Full Paper Asymptotic Effectiveness of the Event-B

Sensors2007,7,16-37sensorsISSN1424-8220©2007byMDPIów,Poland;Email:[e-mail]:10August2006/Accepted:5January2007/6January2007Abstract:Arapidprogressinintelligentsensingtechnologycreatesnewinterestinadevelopmentofanalysisanddesignofnon-conventionalsamplingschemes.Theinvestigationoftheevent-basedsamplingaccordingtotheintegralcriterionispresentedinthispaper.Theinvestigatedsamplingschemeisanextensionofthepurelinearsend-on-delta/level-crossingalgorithmutilizedforreportingthestateofobjectsmonitoredbyintelligentsensors.Themotivationofusingtheevent-basedintegralsamplingisoutlined.Therelatedworksinadaptivesamplingaresummarized.Theanalyticalclosed-formformulasfortheevaluationofthemeanrateofevent-basedtraffic,andtheasymptoticintegralsamplingeffectiveness,arederived.Thesimulationresultsverifyingtheanalyticalformulasarereported.Theeffectivenessoftheintegralsamplingiscomparedwiththerelatedlinearsend-on-delta/level-crossingscheme.Thecalculationoftheasymptoticeffectivenessforcommonsignals,whichmodelthestateevolutionofdynamicsystemsintime,isexemplified.Keywords:dataacquisition,signalsampling,samplingmethods,sampled-datasystems.1.IntroductionIntelligentsensorsaresophisticatedsensingdevicespossessingadvancedfunctionalityintermsofinformationprocessing,reliability,datafusionandintegration.Theintelligentsensorisessentiallyanevent-basedsystemintendedtodetectspecifiedeventsofinterestinasensorfield.Therefore,informationfromintelligentsensorsisusuallyrepresentativeofalargeclassofsystemobservationswhicharecategorizedasirregularobservations[1].Evenifsensorreportsaretransmittedperiodically,theymaynotappeartobeperiodicatthedestinationduetotime-varyingdeliverydelaysorSensors2007,717transmissionerrorshandlinginvariousnetworkprotocols.Moreover,ingeneralthestateoftheobjectsbeingmonitoredevolvesirregularlyintime.Toavoidalossofvaluableinformationbetweensamplinginstants,thesamplingfrequencyinperiodicreportinghastobesetonabasisofpredictingtheworst-casesignalchangeability.However,settingasamplingratebytakingintoaccounttheworst-caseconditionsinastatevariationismoreorlesswastefulsincenumeroususelesssamplesaretaken.Actually,manysignalsinsensoryapplications(e.g.temperaturesensors,speechsignals,electrocardiograms,etc.)showburstystatisticalproperties,i.e.thesesignalsareconstantformostofthetimeandmayvarysignificantlyonlyduringshorttimeintervals.Theirregularobservationsaredefinedtobemessage-basedobservationswithpossiblyinfrequentandnon-periodicmeasurements,wherethemessageindicatesthesourceandthedestination(s),thetimestampofasample,therangeofpossiblemeasurementvalues,andtheprotocolservicespecification(e.g.messageservicetype,optionalauthentication,ornetworktrafficprediction)[1].Forexample,LonWorks/EIA-709technology,thatiswidelyusedfortheinterconnectionofintelligentsensor/actuatordevices,providesprogrammingobjectscalledtheStandardNetworkVariableTypes(SNVT)specifying,bytheconvention,variouscommonphysicalmagnitudes(voltage,temperature,etc.),theirrange,unit,andresolution[2].Aspecialclassofirregularobservationsisconstitutedbytheevent-basedsamplingschemes.Thisclassischaracterizedbythefunctionalrelationshipbetweenthesamplinginstantsandsignalbehavior.Thus,theevent-basedsamplingissignal-dependentwhereastheirregularsamplingencompassesalsosignal-independentschemes.Intheevent-basedsampling,thesignalissampledwhenthesignificanteventoccurs(i.e.asignificantchangeofitsstateisnoted)[13].Themostnaturalsignal-dependentsamplingstrategyisthesend-on-deltaalgorithm[3-6],knownalsoasthelevel-crossingsampling[7,8]ordeadbandsconcept[9].Send-on-deltareportingisused,amongothers,byintelligentsensornodesinLonWorksnetworkedcontrolsystems[10].Accordingtothesend-on-deltastrategy,thesensornodedoesnotbroadcastanewmessageiftheinputsignalremainswithinacertainintervalofconfidence.Thehardwaredesignsoflevel-crossingsamplersequippedwiththelow-powerasynchronousanalog-to-digitalconvertersarediscussedin[7,8].Suchconvertersarenotcontrolledbyanyglobalclock,butareenslavedbytheanaloginputsignal.Furthermore,thetimeintervalsbetweenconsecutiveamplitude-relatedeventsarequantizedinsteadofthesignalamplitude[8].Theasynchronousanalog-to-digitalconvertersestablishanewresearchareaespeciallyattractiveforwirelesssensortechnology.Althoughthemethodsofanalysisanddesignforperiodicallysampledsystemsismature,theevent-basedsystemstheoryis,asAströmandBernhardssonpointedout,stillinitsinfancy[6]withrelativelyfewworksstudyingseveralaspectsofevent-basedfunctionality,seee.g.[5,6,9-13].Generalizingthesend-on-delta/level-crossingconcept,theevent-basedstrategyallowsonetosampleinadomainofvarioussignalmeasures.Thespecialclassesarewhatisknownasconventionaluniformsamplinginatimedomain(periodicsampling)anduniformsamplinginavaluedomain(level-crossingsampling/send-on-delta).However,bytheappropriatedefinitionoftrig