsynchronizationprotocolsproposed,some arebasedonestimation oftheskewand/orosetswithrespecttoareferenceclock,whic hwecall absoluteskew andabsoluteoset .Thistypeofsynchronizationis,therefore,calledas absolutetime synchronization 1 Thealgorithmsproposedin[ 4 5 8 9 20 { 22 ]belongtothiscategory. Especially,algorithmsin[ 7 { 10 20 ]aresimilartoonealgorithmthatwewillpropose, wheretheclaricationofthedierencebetweenouralgorithmandt hoseinliteraturewill begiveninSection 1.5 .Thesealgorithmscanbeadaptedtomobilenetworks.However, littleisknownabouthowsuchanalgorithmwillperforminamobilenetwor kundernoisy measurement. Anotherclassofprotocolsestimatesacommonglobaltimethatmay notberelated tothetimeofanyclockinthenetwork,whichwecall virtualtimesynchronization Thealgorithmsproposedin[ 13 27 28 ]belongtothiscategory.[ 27 28 ]synchronize 1 Notethat,ifthereisnoconfusion,westillcallthemas skew oset and timesynchronization respectively. 18

PAGE 19

skews/osetsusingconsensusalgorithmondoubleintegratorfor staticnetworks.Thelast oneisbasedonacascadeoftwoconsensusalgorithmonskewsando setsrespectively. Thealgorithmsin[ 13 28 ]areprovedtoberobusttocommunicationfailurewithout consideringmeasurementnoise.Thisalgorithmpotentiallycouldbead optedtomobile networks.However,theconvergenceofthosealgorithmswhenm easurementsarenoisyis notanalyzed.Virtualtimesynchronizationmethodsuseconsensu salgorithmstoconstruct virtualskewandosets thateverynodeagreesto.Inmanyapplications,theusersof thesensornetworkareinterestedinthetimeofaneventthatisme asuredinareference clockthatphysicallyexistsinanode,oranUTCtimeprovidedbyaGPSu nitona basestation.Inthesecases,absolutetimesynchronizationispre ferableovervirtualtime synchronization.Inthiswork,weonlyconsiderabsolutetimesynch ronization. Alltimesynchronizationmethodssuerfromtheproblemthatnopr otocolcan providesynchronizationforarbitrarylengthsoftimeiftheprotoc olisstoppedatsome nitetime.Forexample,inabsolutetimedistributedsynchronization ,anyerrorinthe estimateoftheskewwillleadtoalargeerrorinestimatingofthegloba ltimeforlarge valuesoftime,whichiseasilyseenfrom( 1{3 ).Asimilarphenomenonoccursinvirtual timesynchronization,sincestoppingtheupdatingalgorithmatanod ewillmakethe logicaltimeatthatnodedivergefromthecommonvirtualtime.Thisis awellrecognized problem.Unlesstrueskewisobtained,aperiodicre-synchronizatio n[ 29 ]needtobe implementedinordertoboundtheerror j ^ t t j inadesignedrange.Thisreferstoshifting theoriginofthetimeaxistothereferenceclock'scurrenttimeande stimatingthenew osetwithrespecttothisorigin,sincethedenitionoftheosetde pendsonthestart timeofthereferenceclock(butnotoftheskew).Inthesynchro nizationprotocolsthat onlycorrectfortheosetandnottheskew(e.g.,[ 18 30 31 ]),resynchronizationhasto beperformedfrequently.Ifskewsarealsoestimated,thenthee stimateoftheglobaltime basedon( 1{3 )remainsaccurateforlongertimeintervals,sothatresynchroniz ationcanbe 19

PAGE 20

performedlessfrequently.Thisisimportantforpowerconservat ioninsensornetworks.In thiswork,wedonotdiscussdetailsofre-synchronization. 1.5Contributions Thisworksolvesclocksynchronizationprobleminmobilesensornetwo rks,andwe focusondevelopingdistributedestimationalgorithmthatcanbeuse dtoestimateclock skewsandosets,fromrelativemeasurements.Thecontribution sofChapter 2 3 4 areas follows: InChapter 2 ,weintroduceatwo-stepapproachforpairsofnodestoobtainre lative measurementsofskewsandosetsbypairwisecommunication.The twostepsare pairwise synchronization and transferringtheresultsofpairwisesynchronizationtore lativemeasurements .Intherststep,wedonotuseaspecicpairwisesynchronization algorithm, asmanyexistingalgorithmssuchas[ 1 3 { 5 8 9 21 ]canbeadopted.Thisprovidessome rexibility:inthefuture,abetterpairwisesynchronizationalgorithm canbeadopted,and theresultingrelativemeasurementscanbeimmediatelyusedbydistr ibutedestimation algorithmsproposedinthenexttwochapters.Thetwo-stepappr oachwillbeimplemented inthesimulationsectionsofthesetwochapters,therefore,wepic kthepairwisesynchronizationalgorithmin[ 1 ]asanexampletodemonstratethewholeprocedure.Notethat thereexistsrandomdelayduringpairwisecommunication,andweuse thereasonabledelay parametersobtainedfrominformationin[ 19 ]toshowtheresultingaccuracyofrelative measurementsbysimulation. InChapter 3 ,weanalyzeanalgorithm(JAT)fornodevariableestimationfromnois y pairwiserelativemeasurements,whichisaslightmodicationofthealg orithmproposed in[ 6 { 10 ]forskewsandosetsestimation.Thoughthealgorithmisadopted fromthese earlierpapers,theanalysisinthosepaperswaslimitedtostaticnetw orks.Thus,littleis knownabouthowsuchanalgorithmwillperforminamobilenetwork.We analyzethe convergenceofthealgorithmwhenthenetworktopologychanges duetothemotionof thenodesaswellasrandomcommunicationfailure.Wemodeltheres ultingtime-varying 20

PAGE 21

topologyofthenetworkasthestateofaMarkovchain.Technique sfortheanalysisof jumplinearsystemsfrom[ 32 ]areusedtostudytheconvergenceofthealgorithm.We showthatunderfairlyweakassumptionsontheMarkovchain,thep roposedalgorithmis meansquarestableifandonlyiftheunionofthegraphsthatoccuris connected.Mean squarestablemeanstheexpectedvalueandthevarianceofthees timatesobtainedbyeach nodeconvergestoxedvaluesthatdonotdependontheinitialcon ditions.Whenthe relativemeasurementsareunbiased,thenlimitingmeanisthesameas thetruevalueof thevariable,meaningtheestimatesobtainedareasymptoticallyunb iased.Formulasfor thelimitingmeanandvarianceareobtainedbyutilizingresultsfromjum plinearsystems. Furthermore,weshowthealgorithmcanbeimplementedasynchron ously,whichissuitable fortimesynchronizationproblem. Themaincontributionofthischapteristheanalysisoftheperforma nceofthealgorithm;thealgorithmitselfisverysimilartopreviouslyproposedone sandalsobears acloseresemblancetoconsensusalgorithms.Theerrordynamicso ftheestimationalgorithmturnsouttobealeader-followerconsensusalgorithm,whe retheleaderstates -correspondingtotheestimationerrorofthereferencenodesarealways0.Although theliteratureonconsensusisextensive,thetopicofconsensusw ithtime-varyinggraph topolgyandadditivemeasurementnoiseisconsideredonlyinalimitednu mberofpapers, with[ 11 12 ]representingthestateoftheartinthistopic.Therearesevera ldierences betweentheconsensusalgorithmsstudiedinthosepapersandthe errordynamicsexaminedinthispaper,whichprecludeusingexistingresultstoperformth eanalysis.These includerequirementofbalanceddigraphs,pre-designeddecreasin gconsensusgainthat mustbesynchronizedamongallnodes,etc.,whichwillbeexplainedmo reinSection 3.8 (seeRemark 3.1 ).Especially,therequirementofsynchronousdecreasinggainpre cludes thosealgorithmstobeusedintimesynchronizationproblemasthere isnocommon iterationindexamongnodes. 21

PAGE 22

Anothercontributionofthischapteristoprovidejusticationfor theMarkovian switchingtopologyformobilenetworks.TheMarkovianswitchingmod elhasalsobeen usedextensivelyinstudyingconsensusprotocolsinnetworkswithd ynamictopologies[ 11 33 { 35 ].Foranetworkofstaticnodeswithlinkdrops,theMarkovianswitch ingmodel arisesnaturallyfromaMarkovianlinkdropmodel.Inmobilenetworks, though,the onlycasewherewecanprovethatamobilenetworkevolvesaccordin gtoaMarkov chainiswhennodesmoveaccordingtotheso-calledrandomwalkmobilit ymodel[ 36 ]. AlthoughtheMarkovianswitchingassumptionfacilitatesanalysis,th isassumption requiresjusticationformorecomplexmotionmodels.Weuseatech niquefrom[ 37 ] totestifthegraphswitchingisMarkovianifnodesaccordingtothes o-calledRandom DirectionMobility(RD)model.TheRDmodelisoneofthemostwidelyuse dmobility modelsformobilesensornetworks[ 36 ].Weshowthattheresultinggraphswitching processcanindeedbeapproximatedwellbya(rstorder)Markov ianswitchingmodel. IntheJATalgorithm,thevarianceoftheestimateconvergestoan onzeroconstant.If measurementnoiseislarge,theestimatedskewcouldcontainalarge errornomatterhow longthealgorithmisperformed.Inthiscase,theprediction ^ t couldbequitepooraswe canseein( 1{3 ).Thismotivatesustoproposeamoreaccuratealgorithm. Inchapter 4 ,weproposeanewalgorithm(STO)forestimatingnodevariablesfro m relativemeasurementinmobilesensornetworks.TheSTOalgorithma chieveshighaccuracy.Weprovetheestimationerrorconvergesinmeansquaretoa constant,i.e.,themean convergestothelimitingvalueandthevarianceconvergestozero. Whentheswitchesof thenetworktopologiesaredeterministic,theconditionsofthecon vergenceincludethat unionofthegraphtopologiesisconnectedwithinacertainlengthoft imeintervaland atime-averagedLaplacianmatrixexists.Whentheswitchesofthen etworktopologies canbemodeledashomogeneousMarkovchain,theconditionisthatt heMarkovchainis ergodic.Furthermore,ifmeasurementnoiseiszeromean,theest imationerrorconverges 22

PAGE 23

inmeansquaretozero.ThisachievesbetterresultthanthatinJAT algorithm,inwhich varianceofestimationerrorconvergestosomenonzerovalue. AstochasticapproximationapproachisadoptedintheSTOalgorithm todealwith theaccumulationofmeasurementnoise[ 38 ].Thealgorithmwasinspiredbytheleaderless consensusprinciple[ 11 12 ].Themeritisthattheestimationerrorismeansquare convergent(highaccuracy).InChapter 4 ,wemaketwocontributionstakeadvantageof thestochasticapproximationapproach. First,werecallthattheconsensusalgorithmin[ 11 12 ]cannotbedirectlyusedfor ourpurpose,becausetheyrequireagain(stepsize)intheiteratio nupdateformulato decreasesynchronously.Otherwise,theconvergenceofthealg orithmisnotguaranteed. However,nodeshavenoaccesstosuchaglobalindexasclocksinno desarenoteven synchronized.Weproposedanimplementationmethodbylettingnod esstarttheir iterationsaccordingasequenceofpre-speciedlocaltimes.Using thismethod,the algorithmisasynchronouslyimplementable,whileitstillcanbethought ofasperforming updatinginasynchronousfashionduringconvergenceanalysis.On edrawbackofthe methodisthattheiterationintervalwillbeincreasingalongtime.How ever,weshowthat theintervalsincreaseslowly.Astheestimateofanodevariablewillco nvergequickly,the iterationscanstopbeforetheintervalbecomesnoticablylarge. Thesecondcontributionisthatwealsoanalyzethecasewhennoisein relative measurementsofnodevariablesisbiased,whichisnotconsideredby thosealgorithms in[ 11 12 ].Wenotonlyshowestimationerrorisstillmeansquareconvergent, butalso provideaformulatocomputethelimitingvalueofthemeanoftheestim ationerror. Insimulation,wecomparetheproposedJATandSTOalgorithmsalong withthe state-of-the-artATSalgorithm[ 13 ].TocompareJATandSTO,weuseglobaltime estimationerror.NotethatATSperformsvirtualtimesynchroniz ationsuchthatnodes estimateavirtualglobaltimeinsteadofrealglobaltime.Therefor e,tocompareallthree algorithms,synchronizationerror(recallthatitisdenedasthem aximumabsolutetime 23

PAGE 24

estimationerrorbetweenpairsofnodesamongnetwork),isuseda sthemetric.Weshow thattheSTOalgorithmoutperformsJATandATSintermsofthemet rics.However, theconvergencerateoftheSTOalgorithmisslowerthanthatinATS .Wepropose modicationstoimprovetheconvergencerateofSTOandproviden umericalvalidation. InChapter 5 ,weconcludethedissertationbydiscussingthemainresultsineach chapterandtheopenproblemsforfutureresearch. 24

PAGE 25

CHAPTER2 RELATIVEMEASUREMENTSOFSKEWANDOFFSET Inthischapter,weintroducetheproceduretoobtainnoisymeasu rementsoflog u log v and u v forapairofneighboringnodes u and v .Thiscontainstwosteps:1. pairwisesynchronization(Section 2.1 );2.transferringresultsofpairwisesynchronization torelativemeasurement(Section 2.2 ).Inpairwisesynchronization,byexchanginga numberoftime-stampedmessages,node u and v estimatetheso-called relativeskew u;v and relativeoset u;v [ 2 ],where u;v istherelativespeedoftwoclockwithmeasuredby clockofnode v and u;v isthelocalclockreadingofnode u ,whenclockof v isreadas zero.Therelativeskewandosetbetweentwonodes u and v aredenedbythefollowing equationthatrelatetheirlocaltimes: u ( t )= u;v v ( t )+ u;v ; (2{1) Therelationshipbetweentheabsoluteskew&oset u ; u ; v ; v andrelativeskew& oset u;v ; u;v isgivenby u;v := u v u;v := u v u v : (2{2) Thisrelationshipisobtainedbyexpressingthelocaltime u ( t )ofnode u atglobaltime t intermsofthelocaltime v ( t )atnode v atthesametime t byusing( 1{4 ): u ( t )= u ( v ( t ) v v )+ u = u v v ( t )+ u v u v ; andcomparingwith( 2{1 ). Anumberofmethodsareavailableintheliteraturethatallowspairwise synchronizationbetweenanodepairfromtime-stampedmessages[ 1 3 { 5 8 9 21 ].Inthesecond step,thenodepairusestheresultingestimates^ u;v and ^ u;v tocomputethemeasurementsoflog u log v and u v ,byusingtherelationshipbetween u ; u ; v ; v (recall thattheyareabsoluteskews&osetsprecisely)and u;v ; u;v (relativeskew&oset), 25

PAGE 26

whichisgiveninSection 2.2 .Theadvantageofthistwo-stepapproachisthatthenode variableestimationalgorithmsproposedinthenexttwochaptersca nusetherelative measurementsobtainedfrommanysuitablepairwisesynchronizatio nalgorithms.Therefore,inthefuture,theproposedalgorithmswillbereadytousewh enabetterpairwise synchronizationalgorithmisavailable. 2.1PairwiseSynchronization uv (1) v (1) u (2) u (2) v (3) v (3) u (4) u (4) v w (1) u w (3) u w (3 ; 1) v Figure2-1.Two-waymessagesexchangeforpairwisesynchroniza tion. Asmentionedabove,manypairwisesynchronizationalgorithmscanb eusedto estimate u;v and u;v forapairofnode u and v .Thegeneralprocedureisthatnode u and v rstperformtwo-waymessageexchanges(seeFigure 2-1 )andsavelocaltime stampsattheinstantsofthesendingandreceivingofmessages,a ndthencompute^ u;v and ^ u;v usingthoserecordedtimestamps. Inthissection,webrieryintroducethealgorithmproposedin[ 1 ]asanexample todemonstratetheprocedureasthisalgorithmwillbeintegratedin simulationsinthe followingchapters.Inthebeginning,node v sendsamessageto u thatcontainsthevalue ofthelocaltimeat u whenthemessageissent: (1) v .Whennode u receivesthismessage, itrecordsthelocaltimeofreception: (1) u .Afterawaitingperiod w (1) u (measuredinterms oflocaltimeof u ),node u sendsamessagebackto v thatcontainsboth (2) u and (1) u Whenitarrivesat v ,node v againrecordsthelocaltimeofreception: (2) v .Notethat, thereexistsunknownrandomdelaybetweenanode v sendingamessageandanode u receivingthemessage,whichisthereasonwhythepairwisesynchro nizationproblemis challenging.Inordertoestimateboth u;v and u;v ,theabovetwo-waymessageexchange 26

PAGE 27

procedureneedstobeperformedatleasttwice.Node v willthenobtaineighttimestamps f ( i ) u ; ( i ) v g for i =1 ; 2 ; 3 ; 4intheend.Thealgorithmproposedin[ 1 ]canutilizemorethan tworoundsofcommunication(i.e.,morethaneighttime-stampedmes sages).Wedescribe thealgorithmonlyforthecaseofminimalnumber(eight)oftime-sta mps. Wedenotethelocaltimebetweenthetwostarts (3) v (1) v := w (3 ; 1) v forfuture reference,calledwaitingtimebetweentworoundsofmessagesexc hanges.Theprocedure ofthepairwisetwo-waymessageexchangesisshowninFigure 2-1 .Node v usestheeight timestampstoestimate u;v and u;v asdescribedbelow,andthensendstheestimated valuesto u Denotetherandomdelaysduringcommunicationas d ( i ) i =1 ;:::; 4(inthelocal timeofnode v ),therelationshipamongtheeighttimestampsaregivenasfollows: (1) u = u;v d (1) + (1) v + u;v (2) u = u;v ( (2) v (1) v d (2) )+ (1) v + u;v (3) u = u;v ( (3) v (1) v + d (3) )+ (1) v + u;v (4) u = u;v ( (4) v (1) v d (4) )+ (1) v + u;v (2{3) Assuming d ( i ) isGaussiandistributedwithxedmean d andvariance 2 d ,thefollowingformulasderivedin[ 1 ]providesaMaximumLikelihood-likeestimator(MLLE)of u;v and u;v : ^ u;v = ( D ( a ) u ) 2 +( D ( b ) u ) 2 D ( a ) v D ( a ) u + D ( b ) u D ( b ) v ; (2{4) D ( a ) u = (3) u (1) u ;D ( a ) v = (3) v (1) v D ( b ) u = (4) u (2) u ;D ( b ) v = (4) v (2) v and ^ u;v = U (1) + U (3) U (2) U (4) 4 (1) v (2{5) 27

PAGE 28

U (1) = (1) u ; U (2) =^ u;v ( (2) v (1) v ) (2) u ; U (3) = (3) u ^ u;v ( (3) v (1) v ); U (4) =^ u;v ( (4) v (1) v ) (4) u ; Now,combine( 2{3 ),( 2{4 )and( 2{5 ),wehave: ^ u;v = u;v ( D ( a ) v + d ( a ) ) 2 +( D ( b ) v + d ( b ) ) 2 ( D ( a ) v + d ( a ) ) D ( a ) v +( D ( b ) v + d ( b ) ) D ( b ) v = u;v + u;v ( D ( a ) v + d ( a ) ) d ( a ) +( D ( b ) v + d ( b ) ) d ( b ) ( D ( a ) v + d ( a ) ) D ( a ) v +( D ( b ) v + d ( b ) ) D ( b ) v ; (2{6) where d ( a ) = d (3) d (1) and d ( b ) = d (2) d (4) ,whichareGaussiandistributedwithzero meanandvariance2 2 d .Theestimatedrelativeosetis ^ u;v = u;v + 1 4 ( d ( a ) + d ( b ) )^ u;v : (2{7) Notethattheaccuracyof^ u;v dependsonboth D ( j ) v and d ( j ) j = a;b ,whilethat of ^ u;v dependsonlyon d ( j ) .Notethat d ( j ) isnotaparametertobeadjustedsinceit mainlydependsonthecommunicationchannelinsensornodes.Altho ugh D ( j ) v cannotbe determined(sincerandomdelaysareunknown),itincreasesalongw ithcontrollablevalue w (3 ; 1) v (let w (1) u and w (3) u bexed).InSection 2.3 ,wewillevaluatetheeectof w (3 ; 1) v with respecttotheaccuracyofrelativemeasurementsnumerically. 2.2FromPairwiseSynchronizationtoRelativeMeasurement Supposeanode u obtainsnoisyestimates^ u;v ; ^ u;v oftheparameters u;v ; u;v by usingapairwisesynchronizationprotocol.1.Wemodelthenoisyestimateof u;v as ^ u;v = exp ( su;v ) u;v (2{8) 28

PAGE 29

Local clock Local clock u v u;v u;v v uFigure2-2.Thelocalclockmodelof u inthecoordinateoflocalclock v .Thebluesolid lineisrepresentingclocknode u ,whilethedashedlineisrepresentingclockof node v .Therelativeskewandosetaremarkedingure. where exp ( )isexponentialfunctionand su;v isarandomvariable.Iftheestimation errorissmall,then su;v iscloseto0.Takinglogonbothsidesof( 2{2 ),weobtain log^ u;v =log u;v + su;v =log u log v + su;v : (2{9) Withthedenitions s u;v :=log^ u;v ;x si :=log u ; weseethat( 2{9 )canberewrittenas s u;v = x su x sv + su;v ; (2{10) whichmakes s u;v anoisyrelativemeasurementofthenodevariables x su and x sv ; cf.( 1{4 ).Itisimportanttonoticethat s u;v isameasuredquantity{since^ u;v ismeasured{whilethevariables x su ;x sv ,whicharelogarithmsoftheskews,are unknown. 2.Similarly,thenoisyestimate ^ u;v of u;v withrandomestimationerror e ou;v canbe writtenas ^ u;v = u;v + e ou;v = u u v v + e ou;v = u v + ou;v ; (2{11) where ou;v := v (1 u v )+ e ou;v : Withthedenitions o u;v := ^ u;v ;x ou := u ; 29

PAGE 30

weseethat( 2{11 )canberewrittenas o u;v = x ou x ov + ou;v ; (2{12) whichagainfallsunderthecategoryofrelativemeasurementsofno devariables,i.e., hastheform( 1{4 ).Inthiscasethenodevariablesaretheclockosets u 's.The noise ou;v issmallonlyif u v ,otherwiseevenif e 0u;v iszeromean, 0u;v mayhave alargenon-zeromeanandtheeventualestimatesoneobtainsfro mtheserelative measurementswillprobablybequitepoor.Incaseoftimesynchron ization,however, theskewsareallclosetounity,sothat u v 1(anypairofclockswilldriftapartup to100 sec=sec [ 23 ]),whichjustiescalling 0u;v ameasurementnoise/measurement error. 2.3Simulation Intheprevioustwosections,atwo-stepprocedureisintroduced toobtainnoisy measurementsoflog u log v and u v .Sincewewillintegratetheapproachto thenumericalevaluationoftheproposedalgorithmsontimesynchr onizationinthenext twochapters,wenowsimulatethetwo-stepprocedurebypickingr ealisticparameterson pairwisecommunicationandexaminetheresultingaccuracyofrelativ emeasurementsof skewsandosets. Inthesimulation,apairofnode u and v areselectedwithskews1+2 10 5 1 2 10 5 ,andosets10 1 sec, 10 1 secrespectively.Duringmessageexchange,we adoptMAClayertime-stampingmethodtoreducethemainsourceof delaybetween messagesendandreceive[ 13 18 ].MAClayertime-stampingessentiallymeans:atthe senderside,thecurrenttime-stampiswrittenintothemessagerig htbeforethepacketis sentovertheair.Atthereceiverside,thetime-stampisrecorded rightafterthepreamble bytesofanincomingmessagehavebeenreceived.AlthoughMAClaye rcaneliminate largestcomponentsofthedelay,smallrandomdelaystillexists.He re,delayispickedas tobeGaussiandistributedwithmean150 sec andstandarddeviation5 sec ,whichare realisticvaluesbasedoninformationofdierentcomponentsofdela ysin[ 19 ].Weconduct simulationofthetwo-stepapproachbyassumingthepairofnodesc analwaycommutate witheachotherduringthesimulation(about1second).Atlocaltime (1) v = v t 0 + v (correspondingtoglobaltime t 0 ),node v startstherstroundofmessageexchanges. 30

PAGE 31

-1 -0.5 0 0.5 1 1.5 0 100 200 300 400 de su,vNumber (a)Skew -2 -1 0 1 2 3 4 x 10 -4 0 100 200 300 400 de ou,vNumber (b)Oset Figure2-3.Thehistogramsoferrorratios su;v and ou;v ,whereverticalredlinesindicate E [ su;v ]=2 : 3 10 3 and E [ ou;v ]=5 10 5 Thesecondroundofmessageexchangesisstartedby v at (3) v = (1) v + w (3) u ,where w (3) u is0 : 5sec.Thewaitingtime w (1) u = w (3) u =0 : 02secfornode u towaitbeforesending back.Recallthattheerrorofrelativemeasurementsofskewsan dosetsaredenedas su;v =log(^ u;v ) (log( u ) log( v ))and ou;v = ^ u;v ( u v )inSection 2.2 .From 10000MonteCarlosimulations,wecomputemeanandvariance E [ su;v ]=9 10 8 Var [ su;v ]=1 10 10 E [ ou;v ]= 4 : 1 10 6 and Var [ ou;v ]=1 : 3 10 11 Inordertoprovideanideaoftheaccuracyofthepairwisesynchro nization,wedene the errorratio as su;v = su;v log( u ) log( v ) and ou;v = ou;v u v .Figure 2-3(a) and 2-3(b) showthehistogramsoferrorratios su;v and ou;v .FromtheFigure 2-3(a) ,weseethat theerrorratioscouldbequitelargeforrelativemeasurementofsk ewswhilethemean oferrorratioisstillsmall.Thisprovidesthemotivationtocollectmore measurements duringdistributedupdatefornodevariableestimationinChapter 3 and 4 .Furthermore, Figure 2-3(b) showsthatrelativemeasurementofosetscontainsbiasbutthev alueis small,whichisinaccordwiththeoreticalanalysisinSection 2.2 Wealsoinvestigatehowthewaitingtime w (3 ; 1) v betweentworoundsofmessage exchangesaectstheaccuracyof^ su;v and^ ou;v .Wevaries w (3 ; 1) v from0 : 05secto5sec. Foreachvalueof w (3 ; 1) v ,weconduct10000MonteCarlosimulationsinordertond meanandvariance.Figure 2-4(a) and 2-4(b) showsthatthevarianceof su;v decreasesas 31

PAGE 32

0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 x 10 -7 E [ su;v ] w (3 ; 1) v (sec) (a)Mean 0 1 2 3 4 5 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 Var [ su;v ]w (3 ; 1) v (sec) (b)Variance Figure2-4.Themeanandvarianceoferrorofrelativemeasuremen t su;v withrespectto dierentvalueof w (3 ; 1) v w (3 ; 1) v increases.AsseenfromFigure 2-5(a) and 2-5(b) ,theaccuracyof^ ou;v isessentially irrelevantto w (3 ; 1) v Theconclusionsofthesestudiesaresummarizedbelow: 1.Measurementaccuracyofrelativeskewsimproveswhen D ( a ) v and D ( b ) v increases(by increasing w (3 ; 1) v ),whichanswersthequestioninSection 2.1 .However,thewaiting time w (3 ; 1) v cannotbearbitrarilylonginmobilenetworksincecommunicatelink changesfrequently.Therefore,abalancehastobestruck:cho osinglongerwaiting timebetweentimestampsimprovesaccuracyinrelativemeasuremen ts,butmay reducethenumberofmeasurementsthatcanbeobtainedinagiven timeinterval. 2.Theerrorinrelativemeasurementsofskewscanbehigh;seeFigu res 2-3(a) and 2-3(b) .Thiserrorcanbereducedbyusingmorethaneighttime-stampsas describedin[ 1 ],whichwillrequiresmorethantworoundsofcommunication.In highlymobilenetworks,thenumberoftime-stampsthatcanbeexch angedbetween apairnodesislikelytobesmall,especiallyifthewaitingtime w (3 ; 1) v iskeptlarge toimprovemeasurementaccuracy.However,mobilityalsooersdiv ersity-anode maybeabletocollectrelativemeasurementswithmanyneighborsove rtimeeven thougheachmeasurementsmaybehighlynoisy.Thisisoneofthemot ivationsfor thedistributedalgorithmsforskewandosetestimationdescribed inChapter 3 and 4 ,inwhichanodekeepsoncollectingandusingrelativemeasurementsf rom neighborsthatitencountersovertime. 32

PAGE 33

0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 x 10 -6 E [ su;v ] w (3 ; 1) v (sec) (a)Mean 0 1 2 3 4 5 10 -10.9 10 -10.8 Var [ su;v ]w (3 ; 1) v (sec) (b)Variance Figure2-5.Themeanandvarianceoferrorofrelativemeasuremen t ou;v withrespectto dierentvalueof w (3 ; 1) v 33

PAGE 34

CHAPTER3 THEJATALGORITHMINMARKOVIANSWITCHINGNETWORKS Inthischapter,weaddressthenodevariableestimationproblemint roducedin Chapter 1 .WeproposeanalgorithmwhichisaslightmodicationoftheJacobi-ty pe algorithmsin[ 6 { 10 ]forstaticnetworks.Theproposedalgorithmcanbeimplemented asynchronously,whichissuitablefortimesynchronizationproblem. Techniquesforthe analysisofjumplinearsystemsfrom[ 32 ]areusedtostudyconvergenceofthealgorithm. WeshowthatunderfairlyweakassumptionsontheMarkovchain,th eproposedalgorithm ismeansquarestableifandonlyiftheunionofthegraphsthatoccur isconnected.Mean squarestablemeanstheexpectedvalueandthevarianceofthees timatesobtainedbyeach nodeconvergestoxedvaluesthatdonotdependontheinitialcon ditions.Whenthe relativemeasurementsareunbiased,thenlimitingmeanisthesameas thetruevalueof thevariable,meaningtheestimatesobtainedareasymptoticallyunb iased.Formulasfor thelimitingmeanandvarianceareobtainedbyutilizingresultsfromjum plinearsystems. Thecontributionalongwithsimilarities/dierencestootheralgorithm shasbeenstatedin Section 1.5 Therestofthechapterisorganizedasfollows.Section 3.1 statestheproblemusing graphtheory,whichisusedforintroducingandanalyzingthepropo sedalgorithm.In Section 3.2 ,werstdescribetheexistingJacobialgorithmforstaticnetwork sandthen relateittotheproposedalgorithminmobilenetworks.InSection 3.3 ,weshowthe algorithmcanbeimplementedasynchronouslyforapplicationintimesy nchronization. Themainresultoftheconvergenceanalysisforthecasewhenthes witchingofgraphsis modeledasarst-orderMarkovchain,Theorem 3.1 ,isstatedinSection 3.5 .Inaddition, weshowthattheMarkovianmodelingofgraphswitchinmobilenetwork sisreasonablein Section 3.4 .SimulationstudiesarepresentedinSection 3.6 .Apartfromskewandoset estimationaccuracyaswellastheresultingglobaltimeestimationac curacyareexamined 34

PAGE 35

insimulations.Section 3.7 providescommentsandpotentialfuturework.Section 3.8 is devotedtotheproofofthetheorem. 3.1ProblemStatement Weconsidertheproblemofestimatingthescalarnodevariables x u u =1 ;:::;n b where n b isthenumberofnodesinthenetworkthatdonotknowtheirnodeva riables. Weassumethatthereare n r additionalnodesthatknowstheirnodevariables.Anode thatknowsitsnodevariable(thatiswhoknowtheirownskewsando sets)iscalled a referencenode .Weassume n r 1,otherwisetheproblemisindeterminateuptoa constant.Thetotalnumberofsensornodesinthenetworkisthe refore n = n b + n r .These deneanodeset V = f 1 ;:::;n g .Forlaterreference,wedene V b := f 1 ;:::;n b g and V r = f n b +1 ;:::;n b + n r g ,sothat V = V b [ V r .Notethat n = n b + n r .Timeis measuredbyadiscretetime-index k =0 ; 1 ;::: .Themobilenodesdeneatime-varying undirected measurementgraph G ( k )=( V ; E ( k )),where( u;v ) 2 E ( k )ifandonlyif u and v canobtainarelativemeasurementoftheform( 1{4 )duringthetimeintervalbetween thetimeindices k and k +1.Specically,foreach( u;v ) 2 E ( k ),thereisameasurement u;v ( k )= x u x v + u;v ( k )thatisavailabletoboth u and v attime k .Inpractice,one ofthetwonodescomputesthismeasurementfromsensedinforma tion.Weassumethatif u computesthemeasurement u;v ,itthensendsthismeasurementto v sothat v alsohas accesstothesamemeasurement.Wefollowtheconventionthatth erelativemeasurement between u and v thatisobtainedbythenode u isalwaysof x u x v whilethatusedby v isalwaysof x v x u .Sincethesamemeasurementissharedbyapairofneighboring nodes,if v receivesthemeasurement v;u from u ,thenitconvertsthemeasurementto v;u byassigning v;u ( k ):= u;v ( k ).Fortechnicalreason,weassumethat,foranypair u and v ,whocomputes v;u ( k )foralltime k isxedaheadoftime.Thiscanbeachievedby comparingthemagnitudeoftheindexofnodes.Forexample,if u>v ,then u computes v;u ( k )rstandthensendsitto v 35

PAGE 36

The neighbors of u at k ,denotedby N u ( k ),isthesetofnodesthat u hasanedge withinthemeasurementgraph G ( k ).Weassumethatif v 2N u ( k ),then u and v canalso exchangeinformationthroughwirelesscommunicationattime k .Therefore,ifoneprefers tothinkofacommunicationgraph,weassumethatitisthesameasth emeasurement graph. Theproblemistoestimatethenodevariables x u for u =1 ;:::;n byusingtherelative measurements u;v ( k ) ; ( u;v ) 2 E ( k )thatbecomesavailableovertime k =0 ; 1 ;::: .In addition,thealgorithmhastobedistributedinthesensethateachn odehastoestimate itsownvariables,andateverytime k ,anode u canonlyexchangeinformationwithits neighbors N u ( k ). 3.2ProposedJATAlgorithm 3.2.1PriorWork:Jacobi-typeAlgorithms Wenotedearlierthatmanyoftheiterativealgorithmsforclocksync hronization(such asthosein[ 6 { 10 ])arespecialcasesofthe Jacobialgorithm describedin[ 7 ].Thealgorithm wasproposedtosolvedistributedestimationofnodevariablesfrom relativemeasurement instaticnetworks.Inthisalgorithm,anodeobtainsestimatesfrom itsone-hopneighbors andthenupdatesitsownestimate.With^ x u ( k )denotingnode u 'sestimateof x u atthe k -thiteration,thealgorithmcanbewrittenas ^ x u ( k +1)= X v 2N u 1 2 u;v 1 X v 2N u 1 2 u;v (^ x v ( k )+ u;v ) ; (3{1) where N u denotesthe(one-hop)neighborsof u ; 2 u;v isthevarianceofthemeasurement error u;v .TheJacobialgorithmhasasimpleinterpretation.Anode'sinitialest imateis theaverageofrelativemeasurementswithallofitsone-hopneighb ors.Itupdatesthis estimatebyaweightedsumoftheestimatesofitsneighborsateach iteration.Ithasbeen shownthattheestimate^ x u ( k )convergestoanoptimalvaluewithrespecttotheinitial measurement u;v s.Thealgorithmisalsoprovedtoberobusttotemporarilycommunica tionfailure.Itisnaturaltothinkofadoptingthisalgorithmtomobilen etworks.However, 36

PAGE 37

uv sareonlyobtainedonceatthebeginningoftheiteration.Inmobilene tworks,( u;v ) maynotin E (0)initially,butthenin E ( k )forsome k .Inaddition,ifameasurement obtainedinitiallycontainslargeerror,itwillmakeimpactonthenodeva riableestimates asoptimalestimateofanodevariabledependsontheinitialmeasure ments.Also, N u ( k ) maybeemptyforsome k inmobilecase,making( 3{1 )undened.Therefore,wewould liketomakeamodicationonthisalgorithm,whichwillbestatednext.3.2.2AlgorithmforDistributedEstimationfromRelativeM easurement Thealgorithmweconsiderisaminormodicationof( 3{1 )tomakeitapplicableto timevaryingnetworks.Eachnode u maintainsinitslocalmemoryanestimate^ x u ( k )of itsnodevariable x u 2 R .Everynode-exceptthereferencenodes-iterativelyupdates itsestimateaswe'lldescribenow.Theestimatescanbeinitializedtoar bitraryvalues.In executingthealgorithmatiteration k ,node u communicateswithitscurrentneighborsto obtainmeasurements u;v ( k )andtheircurrentestimates^ x v ( k ), v 2N u ( k ).Sinceobtaining measurementsrequireexchangingtime-stampedmessages,thec urrentestimatescanbe easilyexchangedduringtheprocessofobtainingnewmeasurement s.Node u thenupdates itsestimateaccordingto ^ x u ( k +1)= 8><>: w uu ( k )^ x u ( k )+ P v 2N u ( k ) w vu ( k )(^ x v ( k )+ u;v ( k )) w uu ( k )+ P v 2N u ( k ) w vu ( k ) for u 2 V b x u for u 2 V r ; (3{2) wherethe weights w vu ( k )and w uu ( k )arearbitrarypositivenumbers.Theupdate lawiswell-denedevenattimeswhen u hasnoneighbors.Nodescontinuethisiterative updateunlesstheyseelittlechangeintheirlocalestimates,atwhich pointtheycanstop updating.Theupdateprocedureineachnode u 2 V b isspeciedinAlgorithm 1 Eachnode u isallowedtovaryitslocalweights w uv ( k )withtimeandusedistinct weightsfordistinctneighborstoaccountfortheheterogeneityin measurementquality. Betweentwoneighbors p;q ofnode u attime k ,therelativemeasurement u;p ( k )between 37

PAGE 38

Algorithm1 Distributedupdateatnode u 1: Initializeestimate^ x u (0)and k =0 2: while u isperformingiteration do 3: for v 2N u ( k ) do 4: if u doesnothave u;v ( k ) then 5: 1. u and v performpairwisesynchronization; 6: 2. u saves u;v ( k )and^ x v ( k ); v saves v;u ( k )and^ x u ( k ); 7: else 8: u stopstocommunicatewith v atcurrent k ; 9: endif 10: endfor 11: if N u ( k u ) 6 = ; then 12: u updates^ x u ( k +1)using( 3{2 ); 13: else 14: ^ x u ( k +1)=^ x u ( k ); 15: endif 16: k = k +1; 17: endwhile u and p mayhavelowermeasurementerrorthantherelativemeasurement u;q ( k )between u and q .Thisoccurs,forexample,if u and p wereabletoexchangemoretimestamped messagesthan u and q beforecomputingtherelativemeasurements[ 4 26 ].Inthis case,node u shouldchooseitslocalweightsat k sothat w pu ( k ) >w qu ( k ).Duetothe denominatorin( 3{2 ),itisonlytheratiosamongtheweightsthatmatter,nottheir absolutevalues. 3.3AsynchronousImplementationforTimeSynchronization Thedescriptionsofarisintermsofacommonglobaliterationindex k .Inpractice, nodesdonothaveaccesstosuchaglobalindex.Instead,eachno de u keepsalocal iterationindex k u .Aftereveryincrementofthelocalindex,thenodetriestocollect a newsetofrelativemeasurementswithrespecttooneormoreofits neighborswithin apre-speciedtimeinterval.Attheendofthetimeinterval,whethe ritisabletoget newmeasurementsornot,itupdatesitsestimateaccordingtothe updatelaw( 3{2 )with k replacedby k u init-andincrementsitslocaliterationcounter.Theprocessthen 38

PAGE 39

repeats.Itfollowsfrom( 3{2 )thatifanodeisunabletogathernewmeasurementsfrom anyneighbors,thenitsupdatedestimateispreciselythepreviouse stimate. Theglobaliterationindexisusefultodescribethealgorithmfromth epointofviewof anomniscientspectator.Let T thetimeinterval,say,inseconds,betweentwosuccessive incrementsoftheglobalindex k .Theparameter T isarbitrary,aslongasissmallenough sothatnonodeupdatesitslocalestimatemorethanoncewiththet imeinterval T .In thatcase,oneofonlytwoeventsarepossibleforanarbitrarynod e u attheendofthe timeintervalwhentheglobalcounterisincreasedfrom k to k +1:(i) u eitherincreasesits localindexbyone,or(ii) u doesnotincreasesitslocalindex.Ifanodeincreasesitslocal index,boththelocalandglobalindicesincreasebyone.Anodedoes notincreaseitslocal iterationindexifitisnotabletogathernewmeasurements.Intheom niscientspectator's view,thenode'sneighborsetisemptyatthistimeindex;soaccording to( 3{2 ),thenext estimateofthenode'svariableisthesameasthepreviousone.Thus ,anode'slocal asynchronousstateupdatecanbedescribedintermsofthesync hronousalgorithm( 3{2 ); thelatterbeingmoreconvenientforthepurposeofexposition.In theremainderofthe chapterweconsideronlythesynchronousversion. 3.4MarkovianModelofTopologySwitch Inthischapterwemodelthesequenceofmeasurementgraphs fG ( k ) g 1k =0 that appearastimeprogressesastherealizationofa(rstorder)Mar kovchain,whosestate space G = fG 1 ;:::; G N g isthesetofgraphsthatcanoccurovertime.Ingeneral,a stochasticprocess X ( k )iscalled ` -thorderMarkovif P ( X ( k +1) j X ( k ) ;X ( k 1) ;::: )= P ( X ( k +1) j X ( k ) ;X ( k 1) ;:::;X ( k ` +1)),where P ( )denotesprobability.The Markovianswitchingassumptiononthegraphsmeansthat P ( G ( k +1)= G i jG ( k )= G j )= P ( G ( k +1)= G i jG ( k )= G j ; G ( k 1)= G ` ;:::; G (0)= G p )where G i ; G j ; G ` ;:::; G p 2 G WeassumethattheMarkovchainishomogeneous,anddenotethet ransitionprobability matrixofthechainby P ,inwhich p ij isthe( i;j )-thentryof P 39

PAGE 40

HereweexaminethequestionoftheapplicabilityoftheMarkovianmod elofgraph switching.Anexampleinwhichthetimevariationofthegraphssatise sthehomogeneous Markovmodelisanetworkofmobileagentswhosemotionismodeledwit hrstorder dynamicswithrange-determinedcommunication.Inmobilenetworks literaturethisis referredtoasthe randomwalkmobilitymodel [ 36 ].Specically,supposetheposition ofnode u attime k ,denotedby p u ( k ),isrestrictedtolieontheunitsphere S 2 = f x 2 R 3 jk x k =1 g ,andsupposethepositionevolutionobeys: p u ( k +1)= f ( p u ( k )+ u ( k )),where u ( k )isastationaryzero-meanwhitenoisesequenceforevery u ,andE[ u ( k ) v ( k ) T ]=0 unless u = v .Thefunction f ( ): R 3 S 2 isaprojectionfunctionontotheunit-sphere. Inaddition,( u;v ) 2 E ( k )ifandonlyifthegeodesicdistancebetweenthemislessthan orequaltosomepredeterminedvalue.Inthiscase,thegraph G ( k )isuniquelydetermined bythenodepositionsattime k ,andthepredictionof G ( k +1)given G ( k )cannotbe improvedbytheknowledgeofthegraphsobservedpriorto k : G ( k 1) ;:::; G (0).Hence theevolutionofthegraphsequencesatisestheMarkovianprope rty.Ifinaddition randomcommunicationfailureleadstotwonodesnotbeingabletocom municateeven whentheyareinrange,theMarkovianpropertyisretainediftheco mmunicationfailureis i.i.d. Nevertheless,forsomemobilitymodels,itisnotstraightforwardto checkifthesequenceofgraphsgeneratedbythemodelsatisestheMarkovian property.Ageneral methodofcheckingMarkovianswitchingofgraphsisthereforenee ded.Weborrowa methodthatisproposedin[ 37 ]tocheckifastochasticprocessisMarkovfromobservationsoftheprocess.Werstintroducesomestandardnotationf rominformationtheory. Let X beadiscreterandomvariablewithsamplespacen= f 1 ;:::;N g andprobability massfunction p ( x )= P ( X = x ),where x 2 n.Theentropyof X isdenedby H ( X )= X x 2 n p ( x )log p ( x ) : (3{3) 40

PAGE 41

Thedenitionofentropyisextendedtoapairofrandomvariable X;Y ,where X;Y 2 n, asfollows H ( X;Y ):= X x;y 2 n p ( x;y )log p ( x;y ) : (3{4) Theconditionalentropy H ( Y j X )isdenedas H ( Y j X ):= X x;y 2 n p ( x;y )log p ( x j y )= H ( X;Y ) H ( X ) ; (3{5) where p ( x;y )isjointprobabilitymassfunction.Theconditionalentropymeasur esthe conditionaluncertaintyaboutaneventgiventheanotherevent. Considerastochastic process f X 1 ;X 2 ;::: g .Assumingtheprocessisstationary,wedenote H ( single ):= H ( X k ) and H ( double ):= H ( X k ;X k +1 ),forall k =1 ;::: .Itisstraightforwardtoshowthat H ( double )=2 H ( single )ifthesuccessiverandomvariables X k arei.i.d.Inthiscase therandomprocessisazero-orderMarkovprocess.Iftherand omvariables X k arenot independent, H
PAGE 42

0 1 2 3 4 5 0 0.5 1 1.5 2 i ^ H i(a)Independent 0 1 2 3 4 5 0 0.5 1 1.5 2 i ^ H i(b)First-order 0 1 2 3 4 5 0 0.5 1 1.5 2 i ^ H i(c)Second-order Figure3-1.Thestandardshapesofestimatedconditionalentrop yforthreedierentcases: (a) independence, (b) rst-orderdependenceand (c) second-orderdependence. IfaprocessisarstorderMarkovchain,empiricallycomputedcond itional entropywillshowatrendsimilartotheonein(b). X k 2 ;X k 3 ;::: willnothelpmuch.Thisaccordswiththedependencepropertyofa rstorderMarkovchain.Similarly,Figure 3-1(c) indicatesthattheprevioustwovariables X k 2 ;X k 1 arebothimportanttopredict X k .Inthiscasetheprocessisbettermodeledas asecondorderMarkovchain. Inordertoconcludewhethertheevolutionofgraphsisgovernedb yarst-order Markovchain,weadoptthemethoddiscussedaboveasfollows.For aparticularmobility model,weconductasimulationandcollectobservationsofthegraph sequence.Sincethe underlyingsamplespace G ofthestochasticprocess G ( k )isnite,themethoddescribed aboveisapplicable.Wethenusetheapproachabovetocheckwheth ertheplotof ^ H i estimatedfromthecollectedobservationsisclosertothatinFigure 3-1(b) thantothose inFigure 3-1(a) orFigure 3-1(c) .Ifso,wedeclarethatitis reasonabletomodelthegraph switchingprocessasMarkovian Asanillustrativeexample,weconsiderthewidelyused randomwaypoint (RD)mobilitymodel[ 36 ].IntheRDmodel,eachnoderandomlypicksaninitialposition.Then, thenodepicksarandomdirectionwitharandomspeedthatisuniform lydistributed in[ v min ;v max ].Oncenodereachestheboundaryofthenetworkarea,itchoose sanew randomdirectionandspeedtostartover.Weconductasimulationo ftheRDmodelwith 3nodes,where v min v max arechosenas5 m=s and10 m=s andthepositionofthenodes aresampledevery0 : 1 s .Thenodesareallowedtomoveinaregion10 10m.Nodes' 42

PAGE 43

positionsareinitializedrandomly.Thesamplespaceconsistsof8grap hs.Byperforming thesimulationforalongtime(10 4 s ),weobtainalargenumberofobservationsoftheprocess fG ( k ) g .Theprobabilitymassfunctionisempiricallyestimatedfromtheobser vations. Forestimatingconditionalentropies,certainconditionalprobabilit ies,especiallythoseof thetype P ( G ( k )= G 1 jG ( k 1)= G 2 ; G ( k 2)= G 3 ),areproblematicsincetherelevant eventsmaynotbeobservedeveninaverylongsequenceofobserv ations.Inthiscasewe setthecorrespondingprobabilitiesto0anduse0log0=0.Theempiric allyestimated conditionalentropies ^ H i areshowninFigure 3-2 .Clearly,theshapeofcurveissimilar tothatinFigure 3-1(b) .Therefore,weconcludethatthegraphswitchingprocessinRD mobilitycanbereasonablymodeledasa(rstorder)Markovchain. 0 1 2 3 0.5 1 1.5 2 2.5 i ^ H iFigure3-2.Empiricallyestimatedconditionalentropyforthegraph process fG k g with threenodesmovingaccordingtotherandomwaypointmobilitymodel. NotethatinRDmobility,predictionofthefuturenodelocations(and thereforethe graph)basedonknowledgeofpastandpresentmaybemoreaccur ateinsteadofprediction basedononlythepresent.Thereforeitisquitepossiblethatthegr aphswitchingisnot rst-orderMarkov.However,theresultsofthetestabovesho wsthataMarkovmodel quiteaccuratelycapturesthegraphswitchingprocesswithRDmob ility. 3.5ConvergenceAnalysiswithMarkovianSwitching Weanalyzetheasymptoticbehaviorofthealgorithm( 3{2 )whengraphchangesare governedbyanergodic N -statehomogeneousMarkovchain. 43

PAGE 44

Let e u ( k ):=^ x u ( k ) x u betheestimationerroratnode u .Since u;v ( k )= x u x v + u;v ( k ),theupdatelaw( 3{2 ) canberewrittenas e u ( k +1)= 8><>: w uu ( k ) e u ( k )+ P v 2N u ( k ) w vu ( k )( e v ( k )+ u;v ( k )) w uu ( k )+ P v 2N u ( k ) w vu ( k ) for u 2 V b 0for u 2 V r ; (3{6) Therationalebehind( 3{6 )isthattherighthandsideof( 3{6 )isaweightedaverage ofestimationerrorsof x u andmeasurementnoise.Ifthemeasurementnoise u;v ( k )is zero-meanandtheinitialestimatesareunbiased,i.e.E[ e u (0)]=0foreach u 2 V b ,then E[ e u ( k )]=0forall k Themainresultofthechapter-onthemeansquarestabilityofthea lgorithm( 3{6 ) -isstatedbelowasatheorem.Inthestatementoftheorem, e ( k ):=[ e 1 ( k ) ;:::;e n b ( k )] T istheestimationerrorvector.Moreover, ( k ):=E[ e ( k )]isthemeanand Q ( k ):= E[ e ( k ) e ( k ) T ]isthecorrelationmatrixoftheestimationerrorvector,whereE[ ]denotes expectation.Wesaythatastochasticprocess y ( k )is meansquarestable ifE[ y ( k )]and E[ y ( k ) y T ( k )]convergesas k !1 foreveryinitialcondition.The uniongraph ^ G isdened asfollows: ^ G := [ Ni =1 G i =( V ; [ Ni =1 E i ) ; (3{7) where E i issetofedgesin G i .Weassumethatthemeasurementnoise u;v ( k )aectingthe measurementsontheedge( u;v )isawidesensestationaryprocesswithmeanE[ u;v ( k )] andvariance Var [ u;v ( k )].RecallthatE[ u;v ( k )]maynotbezero.Wealsoassumethat themeasurementnoisesequence u;v ( k )andtheinitialcondition^ x u (0),forany u;v;k is independentoftheMarkovchainthatgovernsthetime-varyinggr aphs. Duetotechnicalreasons,wemakeanadditionalassumptionthatt hereexistsatime k 0 afterwhichtheweightsdonotchange.Thechoiceofweightsduring the\transient 44

PAGE 45

period"(upto k 0 )willaectinitialreductionoftheestimationerrorsbutwillnotchan ge theasymptoticbehavior.Theorem3.1. Assumethatthetemporalevolutionofthecommunicationgra ph G ( k ) isgovernedbyan N -statehomogeneousMarkovchainthatisergodic,and p ii > 0 for i =1 ;:::;N .Theestimationerror e ( k ) ismeansquarestable,ifandonlyif ^ G is connected.Ifadditionallyallthemeasurementsareunbias ed,thentheestimatesare asymptoticallyunbiased: lim k !1 ( k )=0 Beforeprovidingtheproofofthetheorem,werstexplainthemea ningofthe theoremandalsoprovidearemarkontheconnectiontoconsensus literature. Theformulasforcomputingthelimitingvalues lim k !1 ( k ) and lim k !1 Q ( k ) are providedlaterinLemma 3.3 inSection 3.8 Theimplicationofthetheoremisthataslong asnodesareconnectedina\time-average"sensecharacterized by ^ G beingconnected,the nodeshaveestimateswhosemeanandvarianceconvergesastimep rogressesirrespective oftheinitialconditions.Thus,afterasucientlylongtime,thenode scanturnothe synchronizationupdateswithoutmuchlossofaccuracy.Theassu mptionofergodicity oftheMarkovchainensuresthatthereisanuniquesteadystated istributionandthat thesteadystateprobabilityofeachstateisnon-zero[ 32 ].Thismeanseverygraphin thestatespaceofthechainoccursinnitelyoften.Sincetheirunio ngraphisconnected, ergodicityimpliesthatinformationfromthereferencenode(s)willr owtoeachofthe nodesovertime.Noneofthegraphsthateveroccurisrequiredto beaconnectedgraph. Theassumption p ii > 0means P ( G ( k +1)= G i jG ( k )= G i ) > 0.Thiscanbeassuredifthe nodesmoveslowlyenough.Remark3.1. Theequation ( 3{6 ) canbeinterpretedasaleader-followingconsensus problem,wheretheconsensusstatefornode u istheestimationerror e u ( k ) for u 2 V b ,whiletheleaderstatesare e r ( k ) 0 for r 2 V r .Consensusproblemaremore interestingwhenthereisnoleader.Resultsfromleaderles sconsensuscanbeusedto achieveconvergenceofconsensuswithleaders,asremarked in[ 11 ].Thepapers[ 11 12 ] 45

PAGE 46

proposeleaderlessconsensusalgorithmusingstochastica pproximationundertime-varying graphs,inwhichnodevariablesconvergetoacommonvaluein meansquaresense. However,therearesignicantdierencesbetweenthealgor ithmanalyzedhereandthose in[ 11 12 ],aswellastheresults.Thealgorithmsin[ 11 12 ]requirethatthenodesusea specictime-varyingweighttoensurecertain ` 2 =` 1 conditions,whichisdiculttoensure inanetworkwherethenodesdonothavesynchronizedclocks. Incontrast,weallowthe nodestovarytheirweightswithtimearbitrarilysubjecton lytotheconditionthattheystop doingsoatacertaintime.Furthermore,theresultsin[ 11 12 ]withMarkovianswitching areestablishedundertheassumptionthattheweightedLapl acianmatricesofthedirected graphs 1 arebalanced.WeallowfullasymmetryintheLaplacianmatri ces,sothateach nodecanchooseitsownweightwithoutcoordinatingwithits neighbors,whichisotherwise requiredtoensurethebalancedcondition.Finally,duetoo uruseofjumplinearsystem theory,weareabletoprovideanexpressionforthelimiting variance,whiletheresults in[ 11 12 ]onlyshowthatalimitingrandomvariablewithnitesecond momentexists. 3.5.1ProofofTheorem 3.1 Werstintroducesomestandardandnon-standardgraph-theo reticterminology.We considera weighteddirected graph ~ G ( k )=( V ; ~ E ( k ) ;W ( k ))associatedwithundirected measurementgraph G ( k )=( V ; E ( k )).Inparticular,thereexistsanundirectededge(u,v) in G ( k ),thenthereexisttwodirectededges( u;v )and( v;u )in ~ G ( k ).The weightmatrix W ( k )denedas W uv ( k ):= 8>>>><>>>>: w vu ( k ) > 0for( u;v ) 2 ~ E ( k ) w uu ( k ) > 0for v = u 0o.w. (3{8) 1 TheLaplacianmatrixof G i isequalto M i N i inthischapter,wherethedenitionof matrices M i and N i aregiveninSection 3.8 46

PAGE 47

1 2 3 G ~ G w 12 w 21 w 32 w 23w 31w 13 w 11 w 22 w 33 Figure3-3.Ameasurementgraph G andthecorrespondingweightgraph ~ G Thus,givenameasurementgraph G ( k )and W ( k ), ~ G ( k )isspecied.SeeFigure 3-3 foran exampleofanundirectedmeasurementgraphandanassociateddir ectedweightedgraph. Thesquare non-negative matrices, D ( k ), M ( k )and N ( k )isdenedasfollows: D ( k )isa n n diagonalmatrixmadeupofthediagonalentriesof W ( k ),and N ( k ):= W ( k ) D ( k ). M ( k )isa n n diagonalmatrixwithentry M uu ( k )= P u 6 = v W uv ( k ).Furthermore,we denethe n b n b basismatrix D b ( k ), M b ( k )and N b ( k )astheprinciplesubmatrixof D ( k ), M ( k )and N ( k )obtainedbyremovingthoserowsandcolumnscorrespondingtoth e referencenodes,i.e.,from( n b +1)-thto n -throwsandcolumns.Now,equation( 3{6 )can becompactlyexpressedas e ( k +1)= J b ( k ) e ( k )+ B b ( k ) ( k ) ; (3{9) where J b ( k ):=( M b ( k )+ D b ( k )) 1 ( N b ( k )+ D b ( k )) n b n b ; B b ( k ):=( M b ( k )+ D b ( k )) 1 A b ( k ) ;A b ( k ):= blkdiag ( N 1 ( k ) ;:::; N n b ( k )) n b n b ( n 1) ; N u ( k ):=[ N u 1 ( k ) ;:::;N u ( u 1) ( k ) ;N u ( u +1) ( k ) ;:::;N un ( k )] 1 ( n 1) ; ( k ):=[ 1 ( k ) T ;:::; n b ( k ) T ] Tn b ( n 1) 1 ; u ( k ):=[ u; 1 ( k ) ;:::; u;u 1 ( k ) ; u;u +1 ( k ) ;:::; u;n ( k )] T( n 1) 1 ; where blkdiag isanoperatortoformablockdiagonalmatrix.Notethatdiagonalm atrix M b ( k )+ D b ( k )isalwaysnon-singularasdiagonalentriesin D b ( k )arealwayspositive. Inaddition,readershouldbeawarethatwhen N uv ( k )=0, u;v ( k )isjustdenedasa 47

PAGE 48

randomvariablewiththesamestatisticalpropertyasthemeasure mentnoise.Moreover, recallthatif( u;v ) 2 ~ E ( k )(sothat u;v ( k )isavailableto u ),thenthemeasurement v;u ( k )= u;v ( k )isavailableto v .Therefore, u;v ( k )= v;u ( k ). Ingeneral,theweight W ( k )attime k isnotcompletelyspeciedbythegraph G ( k ) atthattimeifnodesareallowedtovarytheweightsarbitrarilyovert ime.However,recall thatanadditionalconstraintwasimposedonchoosingweights,tha tthereexistsatime k 0 afterwhichtheweightbetweentwonodesdonotchange.Theresu ltofimposingthis constraintisthatafterthetransientperiod,thegraph G ( k )uniquelydeterminesthe weightmatrix W ( k ).Sincethereare N distinctgraphsin G ,aset W := f W 1 ;:::;W N g is dened,with W i associatedwith G i .Asaresult,for k k 0 ,if G ( k )= G i then W ( k )= W i Therefore, D i ;M i ;N i ;D bi ;M bi ;N bi ;J bi areuniquelydenedby W i ,whichisassociatedwith G i Withthesechoicesstatedabove,thestateofthefollowingsystem isidenticaltothat of( 3{9 )forthesameinitialconditions: e ( k +1)= J b ( k ) e ( k )+ B b ( k ) ( k ) ;k k 0 (3{10) where : Z + !f 1 ;:::;N g istheswitchingprocessthatisgovernedbytheunderlying Markovchain G ( k ).Thereasonforthequalier k k 0 isthatweightsarenotlimitedto theset W before k 0 ,sotechnicallythematrices J b ( k ) and B b ( k ) areuniquelydetermined bytheMarkovchainonlyfor k k 0 .Theerrordynamics( 3{10 )isaMarkovjumplinear system(MJLS)[ 32 ].Toproceedwiththeanalysisofthemeansquareconvergenceof ( 3{10 ),weneedsometerminology. r :=E[ ( k )] ; :=E[ ( k ) T ( k )] ; (3{11) ( k ):=E[ e ( k )] ; Q ( k ):=E[ e ( k ) e T ( k )] : (3{12) NotethatE[ ( k )]existsbecausethedirectionofmeasurementforanypairofnod es u and v isxedsuchthatbiasofany u;v ( k )isinvariant.(seeSectionsec:prob-statement) 48

PAGE 49

Furthermore,westatetwoshortcutnotation:forasetofsqua rematrices X i 2 R ` 1 ` 2 Y ij 2 R ` 1 ` 2 i;j =1 ;:::;N ,denote diag[ X i ]:= 0BBBB@ X 1 ::: 0 ... ... 0 :::X N 1CCCCA ` 1 N ` 2 N ; [ X ij ]:= 266666664 Y 11 Y 12 :::Y 1 N Y 21 Y 22 :::Y 2 N ... ... ... Y N 1 Y N 2 :::Y NN 377777775 ` 1 N ` 2 N : Now,denethematrices J i :=( M i + D i ) 1 ( N i + D i ) 2 R n n ;F i := J i n J i 2 R n 2 n 2 ; J bi :=( M bi + D bi ) 1 ( N bi + D bi ) 2 R n b n b ;F bi := J bi n J bi 2 R n 2b n 2b (3{13) where n denotestheKroneckerproduct.Furthermore,denethematr ices D := P T n I diag [ F i ]=[ p ji F j ] 2 R Nn 2 Nn 2 ; (3{14) D b :=( P T n I ) diag [ F bi ]=[ p ji F bj ] 2 R Nn 2b Nn 2b ; (3{15) C b :=( P T n I ) diag [ J bi ]=[ p ji J bj ] 2 R Nn b Nn b ; where I isanidentitymatrixofappropriatedimension.Recallthat P isthetransition probabilitymatrixoftheMarkovchain. ThekeytoestablishTheorem 3.1 ,isthefollowingtechnicalresult. Lemma3.1. Whenthetemporalevolutionofthegraph G ( k ) isgovernedbyahomogeneousergodicMarkovchainwhosetransitionprobabilityma trix P hasthepropertythat itsdiagonalentriesarestrictlypositive,then ( D b ) < 1 ifandonlyiftheuniongraph ^ G denedin ( 3{7 ) isconnected,where D b isdenedin ( 3{15 ) and ( ) denotesthespectral radius.If ^ G isnotconnected, ( D b )=1 TheproofofLemma 3.1 requiresadditionaltechnicalresultsthatarepresentednext. Recallthatanon-negativematrixiscalledstochasticmatrixifeachr owsumis1.If X is astochasticmatrix,then ( X )=1[ 39 ]. 49

PAGE 50

Proposition3.1. 1.If X isstochasticmatrix, X n X isalsoastochasticmatrix. 2.Thematrices J i and F i i =1 ;:::;N ,denedin ( 3{13 ) arestochasticmatrices. 3.Let K := 0BBB@ F 1 F 2 F N F 1 F 2 F N ... ... ... F 1 F 2 F N 1CCCA Nn 2 Nn 2 ;K b := 0BBB@ F b 1 F b 2 F bN F b 1 F 2 F bN ... ... ... F b 1 F b 2 F bN 1CCCA Nn 2b Nn 2b ; Thereexistsapermutationmatrix X sothat K b isaprincipalsub-matrixof X T KX 4.Forthematrix D denedin ( 3{15 ) ( D ) 1 ProofofProposition 3.1 Thersttwostatementsarestraightforwardtoestablish.The thirdstatementfollowsfromthefactthat J bi isaprincipalsubmatrixof J i .Wetherefore proveonlythefourthstatement.From( 3{14 ), ( D )= ([ p ji F j ]). Since p ji F j isanon-negativesquarematrix,itfollowsfrom[ 40 ,Theorem3.2]that ([ p ji F j ]) ([ k p ji F j k 1 ]).Moreover, k p ji F j k 1 = p ji k F j k 1 .Andsince F j isastochastic matrix, k F j k 1 =1.Wethereforehave ( D ) ([ p ji k F j k 1 ])= ([ p ji ])= ( P T )= ( P )=1 : TheproofofthenextresultisprovidedintheAppendixsinceitrequir esintroduction ofconsiderablenewterminology.Lemma3.2. Let P bethetransitionprobabilitymatrixofan N -stateergodicMarkov chainwhosediagonalentriesarepositive.The D denedin ( 3{14 ) isirreducibleifand onlyiftheuniongraph ^ G denedin ( 3{7 ) isconnected. NowwecanproveLemma 3.1 50

PAGE 51

ProofofLemma 3.1 Sincetheuniongraph ^ G isconnected,itfollowsfromLemma 3.2 that D isirreducible.FromthethirdstatementofProposition 3.1 ,thereexistsapermutationmatrix X ,suchthat D b isaprincipalsubmatrixof X T D X .Thespectralradius ofanirreduciblematrixisstrictlygreaterthanthespectralradius ofanyofitsprincipal submatrices,whichfollowsfromTheorem5.1in[ 39 ].Thereforewehave ( D b ) < ( X T D X ) : FromthefourthstatementinProposition 3.1 andthefactthatpermutationdoesnot changeeigenvalues,itfollowsthat ( X T D X )= ( D ) 1 : Combiningthesetwoinequalitieswegetthatif ^ G isconnectedthen ( D b ) < 1.Toprove necessity,weconstructacounterexample,inparticular,atrivial Markovchainwitha singlestate: G = fG 1 g (sothat P =1)where G 1 isa n -nodegraphwithoutanyconnected edge.Then D b = F b 1 = J b 1 n J b 1 = I ,whichhasaspectralradiusofunity.Thiscompletes theproofofthelemma. Thefollowingdenitionsandterminologyfrom[ 32 ]willbeneededinthesequel.Let R ` 1 ` 2 bethespaceof ` 1 ` 2 realmatrices.Let H ` 1 ` 2 bethesetofallN-sequencesofreal ` 1 ` 2 matrices,sothat V 2 H ` 1 ` 2 means V =( V 1 ;V 2 ;:::;V N )where V i 2 R ` 1 ` 2 for i =1 ;:::;N .Theoperators and^ isdenedtocreateatallvectorbystackingtogether columnsfromthesematrices,asfollows:let( V i ) j 2 R ` 1 bethe j -thcolumnof V i 2 R ` 1 ` 2 then ( V i ):= 0BBBB@ ( V i ) 1 ... ( V i ) n 1CCCCA 2 R ` 1 ` 2 ^ ( V ):= 0BBBB@ ( V 1 ) ... ( V N ) 1CCCCA 2 R N` 1 ` 2 : (3{16) 51

PAGE 52

Similarly,theinversefunction^ 1 : R N` 1 ` 2 H ` 1 ` 2 isdenedsothatitproducesan elementof H ` 1 ` 2 givenavectorin R N` 1 ` 2 Lemma3.3. Considerthejumplinearsystem ( 3{10 ) withanunderlyinghomogeneous andergodicMarkovchain.Thestatevector e ( k ) ofthesystem ( 3{10 ) ismeansquare stableifandonlyif ( D b ) < 1 ,where D b isdenedin ( 3{15 ) .Whenmeansquarestability occurs,then ( k ) and Q ( k ) Q ,where := N X i =1 q i ; Q := N X i =1 Q i ; (3{17) where [ q T 1 ;:::;q T N ] T = q :=( I C b ) 1 2 R Nn b ; ( Q 1 ;:::;Q N )= Q :=^ 1 ( I D b ) 1 ^ ( R ( q )) 2 H n b n b ; and ;R ( q ) aregivenby :=[ T 1 ;:::; T N ] T 2 R Nn b ; j := N X i =1 p ij B bi r i 2 R n b ; R ( q ):=( R 1 ( q ) ;:::;R N ( q )) 2 H n b n b ; R j ( q ):= N X i =1 p ij ( B bi B T bi i + J bi q i r T B T bi + B bi rq T i J T bi )) 2 R n b n b : Moreover, Q ispositivesemi-denite. Proof. ItfollowsfromTheorem3 : 33andTheorem3 : 9of[ 32 ]aswellasremark3.5[ 32 pg.35],thatmeansquarestabilityof( 3{10 )isequivalentto ( D b ) < 1 2 .Theexpressions forthemeanandcorrelation,aswellasthefactthat Q 0,alsofollowfrom[ 32 Proposition3.37,3.38].Theexistenceofthesteadystatedistributio n followsfromthe ergodicityoftheMarkovchain,sotheexpressionsprovidedarewe lldened. 2 Fortheinterestedreader,thematrix D b isreferredtoas A 1 in[ 32 ]. 52

PAGE 53

1 2 3 4 1 2 3 4 1 2 3 4 Figure3-4.Thethreegraphs G 1 ; G 2 ; G 3 thatcomprises G .Node1isthereference. NowwearereadytoproveTheorem 3.1 ProofofTheorem 3.1 (Suciency) :ItfollowsfromLemma 3.1 thatwehave ( D b ) < 1. ItthenfollowsfromLemma 3.3 thatthestateismeansquarestable.Thestatement abouttheasymptoticmeanbeingunbiasedifthemeasurementnoise iszeromeanfollows immediatelyfromtheexpressionforthelimingmeaninLemma 3.3 bypluggingin r =0. (Necessity) :Iftheunionofgraphisnotconnected,fromtheproofofLemma 3.1 ( D b )=1.Thisshowsthat(duetoLemma 3.3 )convergencewillnotoccur. 3.5.2VericationofTheorem 3.1 Inthissection,weperformsimulationsonamade-upscenarioinorde rtoverifythe Theorem 3.1 .A4-nodenetworkisselected,wherethenodesmoveinsuchawayt hat thegraph G ( k )canbeoneofonly3graphsshowninFigure 3-4 .Inaddition,thegraphs changeaccordingtoaMarkovchainwhosetransitionprobabilitymat rixis P = 266664 0 : 300 : 7 0 : 10 : 50 : 4 00 : 50 : 5 377775 : (3{18) Noticethattheunionofthegraphsin G isconnected,thoughnoneofthegraphsis aconnectedgraph,and P isergodic.Nodevariablesarechosenarbitrarily,asingle referencenodeispresent,andthevalueoftheitsnodevariableis0 .Themeanand varianceofmeasurementnoiseoneveryedgearechosenas0and1 0 4 ,respectively.All theedgeweightsareassignedavalueofunityateverytime.MonteCarloexperiments 53

PAGE 54

0 50 100 150 200 0.95 1 1.05 Iteration indexMean Empirical True value (a)Mean 0 50 100 150 200 0 2 4 x 10 -4 Iteration indexVariance Empirical Steady state (b)Variance Figure3-5.Thetimeseriesofthemeanandvarianceoftheestimate ofnode3'snode variable.Theempiricalestimateofmeanandvarianceiscomputedfr om1000 MonteCarloexperiments.In( b ),the\steady-state"correspondstothe limitingstandarddeviationpredictedbyLemma 3.3 areconductedtoempiricallyestimatethemeanandvarianceofthee stimationerror,by averagingover1000sampleruns.Figure 3-5(a) andFigure 3-5(b) showtheempirically estimatedmeanandvarianceofnode3'sestimateofitsnodevariable .Aspredictedby Theorem 4.1 ,themeanoftheestimateconvergestothetruevalue,sincethem easurement noiseis0mean.Thevariancealsoconvergestothetheoreticalste adystatevarianceas predictedbyLemma 3.3 3.6NumericalEvaluationofTimeSynchronizationAccuracy withJAT Inthissection,weevaluatetheperformanceoftheJATalgorithmf orclocksynchronization.Wenotonlyevaluatetheaccuracyoftheestimationo fskewandoset usingthealgorithm,butalsoevaluatethatoftheglobaltimepredict ion.Simulations areperformedfora50-nodemobilenetworkina100 m 100 m squareeld.Thetrue skewsandosetsof49nodesarepickedrandomlyfrom[1 2 10 5 ; 1+2 10 5 ]and [ 10 1 sec ; 10 1 sec]respectivelyaccordingto[ 19 ].Thesinglereferencenodehasskew 54

PAGE 55

1andoset0.Nodes'motionsaregeneratedaccordingtotheRan domDirection(RD) mobilitymodel[ 36 ].Nodes'speed v min v max arechosenas0 : 5 m=s and1 m=s .Apair ofnodescancommunicatewhendistancebetweenthemislessthan1 5 m .Forthesakeof convenience,simulationsarecarriedoutinasynchronousfashion. Nodesperformoneupdateofskewandosetestimationineach k thinterval(socalledsynchronizationperiod), where k =1 ; 2 ;::: .Eachintervalstartsatglobalinstant t k .Thesynchronizationperiod ischosenas1sec,therefore t k +1 t k =1.Thenetworktopologydoesn'tchangewithin eachsynchronizationperiod.Atthebeginningof k -thinterval,pairsofneighboringnodes usethetwo-stepapproachinChapter 2 tocomputerelativemeasurementsofskewsand osets,inwhichthepairwisesynchronizationalgorithmin[ 1 ]isstilladopted.CommunicationdelayisGaussiandistributedwithmean150 sec andstandarddeviation5 sec [ 19 ] asitwasusedinSection 2.3 .Attheendofeach k thinterval,nodesupdateitsownskew andosetestimationusingneighbors'estimatesalongwithrelativem easurements.Allthe edgeweightsareassignedavalueofunityateveryinterval.Allnode sinitializetheirskew andosetestimatesas1and0respectively. Figure 3-6 showstwosnapshotsofthe50-nodenetworkduringoneofthesim ulations, inwhichonlyafewedgesexistinnetworks.Thefollowingplotsshowsimu lationresults, wherethex-axisisdiscreteintime,i.e., t k .Figure 3-7 showsthetimetraceofthe skewestimatesoftwonodesinoneofthesimulations.Themeanandv arianceofthe skewestimationerrorwasempiricallycomputedfrom100MonteCarlo simulations. Figure 3-8(a) and 3-8(b) showmeanandvarianceoftheskewestimationerrorfortwo nodes.Thetrueskewoftherstnodeisabout1+8 10 6 .Ifnosynchronization algorithmisimplemented,therstnodewouldguessitsskewtobe1,w hichleadsto theerror8 10 6 .Byusingtheproposedalgorithm,themeanoftheerrorisabout within1 10 6 ,whichisbetterthantheinnocentguess.Figure 3-9 3-8(a) and 3-8(b) areoneexperimenthistory,meanandvarianceofestimationerror correspondingto oset.Figure 3-11 showstheerrorofglobaltimeestimationalongtime,i.e., ^ t u ( t k ) t k 55

PAGE 56

oftwonodesinoneexperiment.After70s,nodecanestimatetheg lobaltimefairlywell. However,thevarianceofthepredictionerrorseemstoincreasea longtime,whichcanbe analyzedfromthefollowingderivation.Withoutlossofthegenerality ,weassumethe osetisexactlyestimated,i.e., ^ u = u .Then, E [ ^ t u ( t k ) t k ]= E [( u ^ u ( t k ) 1) t k ] 1 u E [ u ^ u ( t k )] t k (3{19) Var [ ^ t u ( t k ) t k ]= Var [( u ^ u ( t k ) 1) t k ] 1 2 u Var [ u ^ u ( t k )] t 2k (3{20) Therefore,asJATalgorithmdrives Var [^ u ( t k ) u ]toconstant, Var [ ^ t u ( t k ) t k ]increases along t k Figure3-6.Twographsthatoccurduringonesimulationwith50node smovingaccording totherandomdirectionmobilitymodel. 0 50 100 150 200 250 300 1 1 1 1 Time (sec)Skew node 1 node 30 true Figure3-7.Thetimeseriesoftheskewestimates^ u ( t k )inoneexperimentin50-node mobilenetworks. 56

PAGE 57

0 50 100 150 200 250 300 -3 -2 -1 0 1 x 10 -6 Time (sec)Mean node 1 node 30 (a)Mean 0 50 100 150 200 250 300 2 4 6 8 10 x 10 -11 Time (sec)Variance node 1 node 30 (b)Variance Figure3-8.Thetimeseriesofthemeanandvarianceoftheskewest imationerror,i.e., E [^ u ( t k ) u ]and Var [^ u ( t k ) u ],inthe50-nodemobilenetwork. 0 50 100 150 200 250 300 -0.1 -0.08 -0.06 -0.04 -0.02 Time (sec)Offset node 1 node 30 true Figure3-9.Thetimeseriesoftheosetestimates ^ u ( t k )inoneexperimentin50-node mobilenetworks. 57

PAGE 58

0 50 100 150 200 250 300 0.005 0.01 0.015 0.02 0.025 Time (sec)Mean node 1 node 30 (a)Mean 0 50 100 150 200 250 300 5 10 15 x 10 -4 Time (sec)Variance node 1 node 30 (b)Variance Figure3-10.Thetimeseriesofthemeanandvarianceoftheosete stimationerror,i.e., E [ ^ u ( t k ) u ]and Var [ ^ u ( t k ) u ],inthe50-nodemobilenetwork. 0 50 100 150 200 250 300 -0.05 0 0.05 Time (sec)Time est. error (sec) node 1 node 30 Figure3-11.Theerrorofglobaltimeestimation,i.e., ^ t u ( t k ) t k ,oftwonodes 3.7ConclusionandOpenProblems Weanalyzedadistributedalgorithmforestimationofclockskewando setofthe nodesofamobilenetworkandexamineditsconvergenceproperties .Thealgorithmallows nodestoputdierentweightsonestimatesreceivedfromdistinctn eighbors,depending ontheaccuracyofthecorrespondingrelativemeasurements.Th etimevariationofthe networkwasmodeledasaMarkovchain,whichmakesthealgorithmaj umplinearsystem. 58

PAGE 59

UndertheassumptionsthattheMarkovchainisergodicandthediag onalentriesofits transitionprobabilitymatrixarepositive,theestimateswereshown tobemeansquare stableaslongastheunionofthegraphsovertimeisconnected.Usin gtheJATalgorithm, weshowsinsimulationthatnodescanestimatetheglobaltimequicklyw ithincertain accuracy. Expressionsfortheasymptoticmeanandcorrelationarealsoprov idedbyusingresultsfromjumplinearsystemsfrom[ 32 ].Evaluatingtheseexpressionsrequiressummation of N terms,where N isthenumberofdistinctgraphsthatcanoccur.Ingeneral N isa verylargenumber,sotheutilityoftheseexpressionsislimitedintheg eneralsetting.For instance,ifnorestrictionisplacedonthemotionofthenodesoredg eformation, N isthe numberofdistinctgraphspossiblewith n nodes,whichis2 1 2 n ( n 1) .Clearly,thisisavery largenumberunless n isextremelysmall.However,inspecialsituations N canbesmaller, e.g.,ifcertainnodesarerestrictedtomoveonlywithincertaingeogr aphicareas. Intime-varyingsystems,therateofchangeisanimportantparam eter.TheassumptionthatMarkovchainsatises p ii > 0providesanupperboundonhowfastnodes canmoveandthenetworkcanchange(comparedtothetimerequir edtoobtainrelative measurementsandcurrentestimates).Thisassumptionwasused toproveTheorem 4.1 However,itispossiblethatmeansquarestabilitycanbeprovedwithw eakerconstraints onthespeedoftopologychange. Wehavenotexaminedthequestionofconvergencerate.Itislikelyt hatthetransition probabilitiesofthechainwillplayaroleintheconvergencerate.Howe ver,precisely characterizingoftheconvergencerateofthealgorithmremainsa nopenproblem.The timetoreachacceptableestimationaccuracycanhoweverberedu cedbymorecareful choiceoftheinitialcondition,e.g.,usingtheraggedinitializationschem eproposedin[ 20 ]. Theaccuracyofglobaltimeestimationmostlydependsontheaccur acyofskew estimation.Theproposedalgorithmareabletodrivethevarianceof skewestimationerror toconstant.Althoughthevarianceissmall,itstillleadstotheincrea seofthevarianceof 59

PAGE 60

globaltimeestimationinalongterm.Amodicationofthealgorithm,wh ichcanachieve thevarianceoftheerrorgoestozerowouldbevaluable. 3.8ProofofLemma 3.2 Allmatricesarenon-negativehereafter;sowewillexplicitlysay\no n-negative"only whenwehavetostressit.Formatrices X 1 ;X 2 ofsamedimension,wesay X 1 and X 2 are congruent ,andwrite X 1 = X 2 ,ifthefollowingholds:( X 1 ) {| 6 =0ifandonlyif( X 2 ) {| 6 =0. Wealsowrite X 1 X 2 ifthefollowingconditionissatised:( X 1 ) {| 6 =0if( X 2 ) {| 6 =0. Thedirectedgraph ~ G ( X )=( V ( X ) ; ~ E ( X ))correspondingtoasquarematrix X 2 R n n isagraphdenedon n nodesinwhich( u;v ) 2 ~ E ( X )ifandonlyif X u;v 6 =0.Adirected graph ~ G iscalled stronglyconnected ifforeachpairofnodes u and v ,thereisasequence ofdirectededgesin ~ E leadingfrom u to v [ 41 ].If ~ G 1 isasubgraphof ~ G 2 ,meaningthat ~ G 2 containsallthenodesandedgesof ~ G 1 ,wewrite ~ G 1 ~ G 2 or ~ G 2 ~ G 1 .Twodirected graphs ~ G 1 and ~ G 2 arecalled congruent iftheiradjacencymatricesarecongruent.We denoteby Adj ( ~ G )theadjacencymatrixofthegraph ~ G .Fora n n squarematrix X wewrite Adj ( X )todenote Adj ( ~ G ( X )),whichisan n n matrixwith {;| -thentryequal to1ifandonlyif X {| > 0,and0otherwise.Essentially,thematrix Adj ( X )replacesthe positiveentriesof X by1andleavesthe0entriesuntouched. Thefollowingstatementsfornon-negativematricescanbeveried inastraightforwardmanner.Allthematricesareofthesamedimension.Proposition3.2. 1. X = Adj ( X ) 2. ~ G ( X 1 ) = ~ G ( X 2 ) ifandonlyif X 1 = X 2 3. ~ G ( X 1 ) ~ G ( X 2 ) if X 1 X 2 4. ~ G ( P `i X i ) = [ `i =1 ~ G ( X i ) Proposition3.3. Thegraph ~ G ( X ) isstronglyconnectedifandonlyif X isirreducible.If ~ G ( X ) isstronglyconnected,then ~ G ( X n X ) isalsostronglyconnectedandthus X n X is irreducible. 60

PAGE 61

Therststatementofthepropositioniswell-known[ 41 ,pp.671].Thesecondstatementfollowsfromtherstinastraightforwardmanner. Nowwedenethe Cartesianproduct oftwodirectedgraphs ~ G 1 =( V 1 ; ~ E 1 )and ~ G 2 =( V 2 ; ~ E 2 ),whichisdenotedby ~ G 1 2 ~ G 2 .TheCartesianproducthasthevertexset equalto V 1 V 2 ,sothatnodesintheproductaredenotedbythepair( u;v ),with u 2 V 1 and v 2 V 2 ,whichisnottobeconfusedwithanedge. Inordertopreventconfusion,we willdenoteanedgefrom u to v inthesequelby u v .TheedgesetoftheCartesian productischaracterizedbythefollowingproperty:thereisanedg e( u 1 ;v 1 ) ( u 2 ;v 2 )in ~ G 1 2 ~ G 2 ifeither u 1 = u 2 and v 1 v 2 2 ~ E 2 or v 1 = v 2 and u 1 u 2 2 ~ E 1 .Cartesian productsofundirectedgraphs G 1 and G 2 aresimilarlydened,exceptthattheresulting productgraphisalsoundirected.Thefollowingpropertieswillbeuse fulinfuture. Proposition3.4. 1.If ~ G 1 and ~ G 2 arestronglyconnected,sois ~ G 1 2 ~ G 2 2.If Adj ( X 1 ) and Adj ( X 2 ) aresymmetric,then, Adj ( ~ G ( X 1 ) 2 ~ G ( X 2 ))= Adj ( X 1 ) n I + I n Adj ( X 2 ) : (3{21) ProofofProposition 3.4 Therststatementisfrom[ 42 ,Table2].Toprovethe secondstatement,weintroducethenotation G ( X ),whichistheundirectedgraph correspondingtoasymmetricmatrix X .Itfollowsthat Adj ( ~ G ( X 1 ) 2 ~ G ( X 2 ))= Adj ( G ( Adj ( X 1 )) 2 G ( Adj ( X 2 ))).From[ 43 ,Section2.3], Adj ( G ( Adj ( X 1 )) 2 G ( Adj ( X 2 )))= Adj ( X 1 ) n I + I n Adj ( X 2 ).Thus,weprove( 3{21 ). ThefollowingresultswillbeusefulintheproofofLemma 3.2 Proposition3.5. Iftheuniongraph ^ G = [ Ni =1 G i isconnected,then [ Ni =1 ~ G ( F i ) isstrongly connected.ProofofProposition 3.5 Recallthat F i = J i n J i ,where J i =( M i + D i ) 1 ( N i + D i ).Since M i + D i ,andthereforeitsinverse,isadiagonalmatrixwithpositiveentries J i = ( N i + D i ). 61

PAGE 62

Byproperty2ofProposition 3.2 [ Ni =1 ~ G ( F i ) = ~ G ( N X i =1 F i ) = ~ G N X i =1 f ( N i + D i ) n ( N i + D i ) g (3{22) Wealsohave ( N i + D i ) n ( N i + D i ) D i n N i + N i n D i (3{23) bydroppingtwotermsintheexpansionusingtheirnon-negativity.S ince D i = I ,weget ( N i + D i ) n ( N i + D i ) I n N i + N i n I ) N X i =1 (( N i + D i ) n ( N i + D i )) I n N X i =1 N i + N X i =1 N i n I: Usingproperty3inProposition 3.2 ,wehave [ Ni =1 ~ G ( F i ) ~ G ( N X i =1 N i ) n I + I n ( N X i =1 N i ) = ~ G ( N X i =1 N i ) 2 ~ G ( N X i =1 N i ) : (3{24) wherethecongruencefollowsfromthesecondstatementinPropo sition 3.4 .Recall thestructureof N i ,itfollowsthat(i) Adj ( G i )= Adj ( N i ),and(ii) Adj ( N i )issymmetric.Asaresult, Adj ( P Ni =1 N i )isalsosymmetric.Since ^ G = [ Ni =1 G i isaconnected undirectedgraph,itsadjacencymatrixisirreducible.Thismeans Adj ( [ Ni =1 G i )= P Ni =1 Adj ( G i )= P Ni =1 Adj ( N i )isirreducible.DuetotherststatementinProposition 3.4 ~ G ( P Ni =1 N i ) 2 ~ G ( P Ni =1 N i )isstronglyconnected.Theresultofthisproposition nowfollowsfrom( 3{24 ). The Kroneckerproduct oftwographs ~ G 1 =( V 1 ; ~ E 1 )and ~ G 2 =( V 2 ; ~ E 2 ),denoted by ~ G 1 n ~ G 2 ,hasthevertexsetequalto ~ V 1 ~ V 2 andanedgesetthatischaracterized bythefollowingproperty:thereisanedge( u 1 ;v 1 ) ( u 2 ;v 2 )in ~ G 1 n ~ G 2 ifandonlyif u 1 u 2 2 ~ E 1 and v 1 v 2 2 ~ E 2 [ 42 ].NotethattheCartesianandKroneckerproducts ~ G 1 2 ~ G 2 and ~ G 1 n ~ G 2 havethesamevertexsetsbutdistinctedgesets.Wehavethe 62

PAGE 63

followingpropertyofKroneckerproductofgraphsfrom[ 44 ]: Adj ( ~ G 1 n ~ G 2 )= Adj ( ~ G 1 ) n Adj ( ~ G 2 ) : (3{25) AdjacencymatricesofbothCartesianandKroneckerproductso ftwographsarerelated totheadjacencymatricesoftheindividualgraphsthroughthema trixKroneckerproduct, cf.( 3{21 )and( 3{25 ). NowwearereadytoprovetheLemma 3.2 ProofofLemma 3.2 (Connectivity ) irreducibility):Herewehavetoprovethatifthe uniongraph ^ G isconnectedthenthematrix D isirreducible.Wewillproveitbyshowing thatthedirectedgraph ~ G ( D )isstronglyconnected.Let Z j and S j bethediagonaland o-diagonalpartsof F j .Since Z i isanon-negativematrixwithpositivediagonal,weget D = [ p ji Z j ]+[ p ji S j ] = [ p ji I n 2 ]+[ p ji S j ] P T n I + diag [ p ii S i ](3{26) wherewehaveusedthefactthat[ p ji I n 2 ]= P T n I anddroppedtheo-diagonalblocksof [ p ji S j ].Therefore ~ G ( D ) ~ G ( P T n I ) [ ~ G ( diag [ S i ]) = f ~ G ( P T ) n ~ G ( I ) g [ ~ G ( diag [ S i ]) wheretheequalityfollowsfromtheproperty( 3{25 )ofKroneckerproductofgraphs.We willnowshowthatthedirectedgraph f ~ G ( P T ) n ~ G ( I ) g S ~ G ( diag [ S i ])isstronglyconnected, whichprovesthat ~ G ( D )isaswell. Firstnoticethatthereare Nn 2 nodesinthegraph ~ G ( D ),soare P T n I and diag [ S i ]. Itisconvenienttoimaginethemas N N blockmatrices,witheachblockbeingof dimension n 2 n 2 .Therefore,weintroduceausefulnewnotation.Let'sindexanod eby thepair( p i ;d ),whichisthe(( i 1) N + )-thnodein ~ G ( D ),where i =1 ;:::;N and =1 ;:::;n 2 .Thisnotationissimilarlysuitablefor P T n I and diag [ S i ].Toprovethat 63

PAGE 64

f ~ G ( P T ) n ~ G ( I ) g S ~ G ( diag [ S i ])isstronglyconnected,weneedtoshowthefollowing Thereisapathfromanarbitrarynode ( p i ;d ) toanotherarbitrarynode ( p j ;d ) inthegraph f ~ G ( P T ) n ~ G ( I ) g [ ~ G ( diag [ S i ]) : (3{27) Thefollowingpropertieswillbeusedtoconstructaproofof( 3{27 ): 1. s1: Thereexistsapathfrom( p i ;d )to( p j ;d )in ~ G ( P T n I )forall i;j =1 ;:::;N and =1 ;:::;n 2 2. s2: If d d h isanedgein ~ G ( S ` ),then( p ` ;d ) ( p ` ;d h )isanedgein ~ G ( diag [ S i ]). Therststatementisprovedasfollows.SincetheMarkovchainiser godic, P -and therefore P T -isirreducible,whichmeans ~ G ( P T )isstronglyconnected.Thus,given arbitrarynodes p and q in ~ G ( P T ),thereisapathconnectingthemin ~ G ( P T ).Call thispath p;u 1 ;u 2 ;:::;u m ;q .Sincetheedge d d 2 ~ G ( I )existsforevery itnowfollowsfromthedenitionofKroneckerproductofgraphsth atthepath ( p;d ) ; ( u 1 ;d ) ; ( u 2 ;d ) ;:::; ( u m ;d ) ; ( q;d )existsin ~ G ( P T ) n ~ G ( I )forevery =1 ;:::;n 2 Thestatement s1 isnowproveduponreplacing p and q by p i and p j .Thestatement s2 istruebecauseofthestructureofthematrix diag [ S i ]andthenodeindexingscheme describedimmediatelybefore( 3{27 ). FromProposition 3.5 ,wehavethat [ Ni =1 ~ G ( F i )isconnected.Since S i istheodiagonalpartof F i [ Ni =1 ~ G ( S i )isconnectedaswell.Therefore,thereisapathfroman arbitrarynode d toanotherarbitrarynode d in [ Ni =1 ~ G ( S i ),forall ; in f 1 ;:::;n 2 g Toprovethestatement( 3{27 ),picksuchapathfromthenode d tothenode d in [ Ni =1 ~ G ( S i ),whereeachedgeinthepathmaylieinanyofthegraphs f ~ G ( S i ) g Ni =1 .Forthe sakeofconcretenessandcompactness,letusconsiderapathof lengthtwo,consistingof thetwoedges d d h and d h d ,whichbelongtothegraphs,say, ~ G ( S ` )and ~ G ( S m ), respectively.From s1 wehaveprovedabove,weknowthatthereisapathfromthenode ( p i ;d )tothenode( p ` ;d )inthegraph ~ G ( P T n I ),callthispath path [( p i ;d ) ; ( p ` ;d )]. From s2 ,wehavethattheedge( p ` ;d ) ( p ` ;d h )existsinthegraph ~ G ( diag [ S i ])due 64

PAGE 65

totheexistenceoftheedge d d h in ~ G ( S ` ).Thus,wehavethepathfrom( p i ;d ) to( p ` ;d h )inthecombinedgraph f ~ G ( P T ) n ~ G ( I ) g S ~ G ( diag [ S i ])byjoiningthepath path [( p i ;d ) ; ( p ` ;d )]withtheedge( p ` ;d ) ( p ` ;d h ).Usingthisidearepeatedly,we constructapathfrom( p i ;d )to( p j ;d )in f ~ G ( P T ) n ~ G ( I ) g S ~ G ( diag [ S i ])asfollows: path [( p i ;d ) ; ( p ` ;d )] ; in ~ G ( P T ) n ~ G ( I ) ( p ` ;d ) ( p ` ;d h ) ; 2 ~ G ( diag [ S i ]) path [( p ` ;d h ) ; ( p m ;d h )] ; in ~ G ( P T ) n ~ G ( I ) ( p m ;d h ) ( p m ;d ) ; 2 ~ G ( diag [ S i ]) path [( p m ;d ) ; ( p j ;d )] ; in ~ G ( P T ) n ~ G ( I ) ; whereeach path [ ]existsduetotheproperty s1 establishedabove,andeachedgeexists duetotheproperty s2 aswellaswiththeassumedexistenceoftheedges d d h and d h d intheuniongraph.Thisargumentcanbeextendedtoapathofanyle ngth between d and d intheuniongraph [ Ni =1 ~ G ( S i ).Thus,thereisapathfrom( p i ;d )to ( p j ;d )in f ~ G ( P T ) n ~ G ( I ) g S ~ G ( diag [ S i ]),whichprovessuciency. (Notconnected ) reducible):Asimplecounterexampleprovesnecessity.Construct a trivialMarkovchainwithasinglestate: G = fG 1 g (sothat P =1)where G 1 isan n -node graphwithoutasingleedge.Then D = F 1 = J 1 n J 1 = I ,whichisreducible. 65

PAGE 66

CHAPTER4 THESTOALGORITHMINTWOTYPESOFSWITCHINGNETWORKS Inthelastchapter,weproposedadistributedalgorithm(JAT)for theestimationof nodevariablesfromnoisyrelativemeasurementsinmobilenetworks. Thetime-variationin thetopologyofthenetworkwasmodeledbythestatesofaMarkov chain.Itwasshown thatthemeanoftheestimationerrorconvergestozero(ifmeasu rementnoiseiszero mean)andthevariancetoapositiveconstantvalue,underthecon ditionthattheunionof allgraphsisconnected. Inthischapter,weproposeanewalgorithmfornodevariableestima tionproblem. Theproposedalgorithm(calledSTOalgorithm)achieveshigheraccur acycomparedto theJATalgorithm.Weprovethattheestimationerrorconvergesin meansquaretoa limitingvalue,i.e.,themeanconvergestothelimitingvalueandthevarian ceconvergesto zeroundertwodierentscenarios:deterministictopologyswitcha ndMarkoviantopology switch.Theproofoftheformeronerequiresthattheunionofthe graphtopologiesis connectedwithinaboundedlengthtimeintervalandatime-average dLaplacianmatrix exists;theproofofthelatteronerequiresthatthegraphtopolo giesaregovernedbya homogeneousergodicMarkovchain.Inaddition,thelimitingvaluecan becalculatedby aformulaiffurtherinformationontopologyswitchingisprovided.Mo reover,ifrelative measurementsareunbiased,weprovethatboththemeanandvar ianceofestimationerror convergetozero.Thealgorithmisinspiredbytheconsensusalgorit hmsproposedin[ 11 12 ],whereastochasticapproximationapproachisadoptedtodealwit htheaccumulation ofmeasurementnoise[ 38 ].Theproposedalgorithmcanbeimplementedasynchronously. WeevaluatetheaccuracyoftheSTOalgorithmontheglobaltimeest imationthrough MonteCarlosimulations.Simulationsindicatetheerrorintheglobaltim eestimate stayscloseto0forlongtimeintervals.WealsocompareitwithJATalgo rithm,aswell asATSalgorithmproposedin[ 13 ].SinceATSdoesnotestimateglobaltimedirectly, wecomparethethreealgorithmsintermsofthemaximumsynchroniz ationerror-the 66

PAGE 67

maximumdeviationinthetimeestimatesoftwoarbitrarynodes.Ittu rnsoutthatthe proposedSTOalgorithmoutperformsJATandATSalgorithminthisme tric.Finally,we introducetwomethodstoimprovetheconvergencerate,andthe improvementisveried bysimulation. Therestofthechapterisorganizedasfollows.Section 4.1 proposesthemaindistributedalgorithm.Section 4.2 statesthemainresultandprovidestheproofofthe theoremundercertainconditions.SimulationstudyispresentedinS ection 4.3 .We comparetheSTOwithJATandATSalgorithminSection 4.4 .Section 4.5 showsthe improvementofconvergencerateforSTOalgorithm.Section 4.6 providestheconclusion andopenproblem. 4.1ProposedSTOAlgorithm Werstpresentaniterativealgorithmthatnodescanusetosolvet heproblemdescribedinSection 3.1 :toestimatescalarnodevariablesfromnoisyrelativemeasurement s inadistributedmanner. Sincenodesdonothavesynchronizedclocks,iterativeupdatesha vetobeperformed asynchronously.Eachnode u 2 V b keepsitslocal iterationindex k u andmaintainsan estimate^ x u ( k u ) 2 R ofitsnodevariable x u initslocalmemory.Theestimatescanbe initializedtoarbitraryvalues.Inexecutingthealgorithm,node u startsits i -thiteration atapre-speciedlocaltime ( i ) ,for i =0 ; 1 ;::: .Then,node u communicateswithits currentneighbors v 2N u ( k u )toobtaintheircurrentestimates^ x v ( k u )alongwiththe measurements u;v ( k u ).Afteraxedlengthoftime t (measuredinlocaltime),node u updatesitsnewestimatebasedoncurrentmeasurementsandneig hbors'estimatesby usingthefollowingupdatelaw: ^ x u ( k u +1)=^ x u ( k u )+ m ( k u ) X v 2N u ( k u ) a uv ( k u )(^ x v ( k u )+ u;v ( k u ) ^ x u ( k u ));(4{1) wherethetimevaryinggains m ( ): Z + R + havetobespeciedtoallnodesa-priori. Notethatif N u ( k u )= ; ,^ x u ( k u +1)=^ x u ( k u ).Thechoiceof m ( k u )willplayacrucialrole 67

PAGE 68

inconvergenceofthealgorithmandwillbedescribedinSection 4.2 .Theweight a uv ( k u )is arbitrarypositivenumber.Thereferencenodestakepartbyhelp ingtheirneighborsobtain relativemeasurements,buttheydonotupdatetheestimatesoft heirnodevariables. Instead,theyarekeptat0foralltime.Aftertheupdate,node u incrementsitslocal iterationindex k u by1.ThealgorithmissummarizedinAlgorithm 2 .Notethat,since obtainingrelativemeasurementsrequiresexchangingtime-stampe dmessages,current estimatescanbeeasilyexchangedduringtheprocessofobtainingn ewmeasurements. Algorithm2 STOalgorithmatnode u 1: while u isperformingtimesynchronization do 2: if Localtime u = ( i ) i =0 ; 1 ;::: then 3: u collectscurrentlocalindices k v fromneighbors v 2N u ( k u ). 4: forall v 2N u ( k u ) do 5: if k u = k v and u doesnothave u;v ( k u ) then 6: 1. u and v performpairwisecommunication; 7: 2. u saves u;v ( k u )and^ x v ( k u ); v saves v;u ( k v )and^ x u ( k v ); 8: else 9: u and v stopthecommunication; 10: endif 11: endfor 12: endif 13: if u = ( i ) + t i =0 ; 1 ;::: then 14: if N u ( k u ) 6 = ; then 15: u updates^ x u ( k u +1)using( 4{1 ); 16: else 17: ^ x u ( k u +1)=^ x u ( k u ); 18: endif 19: u updates, k u = k u +1; 20: endif 21: endwhile Wewilllaterdescribethatthegain m ( )ischosentobeadecreasingfunctionoftime, whichhelpsreducetheeectofmeasurementnoise.Thisisawell-kno wnideainstochastic approximation.However,usingthisideainanetworkofunsynchron izedclockspresentsan uniquechallengesincenonodehasanotionofacommontimeindex,atle astintheinitial phasewhentheydonothavegoodestimates.Ifnodeswaitsforac onstantlengthoftime 68

PAGE 69

(measuredintheirlocalclocks)beforestartinganewiteration,an odewithfasterskew mightnishthe( i +1)-thiterationwhileanodewithslowerskewhasn'tevennished the i -thiteration.Therefore,specifyingafunction m ( )toallthenodesdoesnotensure thatnodesusethesamegainatthesame(global)interval,whichisre quiredbystochastic approximation. Weaddressthisproblembyprovidingthenodesapriorithesequenc eoflocaltime instants ( i ) i =0 ; 1 ::: mentionedearlier.Thissequenceiscalledan iterationschedule andtheformulaforcomputingitisdescribedbelow.Lettheskewsan dosetsofall clocksbelowerandupperboundedbythoseintwoctitiousclocks c L and c H ,suchthat c L u c H c L u c H .Therefore c L ( t ) u ( t ) c H ( t )forall u 2 V .The formulaforcalculating ( i ) is ( i +1) = c H c L ( ( i ) + t c L )+ c H ; (4{2) where (0) hastobechosensuchthat (0) > c H .Thisscheduleensuresthatnodes operatingontheirunsynchronizedlocalclocksstillperformupdat esinaneectively synchronousmanner.Toseethis,dene I ( i ) :=( ( i ) c H c H ; ( i +1) c H c H )asaglobalinterval and I ( i ) u :=( ( i ) u u ; ( i ) + t u u )astheglobaltimeintervalwithrespectto i -thlocal iterationofnode u .Eq.( 4{2 )guaranteesthat,ateach i I ( i ) u I ( i ) forall u 2 V .In otherwords,thereexistsasequenceofglobaltimeintervalssuch thateach i -thglobal intervalcontains,andonlycontains,the i -thlocaliteration(inglobaltime)ofall u 2 V .Figure 4-1(a) showstherelationshipbetweenintervalsoflocaliterationsandthe correspondingglobalintervals.InFigure 4-1(b) ,wepickthe3rdglobalintervalfrom Figure 4-1(a) ,andshowtheglobaltimeintervalswhenlocaliterationupdatesocc ur.We emphasizethat ( i ) isthesameforallnodesandeverynode u startsandendsits i -th iterationatthesamelocaltimeinstants ( i ) and ( i ) + t 69

PAGE 70

0.85 1.89 3.15 4.70 1.23 2.37 3.77 Global timeLocal time Localinterval Globalinterval t max (t)=1.1t+0.3 t min (t)=0.9t-0.3 PSfrag (a) I (3) c H I (3) u I (3) v I (3) c L I (3) (b) Figure4-1.Therelationshipbetweenlocalandglobalinterval.In(a ),theX-axisislabeled bythetimeinstantsofthebeginningofglobalintervalsandtheY-ax isis labeledbythesequenceof ( i ) .Theredsolidslantedlinesrepresenttwo ctitiousclocks c L and c H astheboundsforlocalclocksinnodes.Theblack solidverticallinesdivideglobaltimeintoasequenceof I ( i ) .Each i -thinterval fromblacksolidverticaltoblackdotted-dashverticallineistheinte rval ( ( i ) c H c H ; ( i ) c L c L ),whichcontainstheglobaltimeinstantsof ( i ) forall u 2 V In(b),the3rdglobalintervalispickedasalsocircledin(a).Theseg mentsin thesecondandfthrowscorrespondto I (3) c H and I (3) c L oftwoctitiousclocks respectively.Thesegmentsinthethirdandfourthrowspresent I (3) u and I (3) v ofanytwonodes u;v 2 V accordingly. Althoughwedon'tknowrealskewsandosets,wecanstillpickfairly safevalues for c H c L c L and c H asfollows.Inwirelesssensornodes,apairofclocksinsensornodes usuallydriftapartupto40 sec=sec [ 19 ].Therefore,wecanpick c H = c L 1+4 10 5 Suchachoiceensuresthatthetimeintervalbetweentwosuccess iveiterations, ( i +1) ( i ) willonlyincreasefrom1secondto60secondafterabout10 5 intervals,whichtakesmore than400hours.Tohelpwithpickingreasonablevaluesoftheosetb ounds,thefollowing procedureshouldbeusedtoinitializetheprocedureofthesynchro nization.Thereference noderstbroadcastsamessage(toindicatethebeginningofsync hronization)andsetsits localclocktime t tozerosimultaneously.Anodethatreceivesthismessagesetsitso wn clocktozeroandbroadcastsuchmessageagain.Thenodesthath earthismessagealso settheirlocalclocksto0,andsoforth.Asnodes(excepttheref erencenode)starttheir localclocksafter{butcloseto{theinstantof t =0,theirosetsarenegativeandsmall. 70

PAGE 71

Therefore, c H canbechosenaszeroand c L canbepickedasanestimateofthetimeit takesforallactivenodestoreceivethe\synchronizationstart" signal.Foranodewho wasoutofcommunicationrangeatthebeginningbutjoinsthenetwo rkslater,itcanset thelocaltimetothecurrentlocaltimeofaneighborthathasalread ystartedthetime synchronizationprocess,andrecordneighbor'siterationindexas well.Inthisway,the newlyjoinednodecantakepartinthesynchronizationprocessasif itstartedatthevery beginning. 4.2ConvergenceAnalysis Asdescribedintheprevioussection,duetotheuseoftheiteration schedule,there existsacommoniterationcounter k thatcanbeusedtodescribethelocalupdatesinthe nodeseventhoughnoneofthenodeshasaccesstoit.Inthissect ionwethereforeconsider onlythesynchronousviewofthealgorithmusingglobalindex k .Werewrite( 4{1 )as ^ x u ( k +1)=^ x u ( k )+ m ( k ) X v 2N u ( k ) a uv ( k )(^ x v ( k )+ u;v ( k ) ^ x u ( k )) : (4{3) Nowdenetheestimationerroras e u ( k ):=^ x u ( k ) x u .Eq.( 4{3 )reducestothefollowing using( 1{4 ): e u ( k +1)= e u ( k )+ m ( k ) X v 2N u ( k ) a uv ( k )( e v ( k ) e u ( k )+ u;v ( k )) : (4{4) Inordertopursuefurtheranalysis,weintroducesomestipulation sandnotations.First, welet a uv ( k )=0for v= 2N u ( k ).Then,the n n Laplacianmatrix L ( k )ofthegraph G ( k )isdenedas L uv ( k )= P nv =1 a uv ( k )if u = v ,and L uv ( k )= a uv ( k )if u 6 = v .By removingtherowsandcolumnsof L ( k )withrespecttoreferencenodes,wegetthe n b n b principlesubmatrix L b ( k )(socalledgroundedorDirichletLaplacianmatrix[ 45 ]).Let e ( k ):=[ e 1 ( k ) ;:::;e n b ( k )] T ,thecorrespondingstatespaceformoftheestimationerroris e ( k +1)=( I m ( k ) L b ( k )) e ( k )+ m ( k ) D ( k ) ( k ) ; (4{5) 71

PAGE 72

where ( k ):=[ 1 ( k ) T ;:::; n b ( k ) T ] T ; u ( k ):=[ u; 1 ( k ) ;::: u;n ( k )] T ; D ( k ):= diag ( a 1 ( k ) ;:::; a n b ( k )) ; a u ( k ):=[ a u; 1 ( k ) ;:::;a u;n ( k )] ; (4{6) where u;u ( k ) = 2 u ( k )and a u;u ( k ) = 2 a u ( k ).Notethatwhen a uv ( k )=0, u;v ( k )isapseudo randomvariablewiththesamemeanandvarianceasthemeasuremen tnoiseonany existingedge.Moreover,recallthatonceanode u computesmeasurement u;v ( k ),itsends thismeasurementto v .Thus u;v ( k )= v;u ( k ). Assumption4.1. Measurementnoisevector ( k ) iswithmean E [ ( k )]= r andbounded secondmoment,i.e. E [ k ( k ) k 2 ] < 1 ,where kk denotes2-norm.Furthermore, ( k ) and ( j ) areindependentfor k 6 = j .Inaddition, f ( k ) g isindependentof e (0) ,where E k e (0) k 2 < 1 Assumption4.2. Thereexistsatime k 0 ,suchthattheedge-weightssatisfy a uv ( k )= a vu ( k )= c uv forallnode u;v 2 V when k>k 0 Assumption4.3. Thenon-increasingpositivesequence f m ( k ) g (stepsizeofthestochastic approximation)ischosenas m ( k )= c 1 k + c 2 ,where c 1 ;c 2 areconstantrealnumbers. Therefore, m ( k ) satises P 1k =0 m ( k )= 1 and P 1k =0 m 2 ( k ) < 1 Remark4.1. 1.Inpractice, E [ ( k )] maybetime-varyingevenifalltheunderlyingprocessesarewidesensestationary.Forinstance,theb iasinosetrelative measurementcomputedfromnode u isdierentfromthatcomputedfromnode v ,as v (1 u v ) 6 = u (1 v u ) .Therefore, E [ ( k )] dependsonwhichnodeinitializespairwise synchronizationattime k .Tomeetthisrequirement E [ ( k )]= r inAssumption 4.1 wecanstipulateinSTOalgorithmthatthenodewhocomputes u;v ( k ) betweenapair u and v isxedforalltime k .Thiscanbeachievedbycomparingthemagnitude oftheindexofnodes.Forexample,if u>v ,then u computes u;v ( k ) rstand thensendsitto v .Indeed,thepurposeofthisrequirementistoprovideformu lato computethesteadystatevalueofestimationerror,andthes ystemmaystillachieve convergencewithoutit. 2.Assumption 4.2 isatechnicalrequirementforconvergenceanalysis.There fore,in STOAlgorithm,pairofnodescanpickarbitrarypositivewei ghtsduringupdatingat thebeginning,butthenmaintainthesameweightsaftercert aintime. 72

PAGE 73

3.IncontrasttotheJATalgorithm,theedge-weightsnownee dtobesymmetricforthe convergenceguaranteestohold. 4.2.1DeterministicTopologySwitching Inthissection,weanalyzetheconvergenceof( 4{1 )whentheswitchofthetopology isdeterministic.Assumption4.4. Thereexists d 2 N s.t.forany t 0 ^ G d t := S t + d 1 k = t G ( k )= ( V ; S t + d 1 k = t E ( k )) isconnected,where E ( k ) issetofedgesin G ( k ) Assumption4.5. Thelimits L; L b ; D denedbelowexist: L :=lim t !1 1 t P tk =0 L ( k ) L b :=lim t !1 1 t P tk =0 L b ( k ) D :=lim t !1 1 t P tk =0 D ( k ) Remark4.2. 1.Assumption 4.4 impliesthatinformationcangofromanynodeto therestnodeswithinauniformlyboundedlengthoftime.Fur thermore,as G ( k ) is bidirectional,anotherequivalentassumptionisthat ^ G dt containsaspanningtree.The proposedalgorithmisalsorobusttopermanentlyaddingord eletingnodesincasethe newresultinggraphsatisestheassumptiononconnectivit y. 2.TounderstandthemeaningofAssumption 4.5 ,denethenitestatespace G = fG 1 ;:::; G N g asthesetofgraphsthatcanoccurovertime.Ifthesequenceo f G ( k ) canbedividedintoasequenceofniteintervals I j j =1 ; 2 ;::: ,suchthatthe percentageoftimesthateachstate G k occursisxedinallexceptnitelymanysuch intervals I j ,then L L b and D exist.Anotherexampleisthatthestate G i occurs accordingtoasamplepathofastationaryergodicprocess.I ntheend,denotesetsof matrices L b = f L b 1 ;:::;L b N g and D = f D 1 ;:::;D N g ,where L b i and D i correspond to G i 2 G .Ifthepercentageofallstatesoccurringis = f 1 ; 2 ;:::; N g ,then L b := N X i =1 i L b i ; D := N X i =1 i D i (4{7) Theorem4.1. UnderAssumption 4.1 4.2 4.3 4.4 4.5 e ( k ) in ( 4{5 ) convergesto L 1 b Dr inmeansquare,i.e., lim k !1 E ( k e ( k ) L 1 b Dr k 2 )=0 Thetheoremstatesthatundertheassumptions,thevarianceof theestimationerror decaysto0.Ifadditionallyalltherelativemeasurementsareunbias ed,i.e., r =0,thebias oftheestimatesconvergeto0aswell.Werststatetwolemmas,wh ichbothconsistspart ofthetheoremandfacilitatetheproofofthetheorem.Theproof ofrstlemmawillbe giveninSection 4.7.1 .Thesecondlemmaistakenfrom[ 46 ] 73

PAGE 74

Lemma4.1. Ifrelativemeasurementisunbiased,i.e., r =0 ,underAssumption 4.1 4.2 4.3 4.4 e ( k ) in ( 4{5 ) convergesto 0 inmeansquare,i.e., lim k !1 E ( k e ( k ) k 2 )=0 When r =0,( 4{5 )canberegardedasaleader-followingconsensusproblemwith time-varyingtopologyandzero-meannoisyinput.Theleadersarer eferencenodes u 2 V r whichholdtheirvariableaszero.Then, e u ( k )for u 2 V b isdriventozerobythereference nodesas k goesto 1 inmeansquaresense.TheproofoftheLemma 4.1 isinspiredfrom average-consensusalgorithmsasthatin[ 11 12 ]. Lemma4.2. [ 46 ]DenotebyAanunknownsymmetricandpositivesemi-denite matrix in R n n ,andwehavetosolvetheequationAx=yforanunknown y 2 R n .Assumethat A 1 exists.Wearegivenasequenceofmatrices A k andasequence y k ,where k =0 ; 1 ;::: Inaddition,supposethat lim k !1 k 1 k P ki =1 y i y k =0 lim k !1 k 1 k P ki =1 A i A k =0 and lim k !1 1 k P ki =1 k A i k 2 exists.Considerthesequence x k : x 0 isarbitrary, x k +1 = x k + c 1 k + c 2 ( y k A k x k ) ; (4{8) where c 1 and c 2 areconstantrealnumbers.Then, lim k !1 x k = A 1 y Proof. ProofofTheorem 4.1 Takingexpectationsonbothsidesof( 4{5 )withrespectto measurementnoise ( k ),weobtain ( k +1)=( I m ( k ) L b ( k )) ( k )+ m ( k ) D ( k ) r; (4{9) where ( k )= E [ e ( k )].Bysubstituting( 4{9 )in( 4{5 ),weget ~ e ( k +1)=( I m ( k ) L b ( k ))~ e ( k )+ m ( k ) D ( k ) ( k ) ; (4{10) where~ e ( k )= e ( k ) ( k )and ( k )= ( k ) r .Notethat ( k )iszeromeanandsatises Assumption 4.1 .ByLemma 4.1 ,~ e ( k )convergesto0inmeansquare.Now,weexamine theconvergenceof ( k ),whichwillbeanalyzedunderO.D.E.argumentinstochastic 74

PAGE 75

approximationtheory.Werstrewritethe( 4{9 )as ( k +1)= ( k )+ m ( k )( D ( k ) r L b ( k ) ( k )) : (4{11) ThenweconsidertheO.D.E ( t )= Dr L b ( t ) : (4{12) Fromthedenitionof L b andthesymmetryof L b i L b isasymmetricgroundedLaplacian of ^ G .Therefore, m ( L b ) x T L b x= ( x T x )for x 6 =0[Lemma8.4.3in[ 47 ]],where m ( L b ) denotesthesmallesteigenvalue L b .Since ^ G isconnected,byLemma1in[ 45 ], L b ispositivedenite.Consequently, m ( L b ) > 0.ThismeanstheO.D.E.( 4{12 )isasymptotically stableandtheequilibriumpointis = L 1 b Dr .Notethateach L b ( k )in 4{11 issymmetricpositivesemidenite.Now,itisfollowedbyLemma 4.2 that ( k )convergesto Consequently,lim k !1 E ( k e ( k ) L 1 b Dr k 2 )=0. 4.2.2MarkovianTopologySwitching Inthissection,weconsidertheconvergenceanalysiswhentheswit chofthenetwork topologycanbemodeledasMarkovchain,whichispreciselygiveninthe following assumption.Assumption4.6. Thetemporalevolutionofthemeasurementgraph G ( k ) isgovernedby anN-statehomogeneousergodicMarkovchainwithstatespac e G = fG 1 ;:::; G N g ,whichis thesetofgraphsthatcanoccurovertime.Furthermore ^ G := S Nk =1 G k =( V ; S Nk =1 E k ) is connected,where E k issetofedgesin G k .Inaddition, G ( k ) and ( j ) areindependentfor all k and j Remark4.3. InAssumption 4.6 ,theMarkovianswitchonthegraphsmeansthat P ( G ( k +1)= G i jG ( k )= G j )= P ( G ( k +1)= G i jG ( k )= G j ; G ( k 1)= G ` ;:::; G (0)= G p ) where G i ; G j ; G ` ;:::; G p 2 G .ThereasonablenessofthishasbeenjustiedinChapter 3 for mobilenetworks.TherequirementforergodicityoftheMark ovchainensuresthatthere isanuniquesteadystatedistributionwithnon-zeroentrie s.Thismeanseverygraphin 75

PAGE 76

thestatespaceofthechainoccursinnitelyoften.Since ^ G isconnected,ergodicityimplies thatinformationfromthereferencenode(s)willrowtoeach ofthenodesovertime. Again,notethatnoneofthegraphsthateveroccurisrequire dtobeaconnectedgraph. Denotesetsofmatrices L b = f L b 1 ;:::;L b N g and D = f D 1 ;:::;D N g ,where L b i and D i correspondto G i 2 G .Similartothedeterministicswitchcase,wecandenethes teady statedistributionoftheergodicMarkovchain = f 1 ; 2 ;:::; N g andtheresulting L b and D asthatdenedin ( 4{7 ) Theorem4.2. UnderAssumption 4.1 4.2 4.3 4.6 e ( k ) in ( 4{5 ) ismeansquare convergent,i.e., lim k !1 E ( k e ( k ) E ( e ( k )) k 2 )=0 ,where E ( e ( k )) isexpectationof e ( k ) w.r.t.measurementnoise ( k ) ,and E ( e ( k )) convergesto L 1 b Dr almostsurely. Theproofofthetheoremusesthefollowingtwolemmas:theproofo frstlemmais giveninSection 4.7.2 andthesecondfollowsinastraightforwardmannerfromtheresult in[ 48 ],whichisastochasticversionofLemma 4.2 Lemma4.3. Ifrelativemeasurementsareunbiased,i.e., r =0 ,underAssumption 4.1 4.2 4.3 4.6 e ( k ) in ( 4{5 ) convergesto 0 inmeansquare,i.e., lim k !1 E ( k e ( k ) k 2 )=0 Lemma4.4. [Proposition1in[ 48 ]]Assume f A k ;y k g k =0 ; 1 ;::: ,isstochasticprocess on ( ; F ;P ) ,where A k issymmetricpositivesemidenitein R n n and y k isin R n Considerthestochasticapproximationalgorithmofthefor m x k +1 = x k + c 1 k + c 2 ( y k A k x k ) ; (4{13) where c 1 and c 2 areconstantrealnumbers.If A :=lim k !1 1 k P ki =1 E [ A i ] y := lim k !1 1 k P ki =1 E [ b i ] ,and A ispositivedenite,then lim k !1 x k = A 1 y almostsurely. Proof. ProofofTheorem 4.2 TheproofissimilartothatforTheorem 4.1 .Dene ( k )= E [ e ( k )],wheretheexpectationistakenwithrespecttomeasurementno ise ( k ).Takethe 76

PAGE 77

expectationonbothsidesof( 4{5 ),weget ( k +1)=( I m ( k ) L b ( k )) ( k )+ m ( k ) D ( k ) r: (4{14) Bysubstituting( 4{14 )in( 4{5 ),weget ~ e ( k +1)=( I m ( k ) L b ( k ))~ e ( k )+ m ( k ) D ( k ) ( k ) ; (4{15) where~ e ( k )= e ( k ) ( k )and ( k )= ( k ) r .Notethat ( k )iszeromeanandsatises Assumption 4.1 .ByLemma 4.3 ,~ e ( k )convergesto0inmeansquare.Now,weexaminethe convergenceof ( k ).Werewrite( 4{14 )as ( k +1)= ( k )+ m ( k )( D ( k ) r L b ( k ) ( k )) : (4{16) ItfollowsfromAssumption 4.6 lim k !1 1 k k X i =1 E [ L b ( k )]= N X i =1 i L b i = L b ; lim k !1 1 k k X i =1 E [ D ( k )]= N X i =1 i D i = D (4{17) Fromthedenitionof L b andthesymmetryof L b i L b isasymmetricgroundedLaplacian of ^ G .Therefore, m ( L b ) x T L b x= ( x T x )for x 6 =0[Lemma8.4.3in[ 47 ]],where m ( L b ) denotesthesmallesteigenvalue L b .Since ^ G isconnected,byLemma1in[ 45 ], L b is positivedenite.Consequently, m ( L b ) > 0.Then,byLemma 4.4 that ( k )convergesto L 1 b Dr almostsurely. 4.2.3VericationoftheTheorems4.2.3.1VericationofTheorem 4.1 Weperformsimulationsonamade-upscenariothatallowsnumericalv erication ofthepredictionsofTheorem 4.1 .Anetworkof5nodesischosen.Thetopology G ( k ) switchesperiodicallyamonganodeset G = fG 1 ; G 2 ; G 3 g showninFigure 4-2 according totherule: G ( k )= G 1 when k =5( T 1)+1or5( T 1)+5; G ( k )= G 2 when k =5( T 1)+2or5( T 1)+3; G ( k )= G 3 when k =5( T 1)+4,where T =1 ; 2 ;::: Therefore,thepercentageoftimeeachgraphoccursis =[2 = 5 ; 2 = 5 ; 1 = 5].Notethat 77

PAGE 78

Figure4-2.Allthegraphsthatoccurinsimulation#1. 0 50 100 150 200 250 300 -15 -10 -5 0 iteration indexmean node 1 node 4 theoretical value 50 100 150 200 250 300 0 1000 2000 3000 iteration indexvariance node 1 node 4 Figure4-3.Empiricallyobtainedmeanandvarianceoftheestimatione rrorfortwonodes inthe5-nodedeterministicswitchingnetwork,whichapproachto 4 : 51and 10 : 73respectivelyasseenfromthegure. theunionofthegraphsin G isconnected,thoughnoneofthegraphsisaconnected graph.Nodevariablesarepickedrandomlyaround0.Node5isthesin glereferencenode; itsnodevariablebeing0.Thevarianceofmeasurementnoiseis1andt hebiasinthe measurement r u;v for1 u
PAGE 79

0 50 100 150 200 250 300 -15 -10 -5 0 iteration indexmean node 1 node 4 theoretical value 50 100 150 200 250 300 0 2000 4000 iteration indexvariance node 1 node 4 Figure4-4.Empiricallyobtainedmeanandvarianceoftheestimatione rrorfortwonodes inthe5-nodeMarkovianswitchingnetwork.Themeanintheestimatio nerror ofthetwonodesapproach 4 : 42and 10 : 79,respectively,aspredictedby Theorem 4.1 4.2.3.2VericationofTheorem 4.2 Thetopology G ( k )switchesamonganodeset G = fG 1 ; G 2 ; G 3 g showninFigure 4-2 accordingtoanergodicMarkovchain,whosetransitionprobabilitym atrixis: P = 0BBBB@ 0 : 50 : 50 0 : 300 : 7 0 : 20 : 60 : 2 1CCCCA : (4{18) Therefore,thepercentageoftimeeachgraphoccursis =[0 : 3363 ; 0 : 3540 ; 0 : 3097]. Notethattheunionofthegraphsin G isconnected,thoughnoneofthegraphsisa connectedgraph.Thebiasofmeasurementerror u;v areassignedinthesamefashionas doneinthelastsection.Theorem 4.2 predictsthattheestimationerrorbiasis L 1 b Dr = f 3 : 5080 ; 4 : 4174 ; 4 : 1641 ; 10 : 7907 g .Figure 4-3 indicatesthattheestimationerrors convergesto L 1 b Dr inmeansquare,asTheorem 4.1 predicts. 4.3NumericalEvaluationofTimeSynchronizationAccuracy withSTO WenowexaminetheperformanceoftheSTOalgorithminperformingc locksynchronization.Simulationsareperformedfora50-nodemobilenetwo rkina100 m 100 m 79

PAGE 80

squareeld.Thetrueskewsandosetsof49nodesarepickedunif ormlyfrom[1 2 10 5 ; 1+2 10 5 ]and[ 10 1 sec ; 10 1 sec]respectivelyaccordingto[ 19 ].Thesingle referencenodehasskew1andoset0.Forthesakeofconvenien ce,simulationsarecarried outinasynchronousfashion.Eachnodestartsits k thupdateinterval(synchronization period)atglobaltime t k ,where k =1 ; 2 ::: .Thelengthofanintervalis1second,therefore, t k +1 t k =1.Nodes'motionsarestillgeneratedaccordingtoRandomDirect ion(RD) mobilitymodel[ 36 ](SeeFigure 4-5 forsamplegraphs).Apairofnodescancommunicate whendistancebetweenthemislessthan15 m .Whentheyarewithincommunication rangetheyexchangetimestampedmessagestoperformpairwises ynchronizationtoobtain relativemeasurementsoflogarithmskewsandosetsbythetwo-s tepproceduredescribed inChapter 2 ,wherethesameparametersareusedincludingthatofrandomdela y.The stepsizeofstochasticapproximationischosenas m ( k )= 1 : 5 k +3 .Allnodesinitializetheir skewandosetestimatesas1and0respectively. Weconduct100MonteCarlosimulations.Figure 4-5 showstwosnapshotsofthe networkduringonesimulation.Aswecansee,onlyalimitednumberofn odescancommunicatewitheachother.Thefollowingplotsshowsimulationresults, wherethex-axisis discreteintime,i.e., t k .Figure 4-6 showsskewestimationhistoryoftwonodesalongtime. Figure 4-7(a) and 4-7(b) showthemeanandvarianceofestimationerrorofskews.Both themeanandvarianceareseentoconvergeto0.Figure 4-8 4-9(a) and 4-9(b) demonstratesthecorrespondingresultsofosetestimation.Itseems thatbiasexistsaspredicted bytheoreticalanalysisinChapter 2 ,whichmaybeduetothebiasintroducedinrelative measurementgeneration.Figure 4-10 showstheglobaltimeestimationerror, ^ t u ( t k ) t k ,as afunctionof t k fortwonodes.Afteraninitialtransientperiod,theerrorinthees timates oftheglobaltimeisquitesmall.InSection 3.6 ,thevarianceofskewestimationleads totheincreasingofthevarianceofglobaltimeestimation.Here,th eextremelyaccurate skewestimates(variancetozero)obtainedbySTOiscrucialingett inggoodglobaltime estimates. 80

PAGE 81

Figure4-5.Twographsthatoccurduringonesimulationwith50node smovingaccording totherandomdirectionmobilitymodel. 0 50 100 150 200 250 300 1 1 1 1 Time (sec)Skew node 1 node 30 true Figure4-6.Theskewestimates^ u ( t k )inoneexperimentinthe50-nodemobilenetwork. 0 50 100 150 200 250 300 -4 -2 0 2 x 10 -6 Time (sec)Mean node 1 node 30 (a)Mean 0 50 100 150 200 250 300 0.5 1 1.5 2 2.5 x 10 -10 Time (sec)Variance node 1 node 30 (b)Variance Figure4-7.Themeanandvarianceoftheskewestimationerror,i.e., E [^ u ( t k ) u ]and Var [^ u ( t k ) u ],inthe50-nodemobilenetwork. 81

PAGE 82

0 50 100 150 200 250 300 -0.1 -0.08 -0.06 -0.04 -0.02 Time (sec)Offset node 1 node 30 true Figure4-8.Theosetestimates ^ u ( t k )inoneexperimentinthe50-nodemobilenetwork. 0 50 100 150 200 250 300 0 0.02 0.04 Time (sec)Mean node 1 node 30 (a)Mean 0 50 100 150 200 250 300 2 4 6 8 10 12 x 10 -3 Time (sec)Variance node 1 node 30 (b)Variance Figure4-9.Themeanandvarianceoftheosetestimationerror,i.e ., E [ ^ u ( t k ) u ]and Var [ ^ u ( t k ) u ],inthe50-nodemobilenetwork. 0 50 100 150 200 250 300 -0.05 0 0.05 Time (sec)Time est. error (sec) node 1 node 30 Figure4-10.Theerroringlobaltimeestimation,i.e., ^ t u ( t k ) t k ,oftwonodes. 82

PAGE 83

4.4NumericalComparisonamongSTO,JATandATSAlgorithm Inthissection,wecomparetheSTOalgorithmwithJATalgorithminCha pter 3 and theATSalgorithmof[ 13 ]forvirtualtime-synchronization. 4.4.1BriefDiscussionofATSAlgorithms ATS[ 13 ]isanaverageconsensustypeoftimesynchronizationprotocol.I nATS, eachnode u estimatesthevirtualglobaltimeusing ^ t ru ( t )=^ % u ( k ) u ( t )+^ o u ( k )for t k t t k +1 ,wherevariables^ % u ( k )and^ o u ( k )canbethoughtofastheskewandosetof avirtualglobaltimerespecttothelocaltimeof u duringinterval k .Here,wepresenta synchronousversionofATSalgorithmthatweuseinsimulationinorde rtobeconsistent withourproposedalgorithms.Node u updates^ % u ( k )and^ o u ( k )using^ % v ( k ), ^ t rv ( k ),^ uv ( k ) fromitsneighbors,where^ uv ( k )istheestimatedrelativeskew.^ uv ( k )isobtainedby pairwisecommunication,aspartoftheATSalgorithm,between u and v during k -th updateinterval.Thisisdoneasfollows.Twotime-stampedmessages aresentfromnode v tonode u :oneatthebeginningof k -thintervalandtheotheroneinthemiddleofthe interval.UsingthesamenotationasthatintroducedinSection 2.1 ,node u obtainsfour timestamps: f ( i ) u ; ( i ) v g for i =1 ; 3andthensendsbackto v .Figure 4-11 demonstrates theprocedureofpairwisecommunicationforaninterval k .Theestimationof u;v is performedbyalow-passlter: ^ u;v ( k )= ^ u;v ( k 1)+(1 ) (1) v (3) v (1) u (3) u ; (4{19) where isatuningparameter,chosenas0 : 2[ 13 ].^ % u ( k )and^ o u ( k )areupdatedasfollows: atthebeginningof k thiteration,let^ % tpu ( k )=^ % u ( k 1),^ o tpu ( k )=^ o u ( k 1)forallnodes.A node u performthefollowingupdatesforeachofitsneighbors, ^ % tpu ( k )= v ^ % tpu ( k )+(1 v )^ u;v ( k )^ % tpv ( k ) ^ o tpu ( k )=^ o tpu ( k )+(1 o )( ^ t rv ( t k ) ^ t ru ( t k )) ; (4{20) 83

PAGE 84

where v and u aretuningparameterschosenas0 : 5.Onceallnodesnishedupdates, ^ % u ( k )=^ % tpu ( k )and^ o u ( k )=^ o tpu ( k ).Undertheconditionthatthereisnorandomdelay duringcommunication,ithasbeenshownthatlim k !1 u ^ % u ( k )= ,lim k !1 ^ o u ( k )+ u ^ % ( k )= ,where and istheskewandosetofthevirtualclockwithrespectto globaltime.TheATSalgorithmensuresthattheestimatedvirtualg lobaltimesfromall nodesareeventuallyequal,i.e.,lim t !1 ^ t ru ( t )= ^ t rv ( t )forall u and v uv (1) v (1) u (3) v (3) u w (3 ; 1) v Figure4-11.Pairwisetime-stampexchangesinATSalgorithm.4.4.2Comparison Inordertocomparetheperformanceofthethreealgorithms(ST O,JAT,andATS), simulationsshouldbeconductedunderidenticalconditions.Recallt hattheperformance guaranteesontheATSalgorithmwereobtainedundertheassumpt ionthatthetime stampsareexchangedwithoutrandomdelay.Tocomparetheperf ormanceofATSwith theproposedJATandSTOalgorithmsunderthesameconditions,we addrandomdelays d ( i ) i =1 ; 3,suchthat (1) u = u;v ( (1) v + d (1) )+ u;v (3) u = u;v ( (3) v + d (3) )+ u;v (4{21) Thedelayparameters d ( i ) arethesameasthoseusedduringtheprevioussimulations. Inaddition,sinceATSdoesnotestimatetheglobalclocktimeatanyo fthenodes,we usethemetric syncerror ^ t ( t k )tocompareATSwiththeothertwo.ForJATand STOalgorithm,theyaredenedas ^ t ( t k )=max u;v j ^ t u ( t k ) ^ t v ( t k ) j forall u and v ;for 84

PAGE 85

ATS,itisdenedas ^ t ( t k )=max u;v j ^ t ru ( t k ) ^ t rv ( t k ) j .A10-nodenetworkswithin[10 m 10 m ]aresimulatedwithcommunicationrange5m.Thesameskew,osetp arametersare usedasthatinprevioussimulations.Weimplement100MonteCarlosimu lationinorder toobtainstatisticssuchasmeanandvariance.Forthesakeofcon sistency,theswitches ofthenetworktopologyineachrunarekeptthesameforthreealg orithms.Figure 4-12 showstwosamplenetworksduringonesimulation. Figure4-12.Twographsthatoccurduringonesimulationwith10nod esmoving accordingtotherandomdirectionmobilitymodel. WerstprovidecomparisonresultsbetweenJATandSTOalgorithm. Figure 4-13(a) 4-13(b) 4-14(a) 4-14(b) showthecomparisonofthemeanandvariance ofskewandosetestimationerror.Obviously,theSTOalgorithmpr ovidesmoreaccurate skewestimationsincethevarianceoftheerrorconvergestozero .However,theconvergencerateisslowercomparedtotheJATalgorithm.Figure 4-15 4-16(a) and 4-16(b) comparetheperformanceofthetwoalgorithmsintermsoftheglob altimeestimation.It turnsoutthatSTOalgorithmachievesmuchbetteraccuracythan thatofJATalgorithm duetoaccuracyskewestimation. Figure 4-17 4-18(a) and 4-18(b) providescomparisonofthethreealgorithmsin termsofsyncerror.Ontheonehand,STOalgorithmshowsitsadva ntageoflongterm accuracy:synchronizationerrorbecomessmallerthanthatofth eothersafterabout200 seconds.Thereasonisthat,bothJATalgorithmandATScannotdr ivethevarianceof skewestimates(skewsofvirtualclockw.r.tlocalclocksforATS)to zero,whichresultsin largesynchronizationerroreventually.Ontheotherhand,itcanb eseenfromthegures thatATSconvergesmuchfasterthanthatoftheJATandSTOalgo rithm. 85

PAGE 86

0 100 200 300 400 500 600 700 800 -5 0 5 x 10 -6 Time (sec)Mean JAT STO True (a)Mean 0 100 200 300 400 500 600 700 800 10 -12 10 -10 Time (sec)Variance JAT STO (b)Variance Figure4-13.Themeanandvarianceoftheskewestimationerrorof node2,i.e., E [^ 2 ( t k ) 2 ]and Var [^ 2 ( t k ) 2 ],inthe10-nodemobilenetwork. 86

PAGE 87

0 100 200 300 400 500 600 700 800 0 10 20 x 10 -3 Time (sec)Mean JAT STO True (a)Mean 0 100 200 300 400 500 600 700 800 10 -10 Time (sec)Variance JAT STO (b)Variance Figure4-14.Themeanandvarianceoftheosetestimationerroro fnode2,i.e., E [ ^ 2 ( t k ) 2 ]and Var [ ^ 2 ( t k ) 2 ],inthe10-nodemobilenetwork. 0 100 200 300 400 500 600 700 800 -0.02 -0.01 0 0.01 Time (sec)Time est. error (sec) JAT STO True Figure4-15.Theerroringlobaltimeestimationbynode2,i.e., ^ t 2 ( t k ) t k 87

PAGE 88

0 100 200 300 400 500 600 700 800 -20 -10 0 x 10 -3 Time (sec)Mean JAT STO True (a)Mean 0 100 200 300 400 500 600 700 800 10 -5 Time (sec)Variance JAT STO (b)Variance Figure4-16.Themeanandvarianceoftheglobaltimeestimationerr orofnode2,i.e., E [ ^ t 2 ( t k ) t k ]and Var [ ^ t 2 ( t k ) t k ],inthe10-nodemobilenetwork. 0 100 200 300 400 500 600 700 800 10 -4 10 -2 10 0 Time (sec)Sync error (sec) ATS JAT STO Figure4-17.Syncerror ^ t ( t k )alongtimeinoneexperiment. 88

PAGE 89

0 100 200 300 400 500 600 700 800 10 -2 Time (sec)Mean ATS JAT STO (a)Mean 0 100 200 300 400 500 600 700 800 10 -6 10 -4 Time (sec)Variance ATS JAT STO (b)Variance Figure4-18.Themeanandvarianceofthesynerror ^ t ( t k )inthe10-nodemobile network. 4.5ImprovementofConvergenceRate Asshownintheprevioussection,theconvergencerateofthethr eealgorithms areintheorder:STO < JAT < ATS.Slowconvergenceofglobaltimeestimationand clocksynchronizationcausesnodestowaitforalongtimebeforeac curatetime-related operationcanbeconducted.Inthissection,weproposetwostra tegiestoimprovethe convergencerateofthetwoproposedalgorithms(mainlyforSTO) 4.5.1UpdatewithAverageDistance Intheproposedalgorithms,informationoftheglobalclockspread soutfromthe referencenode.Therefore,anodethatstaysclosetotherefe rencenodemostofthetime isabletoobtainmoreaccurateskewandosetestimatesthananod efarawayfrom thereferencenode.Supposethosetwonodesatsomepointbeco meneighborsdueto mobility.Thelatternodeshouldusetheestimateoftheformernode toupdatewhilethe formeroneshouldn'tupdateusingtheestimateofthelatter.Ther efore,inthissection, 89

PAGE 90

weintroduceamethodthatanodeuseanindex,calledaveragedista nce,tochoosethe \good"neighborsthathavemoreaccurateestimatesthantheyt hemselvesdo.Inthis approach,eachnode u maintainsanaveragedistance y u ( k ),whichisthetime-averaged numberofhopstothereferencenode.Sincenodesaremobile,itisp ossiblethatanode physicallyfarawaythereferencenodeactuallyhassmalleraverage distancetothe reference.Whenapairofnode u and v areneighboringatinterval k ,theycomparethe valuesofaveragedistance.Essentially,thenode u withhighervalue y u ( k )willupdate usingtheestimatefrom v withlowerdistancevalue.Theaveragedistanceofthereference nodekeepsaszeroallthetime.Theprecisedescriptionhowtheave ragedistanceis updatedisgiveninAlgorithm 3 .Anode u updatesusingtheestimatesandrelative measurementsfromitsneighborsinset S u ( k )= f v j y u ( k ) y v ( k ) ;v 2N u ( k ).In addition, y u ( k )updatesbyaveragingthe y v ( k )from v 2N u ( k ).If N u ( k )isempty, y u ( k )is updatedbyaddingatuningconstant(e.g.0 : 25).Thereasonisthatifanodehaven'tbeen updatingforawhile,itsestimatestartstogetworsecomparedtot heestimatesinother nodeswhoare. Algorithm3 Averagedistancealgorithmatnode u 1: Initialize y u (0)=0for u 2 V r and y u (0)= 1 for u 2 V b 2: while u isperformingiteration do 3: for v 2N u ( k ) do 4: if y u ( k ) y v ( k )and y v ( k ) 6 = 1 then 5: S u ( k ) v ; 6: endif 7: endfor 8: if S u ( k ) 6 = ; then 9: y u ( k +1)= P v 2S u ( k ) y v ( k ) jS u ( k ) j ; 10: else 11: y u ( k +1)= y u ( k )+0 : 25; 12: endif 13: k = k +1; 14: endwhile 90

PAGE 91

Notethattheupdatesforapairofnodesareasymmetricnow.Con sequently, Assumption 4.2 ,thatapairofnodesalwaysupdatessymmetrically,isnotsatised anymore.Webelievethatthisassumptionisnotanecessaryconditio nforconvergence tooccur.Weprovidenumericalvalidationoftheaveragedistancea pproachwithout providingaproofofconvergence.4.5.2CombinationofJATandSTOAlgorithms InSTOAlgorithm,thestepsizeneedtodecreaseinacertainrate.O ntheone hand,decreasingstepsizeattenuatesthemeasurementnoisean densurestheconvergence ofthevarianceto0.Ontheotherhand,itleadstoslowconvergenc easshowninthe simulations.NotethattheestimationerrorofJATstilldecreasesf asterthanthatofSTO initially.Inpractice,wecancombinethetwoproposedalgorithms:th enetworksrstuse JATalgorithmtoachievefasterconvergencerateinthebeginning. Aftercertainnumber ofiterations,nodesswitchtoSTOAlgorithm,whichensuresasympt oticconvergenceof thevarianceto0.Therefore,wedeneanewstepsize m 1 ( k )as: m 1 ( k )= 8><>: 1 1+ jN u ( k ) j for k
PAGE 92

4.5.3NumericalComparison Inthissection,wenumericallyvalidatetheimprovementofconverge nceratebyusing thetwoproposedapproachesabove.Foreaseofdiscussion,wer efertotheJATalgorithm augmentedwiththemethodinSection 4.5.1 (averagedistance)asthe JAT-improved algorithm ,andtheSTOalgorithmaugmentedbyusingboththemethods(aver age distanceandnewstepsize m 1 ( k ))asthe STO-improvedalgorithm .Wesimulatethesame 10-nodemobilenetworkwiththesameparametersasthatinSection 4.4.2 k 1 ispickedas 20.Weconduct100MonteCarlosimulation.Figures 4-19(a) 4-19(b) 4-20(a) 4-20(b) show themeanandvarianceofglobaltimeestimationerrorbetweentheo riginalalgorithmsand theirimprovedversions.Figures 4-21(a) 4-21(b) 4-22(a) 4-22(b) 4-23(a) 4-23(b) show themeanandvarianceofsyncerrorbetweenoriginalandimproved algorithms,aswellas thesyncerrorinoneexperiment.Thoseguresshowthatconver genceratesofJATand STOalgorithmareimproveddramaticallywiththemodicationsdescrib edabove. 0 200 400 600 800 -10 -5 0 5 x 10 -3 Time (sec)Mean JAT STO True (a)Original 0 200 400 600 800 -10 -5 0 5 x 10 -3 Time (sec)Mean JAT-improved STO-improved True (b)Improved Figure4-19.Themeanoftheglobaltimeestimationerrorofnode2, i.e., E [ ^ t 2 ( t k ) t k ] and Var [ ^ t 2 ( t k ) t k ],inthe10-nodemobilenetwork. 92

PAGE 93

0 200 400 600 800 10 -6 10 -5 10 -4 Time (sec)Variance JAT STO (a)Original 0 200 400 600 800 10 -7 10 -6 10 -5 10 -4 Time (sec)Variance JAT-improved STO-improved (b)Improved Figure4-20.Thevarianceoftheglobaltimeestimationerrorofnod e2,i.e., E [ ^ t 2 ( t k ) t k ] and Var [ ^ t 2 ( t k ) t k ],inthe10-nodemobilenetwork. 0 200 400 600 800 10 -4 10 -3 10 -2 10 -1 Time (sec)Sync error (sec) ATS JAT STO (a)Original 0 200 400 600 800 10 -4 10 -3 10 -2 10 -1 10 0 Time (sec)Sync error (sec) ATS JAT-improved STO-improved (b)Improved Figure4-21.Thetimeseriesofthesynchronizationerror ^ t ( t k )inthe10-nodemobile network. 93

PAGE 94

0 200 400 600 800 10 -3 10 -2 Time (sec)Mean ATS JAT STO (a)Original 0 200 400 600 800 10 -3 10 -2 Time (sec)Mean ATS JAT-improved STO-improved (b)Improved Figure4-22.Themeanofthesynchronizationerror ^ t ( t k )inthe10-nodemobilenetwork. 0 200 400 600 800 10 -7 10 -6 10 -5 10 -4 Time (sec)Variance ATS JAT STO (a)Original 0 200 400 600 800 10 -8 10 -7 10 -6 10 -5 10 -4 Time (sec)Variance ATS JAT-improved STO-improved (b)Improved Figure4-23.Thevarianceofthesynchronizationerror ^ t ( t k )inthe10-nodemobile network. 4.6ConclusionandOpenProblems Weproposedadistributedprotocol(STO)fornodevariableestima tionfromnoisy relativemeasurementsinamobilenetwork,whichachieveshighaccur acyandcanbe 94

PAGE 95

implementedasynchronously.Thealgorithmisinspiredbyrecentwor kintheconsensus literaturewherestochasticapproximationideasareusedtoreduc etheeectofmeasurementnoise.Weprovedthatthevarianceofestimationerrora symptoticallygoesto zeroundercertainassumptionsfortwotypesofnetworkswitchin g:deterministicand Markovian.Ifrelativemeasurementsareunbiased,themeanofes timationerrorconverges tozero.Ifrelativemeasurementsarebiased,themeangoestoalim itingvaluethatisa functionofthebiasinmeasurementnoiseandnetworktopologies.M oreover,withfurther knowledgeonthepercentageoftimesthateachtopologyoccurs, thelimitingvaluecanbe calculated.Appliedtothetimesynchronizationproblem,thepropos edalgorithmprovides veryaccurateglobaltimeestimationandlowsyncerror.Twoimprov edmethodsfurther improvetheconvergencerateofSTO. Thereareafewaspectsthatcouldbefurtherinvestigated.First ,theupdating governedbyaveragedistancedescribedinSection 4.5.1 isseentoimprovetheconvergence rateinsimulations.However,theconvergenceanalysisislackingfor thiscasesinceit involvesasymmetricupdates.Secondly,theasynchronousimpleme ntationoftheSTO algorithmleadstotheiterationintervalincreasingwithoutboundalo ngtime.Thiscould bereducedifnodesuseup-to-dateboundsonskewsandosetsf romtheirestimated values.Theperformanceofthisapproachneedstobeanalyzed.T hirdly,itisalsoof interesttodirectlyanalyzetheconvergenceofthealgorithmwhen thegain m ( k )decrease asynchronously. 4.7TechnicalProofs 4.7.1ProofofLemma 4.1 Thefollowingterminologywillbeneededforthesubsequentanalysis. Let( ; F ;P ) beaprobabilityspace,and fF ( k ) g isasequenceofsub-algebrasof F suchthat F ( k ) F ( k +1),forall k .Arandomsequence f ( k ) g iscalledmartingaledierencesw.r.t. fF ( k ) g if ( k )is F ( k )-measurable, E j ( k ) j 2 < 1 and E [ ( k +1) jF ( k )]=0,forall k [ 38 ].Bythetowerpropertyofexpectations, E [ ( k )]=0and E [ ( k ) ( j ) T ]=0,if 95

PAGE 96

k 6 = j .Therefore,byAssumption 4.1 alongwiththemean r =0,thesequenceofvectors f ( k ) g ismartingaledierencesw.r.t. F ( k ),where F ( k )isa -algebrageneratedby f e (0) ; (0) ;:::; ( k ) g Proof. Recallthat e ( k )isthevectoroftheestimationerrorsatindex k ,andwedene V ( k )= e ( k ) T e ( k ).Toprovethetheorem,wewouldliketoshowlim k !1 E [ V ( k )]=0, whichimplies E [ e 2u ( k )] 0as k !1 forall u 2 V b .Let d 1 rd = Q ( r +1) d 1 j = rd ( I m ( j ) L b ( j )), where Q nj = t A t := A n A n 1 ;:::;A t istheorderedproductsofmatrices.Now,( 4{5 )canbe rewrittenas e (( r +1) d )= d 1 rd e ( rd )+ d 1 rd ; (4{23) where d 1 rd = P ( r +1) d 1 j = rd d 1 j +1 m ( j ) D ( j ) ( j ).Itfollowsthat, V (( r +1) d )= e T (( r +1) d ) e (( r +1) d ) = V ( rd ) 2 e T ( rd ) ( r +1) d 1 X j = rd f m ( j ) L b ( j ) g e ( rd )+2 e T ( rd )( d 1 rd ) T d 1 rd +( d 1 rd ) T d 1 rd + e T ( rd ) f ( d 1 rd ) T d 1 rd I +2 ( r +1) d 1 X j = rd ( m ( i ) L b ( i )) g e ( rd ) : (4{24) Usebinomialexpansion, ( d 1 rd ) T d 1 rd = I 2 ( r +1) d 1 X j = rd ( m ( j ) L b ( j ))+ C (2 d; 2) X j 1 ;j 2 2 I d 1 rd ( m ( j 1 ) m ( j 2 ) L b ( j 1 )) L b ( j 2 )) C (2 d; 3) X j 1 ;j 2 ;j 3 2 I d 1 rd ( m ( j 1 ) m ( j 2 ) m ( j 3 ) L b ( j 1 )) L b ( j 3 )) L b ( j 3 ))+ H:O:T; (4{25) where P C (2 d; 2) j 1 ;j 2 2 I d 1 rd ( m ( j 1 ) m ( j 2 ) L b ( j 1 )) L b ( j 2 ))isthesummationofthetermsofallcombinationsof m ( j 1 ) L b ( j 1 )and m ( j 2 ) L b ( j 2 ),where I d 1 rd = f j j rd j ( r +1) d 1 g and C (2 d; 2) isthetotalnumberoftermsinsidethesummation,whichisequaltot henumberof2-nd combinationofa2 d elementsset. P C (2 d; 3) j 1 ;j 2 ;j 3 2I d 1 rd ( : )canbeinterpretedinthesamefashion. 96

PAGE 97

Therefore, k ( d 1 rd ) T d 1 rd I +2 ( r +1) d 1 X j = rd ( m ( i ) L b ( i )) k m 2 ( rd ) C (2 d; 2)max j 2 I d 1 rd k L b ( j ) k 2 + m 3 ( rd ) C (2 d; 3)max j 2 I d 1 rd k L b ( j ) k 3 + H:O:T m 2 ( rd ) Q d ; (4{26) where Q d =(2 d 1) max 0 j 2 d C (2 d;j ) max(sup j> 0 k L b ( j ) k 2 d ; 1) : (4{27) Forconciseness,let L d 1 rd := P rd + d 1 j = rd L ( j )andsimilarly L b d 1 rd := P rd + d 1 j = rd L b ( j ).Since L b d 1 rd issymmetric, m ( L b d 1 rd ) x T L b ( r +1) d 1 rd x= ( x T x )for x 6 =0[Lemma8.4.3in[ 47 ]], wherewerecallthat m meansthesmallesteigenvalue.So, V (( r +1) d ) 1 2 m ( L b d 1 rd ) m (( r +1) d 1)+ m 2 ( rd ) Q d V ( rd ) +2 e T ( rd )( d 1 rd ) T d 1 rd +( d 1 rd ) T d 1 rd : (4{28) AsthegroundedLaplacianmatrix L b d 1 rd isaprincipalsubmatrixof L d 1 rd ,andthe correspondingundirectedgraph ^ G d rd isconnected,byLemma1in[ 45 ], L b d 1 rd ispositive denite.Therefore, m ( L b d 1 rd ) > 0.Since e ( rd )is F ( rd 1)-measurableand f ( k ) g is martingaledierencesequencew.r.t. F ( k ),itfollowsthat E 2 e T ( rd )( d 1 rd ) T d 1 rd jF ( rd 1) =2 e T ( rd )( d 1 rd ) T E d 1 rd jF ( rd 1) =0 ; (4{29) Bythetowerpropertyofconditionalexpectation, E 2 e T ( rd )( d 1 rd ) T d 1 rd =0.In addition,bypropertyofmartingaledierence, E [ ( i ) ( j ) T ]=0for i 6 = j .Thus, E ( d 1 rd ) T d 1 rd = ( r +1) d 1 X j = rd m 2 ( j ) E h ( j ) T D ( j ) T ( ( j ) j +1 ) T ( j ) j +1 D ( j ) ( j ) i F d ( r +1) d 1 X j = rd m 2 ( j ) ; (4{30) 97

PAGE 98

where ( j )=( r +1) d j 2and F d =sup j 0 k D ( j ) k 2 sup j 0 k ( j ) j +1 k 2 E [ k ( j ) k 2 ].Note thatlimsup k !1 m ( k ) =m ( k +1) < 1 ,sothereexists r>r 0 suchthat m ( rd ) s d m (( r + 1) d ),wherescalar s d isafunctionof d ,and m ( rd ) 1.Consequently, m 2 ( rd ) Q d m 2 (( r +1) d ) Q d s 2d for r>r 0 .Let dinf =inf t 0 ( m ( L b d 1 t )),then, E [ V (( r +1) d ] 1 2 dinf m (( r +1) d )+ m 2 (( r +1) d ) Q d s 2d E [ V ( rd )]+ F d ( r +1) d 1 X j = rd m 2 ( j ) : (4{31) Let q ( r )=2 dinf m (( r +1) d ) m 2 (( r +1) d ) Q d s 2d .As dinf > 0,thereexists r>r 1 suchthat dinf m (( r +1) d ) >m 2 (( r +1) d ) Q d s 2d ..Hencethereexists r> max f r 0 ;r 1 g suchthat 1 X r =0 q ( r ) 1 X r =0 dinf m (( r +1) d )= 1 : (4{32) and0 0,thereexists r 2 suchthatwhen r>r 2 ,both E [ V ( rd )] < and m 2 ( rd ) < .Similartothederivationfor( 4{39 ),wealso obtain, E [ V ( rd + ` )] I 2 `inf m (( rd + ` 1)+ m 2 ( rd ) Q ` E [ V ( rd )]+ F ` rd + ` 1 X j = rd m 2 ( j ) ; (4{33) where0 <` 0and0 <`r 2 suchthat E [ V ( rd + ` )] ( I + m 2 (0) Q ` ) + F ` ` =( I + m 2 (0) Q ` + F ` ` ) : (4{34) Togetherwith E [ V ( rd )] < ,weprove E [ V ( k )] 0as k !1 .Therefore, e u ( k )ismean squareconvergentto0forall u 2 V b 98

PAGE 99

4.7.2ProofofLemma 4.3 TheproofissimilartothatofLemma 4.1 ,therefore,wemainlyemphasizethe dierence.Notethat,thesequenceofvectors f ( k ) g isamartingaledierencew.r.t. F ( k ), where F ( k )isa -algebrageneratedby f e (0) ; (0) ;:::; ( k ) ; G (0) :::; G ( k ) g Proof. BythesameprocedureasthatinSection 4.7.1 ,weareabletondtheboundon V ( k )= e ( k ) T e ( k ), V (( r +1) d ) 1 2 m ( L b d 1 rd ) m (( r +1) d 1)+ m 2 ( rd ) Q d V ( rd ) +2 e T ( rd )( d 1 rd ) T d 1 rd +( d 1 rd ) T d 1 rd : (4{35) SincetheMarkovchain G ( k )isergodic,thereexists d> 0suchthat P ( G ( t + d )= j jG ( t )= i ) > 0forall i;j 2 G and t> 0.Let G d 1 rd = S rd + d 1 j = rd G ( j ),then P ( G d 1 rd isconnected) > 0forall r> 0.UsingthesameargumentthatisusedintheproofofLemma 4.1 E [ m ( L b d 1 rd ) jG d 1 rd isconnected] > 0.Therefore, E [ m ( L b d 1 rd )]= E [ m ( L b d 1 rd ) jG d 1 rd isconnected] P ( G d 1 rd isconnected) > 0(4{36) Since e ( rd )is F ( rd 1)-measurable, f ( k ) g ismartingaledierencesequencew.r.t. F ( k )and ( k )isindependentwith G ( j )forall k;j ,itfollowsthat E 2 e T ( rd )( d 1 rd ) T d 1 rd jF ( rd 1) =2 e T ( rd ) ( r +1) d 1 X j = rd E ( d 1 rd ) T d 1 j +1 m ( j ) D ( j ) jF ( rd 1) E [ ( j ) jF ( rd 1)]=0 ; (4{37) Bythetowerpropertyofconditionalexpectation, E 2 e T ( rd )( d 1 rd ) T d 1 rd =0.In addition,bypropertyofmartingaledierence, E [ ( i ) ( j ) T ]=0for i 6 = j .Thus, E ( d 1 rd ) T d 1 rd = ( r +1) d 1 X j = rd m 2 ( j ) E h ( j ) T D ( j ) T ( ( j ) j +1 ) T ( j ) j +1 D ( j ) ( j ) i F d ( r +1) d 1 X j = rd m 2 ( j ) ; (4{38) 99

PAGE 100

where ( j )=( r +1) d j 2and F d =sup j 0 E [ k ( j ) j +1 D ( j ) k 2 sup j 0 E [ k ( j ) k 2 ].Note thatlimsup k !1 m ( k ) =m ( k +1) < 1 ,sothereexists r>r 0 suchthat m ( rd ) s d m (( r + 1) d ),wherescalar s d isafunctionof d ,and m ( rd ) 1.Consequently, m 2 ( rd ) Q d m 2 (( r +1) d ) Q d s 2d for r>r 0 .Let dinf =inf t 0 ( E [ m ( L b d 1 t )]),then, E [ V (( r +1) d ] 1 2 dinf m (( r +1) d )+ m 2 (( r +1) d ) Q d s 2d E [ V ( rd )]+ F d ( r +1) d 1 X j = rd m 2 ( j ) : (4{39) Then,bythesameargumentasthatinSection 4.7.1 E [ V ( rd )] 0as r !1 .Consequently,forany > 0,thereexists r 2 suchthatwhen r>r 2 ,both E [ V ( rd )] < and m 2 ( rd ) < .Similartothederivationfor( 4{39 ),wealsoobtain, E [ V ( rd + ` )] I 2 E [ m ( L b `rd 1)] m (( rd + ` 1)+ m 2 ( rd ) Q ` E [ V ( rd )]+ F ` rd + ` 1 X j = rd m 2 ( j ) ; (4{40) where0 <` 0and0 <`r 2 suchthat E [ V ( rd + ` )] ( I + m 2 (0) Q ` ) + F ` ` =( I + m 2 (0) Q ` + F ` ` ) : (4{41) Togetherwith E [ V ( rd )] < ,weprove E [ V ( k )] 0as k !1 .Therefore, e u ( k )ismean squareconvergentto0forall u 2 V b 100

PAGE 101

CHAPTER5 CONCLUSIONANDFUTUREWORK Thischapterdiscussestheresultsineachchapterandopenproble msforfuture research. 5.1Conclusion Inasensornetwork,eachnodecanestimatetheglobaltimefromit slocaltimeas longasitknowsitsskewandoset(withrespecttotheglobalclock) .Therefore,the timesynchronizationprobleminasensornetworkistoestimatethec lockskewsand osets.Insensornetworks,thecommunicationcapabilityislimiteds othateachnodecan onlycommunicatewithitsneighbors.Consequently,theskewsando setsarerequired tobeestimateddistributively.Theproblemthatwepostisthatnode sdistributively estimateskewsandosetsusingrelativemeasurementsbylocalco mmunicationwiththeir neighbors.Ourfocusisonmobilenetworks. InChapter 2 ,weintroduceatwo-stepprocedureforpairsofnodestocomput erelativemeasurementsofskewsandosetsbypairwisecommunication. Thetwostepsarethe pairwisesynchronizationandthetransformationoftheresultsof pairwisesynchronization torelativemeasurements.Therststepenablesustoselectoneo fthemanyexisting algorithmssuchas[ 1 3 { 5 8 9 21 ]toperformpairwisesynchronization.Theadvantage ofthismethodisrexibility.Inthefuture,ifabetterpairwisesynchr onizationalgorithmis available,itcanbeusedtogeneraterelativemeasurements.Inthis dissertation,wechoose thepairwisesynchronizationalgorithmin[ 1 ]andparameterswithpracticalvalues,and computetheresultingrelativemeasurementsofskewandosetby simulation. Toperformskewandosetestimationinnetworks,wereformulate theskewand osetestimationproblemsasanodevariableestimationproblem.InC hapter 3 ,we proposeanalgorithm(JAT)todistributivelyestimatenodevariables fromnoisyrelative measurements(ofnodevariables),whichisverysimilartothedistrib utedestimation algorithmproposedin[ 6 { 10 ].Weanalyzetheconvergenceofthealgorithmwhenthe 101

PAGE 102

networktopologychangesduetothemotionofthenodes,whichisn otanalyzedinthe earlierpapers.Weassumethatthetime-varyingtopologyofthene tworkisgoverned byrstorderhomogeneousergodicMarkovchain.Weshowthatun derfairlyweak assumptionsonthechain,theproposedalgorithmismeansquarest ableifandonlyif theunionofthegraphsthatoccurisconnected.Whentherelative measurementsare unbiased,thelimitingmeanisthesameasthetruevalueofthevariable ,i.e.,theestimates obtainedareasymptoticallyunbiased.Furthermore,weshowthea lgorithmcanbe implementedasynchronously,whichmakesthealgorithmsuitablefor timesynchronization problem.Oneweaknessofthisalgorithmisthatthevarianceofthee stimateconvergesto anonzeroconstant.Therefore,theresultingglobaltimeestimat esbecomepoorastime increases.Thismotivatesustodevelopmoreaccuratealgorithms. Inchapter 4 ,weproposetheSTOalgorithmtosolvethenodevariableestimation problem.Weprovethattheestimationerrorconvergesinmeansqu aretoaconstant,i.e., themeanconvergestoalimitingvalueandthevarianceconvergesto zero.Furthermore, ifmeasurementnoiseiszeromean,theestimationerrorconverges inmeansquareto zero.ThisachievesmoreaccurateestimatesthanthatbytheJAT algorithm,inwhich varianceofestimationerrorconvergestosomenonzerovalue.Th ealgorithmisinspired bystochasticapproximationtypeconsensusprotocolsin[ 11 12 ],whichcandealwith theaccumulationofmeasurementnoise.Themeritofsuchconsens usalgorithmsisthat theestimationerrorismeansquareconvergent(highaccuracy). However,twoobstacles hinderustotakeadvantageofitdirectlyintimesynchronizationpro blem.First,the algorithmrequiresasynchronouslydecreasingstepsizeintheupda teformula.Otherwise, theconvergenceofthealgorithmisnotguaranteed.However,no deshavenoaccessto suchaglobalindexasclocksinnodesarenotevensynchronized.We proposeamethod toenablenodestoupdatethestepsizebasedontheirownclocksins uchawaythat theconvergenceanalysiscanstillbeconductedinacommonglobalt ime.Secondly,the measurementscanbebiased.Thisisnotcoveredintheanalysisprov idedin[ 11 12 ].We 102

PAGE 103

showthattheestimationerrorisstillmeansquareconvergent,an dprovideaformulato computethelimitingvalueofthemeanoftheestimationerror. WealsocomparethetwoproposedalgorithmswiththetheATSalgorit hminterms ofthesyncerror.WedemonstratethattheSTOalgorithmoutper formsJATintermsof bothglobaltimeestimationerrorandsyncerror,andoutperform sATSintermsofsync error(therstmetricisundenedforATS).Theconvergencera teoftheproposedalgorithmisslowerthanthatinATS.Weproposetwomethodstoimprovet heconvergence rateandverifytheimprovementthroughsimulations. 5.2FutureWork Twodistributedestimationalgorithmsareproposedtoestimateske wsandosets fortimesynchronizationinmobilesensornetworks.Bothalgorithms canbeimprovedin dierentaspects.5.2.1AsymmetricUpdate Insection 4.5.1 ,weshowedthatbyusingaveragedistancetothereferencenode ,the convergencerateofbothJATandSTOalgorithmscanbeimproved. Thisrequirespairs ofnodesupdateasymmetricallyateachiterationbecauseanodeon lyusestheestimates fromitsneighborsthathaveasmalleraveragedistancetotherefe rencenodethanitself. Thoughtheimprovementisvalidatedbysimulation,aconvergencean alysishasnot beenconduced.Althoughtheoriginalsucientconditionforconve rgenceanalysisisnot satisedanymore,webelieveconvergencecanstillbeachieved,as seeninsimulations. 5.2.2AsynchronousDecreaseoftheStepSize Itisrequiredbycurrentstochasticapproximationtypeconsensu salgorithmsthat thestepsizedecreasessynchronously.However,anasynchron ouslydecreasingstepsize wouldbebenecialfortheSTOalgorithmduetothreereasons.Firs t,inSection 4.5.2 theimprovementwouldbebetterifthe k 1 ischosenbasedonreal-timeestimates,which requiresasynchronousdecreaseofthestepsize.Secondly,ifth estepsizecandecease locallyonlywhenanodeupdates(insteadofbeinggovernedbythepr e-speciedsequence 103

PAGE 104

ofiterationintervals),webelieveitwouldimprovetheconvergencer ateaswell.Thirdly, asynchronousdecreaseinthestep-sizeremovestheneedforus ingthelocaliteration intervalsthatincreasealongtimewithoutbound.5.2.3ImplementationinRealSensorNetworks Althoughtheproposedalgorithmshavebeensimulatedusingparame tersthatwe believetobeclosetoreal-worldvalues,itisstillvaluabletoseehowthe sealgorithms performinarealsensornetwork.Thecomparisonwithotheralgor ithmswillbemore persuasiveinthiscaseaswell. 104

PAGE 105

REFERENCES [1]K.-L.Noh,Q.M.Chaudhari,E.Serpedin,andB.W.Suter,\Nove lclockphaseoset andskewestimationusingtwo-waytimingmessageexchangesforwir elesssensor networks," IEEETransactionsonCommunications ,vol.55,no.4,pp.766{777,Apr 2007. [2]S.Yoon,C.Veerarittiphan,andM.L.Sichitiu,\Tiny-sync:Tight timesynchronizationforwirelesssensornetworks," ACMTransactionsonSensorNetworks ,vol.3, no.2,pp.1{34,Jun2007. [3]K.-L.Noh,E.Serpedin,andK.Qaraqe,\Anewapproachfortime synchronizationin wirelesssensornetworks:Pairwisebroadcastsynchronization," IEEETransactionon WirelessCommunications ,vol.7,no.9,pp.3318{3322,Sep2008. [4]M.LengandY.-C.Wu,\Onclocksynchronizationalgorithmsforwir elesssensor networksunderunknowndelay," IEEETransactionsonVehicularTechnology ,vol.59, no.1,pp.182{190,Jan2010. [5]N.Freris,S.Graham,andP.Kumar,\Fundamentallimitsonsync hronizingclocks overnetworks," AutomaticControl,IEEETransactionson ,vol.56,no.6,pp.1352 {1364,june2011. [6]R.Karp,J.Elson,D.Estrin,andS.Shenker,\Optimalandgloba ltimesynchronizationinsensornets,"CenterforEmbeddedNetworkedSensing,Univ .ofCalifornia,Los Angeles,Tech.Rep.,2003. [7]P.BarooahandJ.P.Hespanha,\Estimationfromrelativemeasu rements:Error boundsfromelectricalanalogy,"in Proc.ofthe2ndInternationalConferenceon IntelligentSensingandInformationProcessing(ICISIP) ,January2005,pp.88{93. [8]A.GiridharandP.R.Kumar,\Distributedclocksynchronizationin wirelessnetworks:Algorithmsandanalysis(I),"in 45thIEEEConferenceonDecisonand Control ,December2006,pp.4915{4920. [9]R.Solis,V.S.Borkar,andP.R.Kumar,\Anewdistributedtimesy nchronization protocolformultihopwirelessnetworks,"in Proc.ofthe45thIEEEConferenceon DecisonandControl ,December2006,pp.2734{2739. [10]P.BarooahandJ.P.Hespanha,\Estimationfromrelativemeas urements:Electrical analogyandlargegraphs," IEEETransactiononSignalProcessing ,2008. [11]M.Huang,S.Dey,G.N.Nair,andJ.H.Manton,\Stochasticco nsensusovernoisy networkswithMarkovianandarbitraryswitches," Automatica ,vol.46,no.10,pp. 1571{1583,Oct.2010. [12]T.LiandJ.Zhang,\Consensusconditionsofmulti-agentsyst emswithtime-varying topologiesandstochasticcommunicationnoises," AutomaticControl,IEEETransactionson ,vol.55,no.9,pp.2043{2057,2010. 105

PAGE 106

[13]L.SchenatoandF.Fiorentin,\Averagetimesynch:Aconsens us-basedprotocolfor clocksynchronizationinwirelesssensornetwork," Automatica ,vol.47,no.9,pp.1878 {1886,2011. [14]P.Juang,H.Oki,Y.Wang,M.Martonosi,L.-S.Peh,andD.Rube nstein,\Energyefcientcomputingforwildlifetracking:Designtradeosandearlyex perienceswith zebranet,"in ASPLOS-Xconference ,October.2002. [15]J.KnuthandP.Barooah,\Distributedcooperativevision-bas edlocalizationofmobile robotteams,"in 47thAnnualAllertonConferenceonCommunication,Control and Computing ,September2009,pp.314{321. [16]J.ElsonandK.Rmer,\Wirelesssensornetworks:Anewregimef ortimesynchronization,"in INPROCEEDINGSOFTHEFIRSTWORKSHOPONHOTTOPICSIN NETWORKS(HOTNETS-I ,2002. [17]J.Elson,L.Girod,andD.Estrin,\Fine-grainednetworktimesy nchronizationusing referencebroadcasts,"in theFifthSymposiumonOperatingSystemsDesignand Implementation(OSDI) ,2002. [18]S.Ganeriwal,R.Kumar,andM.B.Srivastava,\Timing-syncpro tocolforsensor networks,"in ACMConferenceonEmbeddedNetworkedSensorSystems(SenSy s) 2003. [19]M.Maroti,B.Kusy,G.Simon,and A.Ledeczi,\Theroodingtimesynchronization protocol,"in ACMConferenceonEmbeddedNetworkedSensorSystems(SenSy s) 2004. [20]P.Barooah,N.M.daSilva,andJ.P.Hespanha,\Distributedop timalestimation fromrelativemeasurementsforlocalizationandtimesynchronizatio n,"in InternationalConferenceonDistributedComputinginSensorSys temsDCOSS'06 ,San Francisco,June2006. [21]N.Freris,V.Borkar,andP.Kumar,\Amodel-basedapproach toclocksynchronization,"in DecisionandControl,2009heldjointlywiththe200928thCh ineseControl Conference.CDC/CCC2009.Proceedingsofthe48thIEEEConf erenceon ,dec.2009, pp.5744{5749. [22]M.LengandY.-C.Wu,\Distributedclocksynchronizationforwir elesssensor networksusingbeliefpropagation," SignalProcessing,IEEETransactionson ,vol.59, no.11,pp.5404{5414,nov.2011. [23]J.R.Vig,\Introductiontoquartzfrequencystandards,"Ar myResearchLaboratory, Tech.Rep.,1992. [24]Q.LiandD.Rus,\Globalclocksynchronizationinsensornetwo rks," Computers, IEEETransactionson ,vol.55,no.2,pp.214{226,feb.2006. 106

PAGE 107

[25]B.M.SadlerandA.Swami,\synchronizationinsensornetworks :anoverview,"in IEEEMILCOM ,October2006,pp.1{6. [26]Y.-C.Wu,Q.Chaudhari,andE.Serpedin,\Clocksynchronizatio nofwirelesssensor networks," IEEESignalProcessingMagazine ,vol.28,no.1,pp.124{138,jan.2011. [27]R.CarliandS.Zampieri,\Networkedclocksynchronizationbas edonsecondorder linearconsensusalgorithms,"in DecisionandControl(CDC),201049thIEEE Conferenceon ,dec.2010,pp.7259{7264. [28]R.Carli,E.D'Elia,andS.Zampieri,\Apicontrollerbasedonasymm etricgossip communicationsforclockssynchronizationinwirelesssensorsnetw orks,"in DecisionandControlandEuropeanControlConference(CDC-ECC) ,201150thIEEE Conferenceon ,dec.2011,pp.7512{7517. [29]J.VanGreunenandJ.Rabaey,\Lightweighttimesynchronizat ionforsensornetworks,"in Proceedingsofthe2ndACMinternationalconferenceonWire lesssensor networksandapplications ,ser.WSNA'03.NewYork,NY,USA:ACM,2003,pp. 11{19. [30]Q.LiandD.Rus,\Globalclocksynchronizationinsensornetwo rks," IEEETransactionsonComputers ,vol.55,p.214226,2006. [31]M.LengandY.-C.Wu,\Distributedclocksynchronizationforwir elesssensor networksusingbeliefpropagation," IEEETransactionsonSignalProcessing ,vol.59, no.11,pp.5404{5414,November2011,onlyosetcorrection. [32]O.Costa,M.Fragoso,andR.Marques, Discrete-TimeMarkovJumpLinearSystems ser.ProbabilityanditsApplications.Springer,2004. [33]V.Gupta,B.Hassibi,andR.M.Murray,\Stabilityanalysisofsto chasticallyvarying formationsofdynamicagents,"in Proceedings.ofthe42ndIEEEConferenceon DecisionandControl ,vol.1,dec.2003,pp.504{509. [34]Y.ZhangandY.-P.Tian,\Consentabilityandprotocoldesignof multi-agentsystems withstochasticswitchingtopology," Automatica ,vol.45,no.5,pp.1195{1201,2009. [35]S.KarandJ.M.F.Moura,\Distributedconsensusalgorithmsin sensornetworks: Quantizeddataandrandomlinkfailures," IEEETransactionsonSignalProcessing vol.58,no.3,p.13831400,March2010. [36]T.Camp,J.Boleng,andV.Davies,\Asurveyofmobilitymodelsfo radhocnetwork research," WirelessCommunicationsandMobileComputing ,vol.2,no.5,pp.483{ 502,Aug2002. [37]C.Chateld,\StatisticalInferenceRegardingMarkovChainM odels," Applied Statistics ,vol.22,no.1,pp.7{20,1973. 107

PAGE 108

[38]V.Borkar, Stochasticapproximation:adynamicalsystemsviewpoint .Cambridge UniversityPress,2008. [39]H.Minc, NonnegativeMatrices .Wiley-Interscience,1988. [40]M.-Q.ChenandX.Li,\Anestimationofthespectralradiusofap roductofblock matrices," LinearAlgebraanditsApplications ,March2004. [41]C.D.Meyer, MatrixAnalysisandAppliedLinearAlgebra .SIAM:Societyfor IndustrialandAppliedMathematics,2001. [42]F.HararyandJ.Trauth,CharlesA.,\Connectednessofprod uctsoftwodirected graphs," SIAMJournalonAppliedMathematics ,vol.14,no.2,pp.pp.250{254,1966. [43]B.Yackley,E.Corona,andT.Lane,\Bayesiannetworkscore approximationusing ametagraphkernel21,"in AdvancesinNeuralInformationProcessingSystems D.Koller,D.Schuurmans,Y.Bengio,andL.Bottou,Eds.,2009,pp .1833{1840. [44]P.M.Weichsel,\Thekroneckerproductofgraphs," ProceedingsoftheAmerican MathematicalSociety ,vol.13,no.1,pp.pp.47{52,1962. [45]P.BarooahandJ.P.Hespanha,\Grapheectiveresistances anddistributedcontrol: Spectralpropertiesandapplications,"in Proc.ofthe45thIEEEConferenceon DecisionandControl ,December2006,pp.3479{3485. [46]L.Gyr,\Stochasticapproximationfromergodicsampleforline arregression," ProbabilityTheoryandRelatedFields ,vol.54,pp.47{55,1980. [47]D.Bernstein, Matrixmathematics:theory,facts,andformulas .PrincetonUniversity Press,2009. [48]M.Kouritzin,\Ontheconvergenceoflinearstochasticapprox imationprocedures," InformationTheory,IEEETransactionson ,vol.42,no.4,pp.1305{1309,jul1996. 108

PAGE 109

BIOGRAPHICALSKETCH ChendaLiaowasborninNovember,1984inMianyang,Sichuanprovinc e,China. HereceivedhisBachelorofEngineeringdegreeinautomationin2007f romZhejiang University,Hangzhou,China.Duringthefollowingoneyear,hestud iedinthemaster's programincontroltheoryandcontrolengineeringinShanghaiJia oTongUniversity, Shanghai,China.Infall2008,hejoinedtheDistributedControlSy stemlabinthe DepartmentofMechanicalandAerospaceEngineeringattheUnive rsityofFloridato pursuehisdoctoraldegreeundertheadvisementofDr.PrabirBa rooah. 109