Table Of ContentTracking Filter
Engineering
The Gauss–Newton and
Polynomial Filters
Norman Morrison
TheInstitutionofEngineeringandTechnology
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Why this book?
FormanyyearstheKalmanfilterhasbeentheacceptedapproachtotrackingfilter
engineering.
Thepurposeofthisbookistopresentanalternateapproach,onebasedontheGauss–
Newtonfilterswhichwebelieveoffercertainimportantadvantages.Wealsoprovide
afulldiscussionofthepolynomialfilters.
◊ InChapters1to7wecoverthenecessarybackground.
◊ InChapters8and9wediscussthedetailsoftheGauss–Newtonfilters.
◊ Chapter10isdevotedtotheMasterControlAlgorithms(MCAs)thatareusedto
controltheGauss–Newtonfilterswhenmanoeuvringtargetsarebeingtracked.
◊ InChapter11weprovideabriefdiscussionontheKalmanandSwerlingfilters.
◊ Chapters12and13aredevotedtothepolynomialfilters.
Welcome, then, to tracking filter engineering based on the Gauss–Newton and
polynomialfilters.
Organisation
As shown below, the material comes in two parts – the printed part and the
supplementarymaterial.
Printedpart Supplementarymaterial
Preface Computerprograms
Contents Miscellaneous
Chapters1to13 Problemsandsolutions
andtheirappendices
Projects
References
Videoclips
Index
Theprintedpartisthebookthatyouarenowreading.Itiscompletelyself-contained
andispublishedbytheInstitutionofEngineeringandTechnology(IET).
Thesupplementarymaterialisforthosereaderswhowishtoexplorethesubjectin
greaterdepth.
PleasenotethattheIETisnotresponsibleforanyofthesupplementarymaterial,and
thattheauthorhasthesoleresponsibilityforitscreationandmaintenance.
ThesupplementarymaterialishousedintheUniversityofCapeTown’sOpenAccess
Databaseandcanberetrievedfromthefollowing(case-sensitive)URL:
http://bit.ly/MorrisonSSP
Thematerialcanalsoberetrievedfromthefollowing(alsocase-sensitive)URL:
http://goo.gl/DD82m
Kindlysendtheauthoranemailatthefollowingaddress
norman.morrison@uct.ac.za
Organisation 3
sothathecannotifyyouwhenanewreleaseofthesupplementarymaterialbecomes
available. Please include the words Tracking Filter Engineering in the title of
youremail.
(cid:1)(cid:2)
Thesupplementarymaterialconsistsofthefollowing:
◊ Computerprograms:Fourteencomputerprograms,allofwhicheitherverifyor
demonstratesomethingimportantthatisdiscussedinthebook.Includedareboth
theexecutablesandthesourcecodelistings.Documentationforrunningeachof
the programs has been included in the form of a short Quickstart and also, in
manycases,amoredetailedReadme.
◊ Miscellaneous:Containsanumberoffiles,executablesanddocumentsthatare
referredtoasneeded.
◊ Problemsandsolutions:Containssetsofproblemsforeachofthe13chapters.
These enable you to test your understanding of the concepts discussed there,
and some of the problems also include further theoretical material. Complete
solutionstotheproblemshavebeenincluded.
◊ Projects:Containsprojectsformostofthe13chapters. Someoftheseinvolve
writing short computer programs and others involve reading the material that
hasbeenprovided.Allofthemareintendedtodeepenyourunderstandingofthe
materialcontainedintheprintedchapters.
◊ Video clips: Contains two sets of video clips, one set with, and the other set
withoutMCAcontrol.(ThelatterisdiscussedinChapter10.)ThesetwithMCA
controlshowstheoutputofaGauss–Newtonfilterthatisbeingusedtotracka
varietyofmanoeuvringtargetssuchasfixed-wingaircraft,helicoptersandcruise
missiles. Intheprefacewesuggestedthatyouviewthetrack-while-scanvideo
clips(thesearetheoneswithMCAcontrol)beforedoinganyfurtherreading.If
youmissedthat,thenperhapsnowwouldbeagoodtimetodoso.
However, first please read the document Video_Clips\TWS\Documents\
Readme.pdf so that you understand fully what you will be viewing.Then run
thevideoclipsinthefolderVideo_Clips\TWS\Flights.
Preface
Why would anyone write a book about Gauss–Newton tracking filters? There are
manyreasons.Herearejustafew.
◊ TheKalmanfilterhasbecomeanextremelycomplextopicwithabewildering
numberofmutations.Bycontrast,Gauss–Newtonfiltersareeasytounderstand
and easy to implement. What we are offering here is a simpler, more practi-
calapproachtotrackingfilterengineeringthatwillappealtobothexperienced
practitionersandtonewcomers.
◊ The Gauss–Newton filters work, and they work extremely well. We and our
colleagues have established that – by using them in a variety of operational
hardwareaswellasinsimulations.
◊ Once launched, Kalman filters1 have only one type of memory – expanding –
somethingthatisimpossibletochange.
Bycontrast,theGauss–Newtonfilterscanhavefixedlength,variablelengthor
expandinglengthmemoriesandanyofthesecanbechangedatwillduringfilter
operation.
TheGauss–Newtonfiltermodelscanalsobechangedduringfilteroperation–
ifneedbefromonecycletothenext.
Thisabilitytochangetheirmemorylengthsandfiltermodelsopensuppower-
fulnewwaysbywhichtotrackmanoeuvringtargets, waysthataredifficultto
implementwiththeKalmanfilters.
◊ The Gauss–Newton filters can be used when the observations are stage-wise
correlated,somethingthatcannotbedonewiththeKalmanfilters,andwhileit
istruethatobservationsinfilterengineeringareoftenstage-wiseuncorrelated,
thatisnotalwaysthecase.
1Thewords‘Kalmanfilter’meandifferentthingstodifferentpeople.Whenweusethemhereweare
referringspecificallytothemainstreamversionappearinginReferences3,6,22,23,24,25,30,31,48,
50,53,54,55,56,57,173and174.TheequationsofthemainstreamversionappearinAppendix11.1.
xviii Trackingfilterengineering:theGauss–Newtonandpolynomialfilters
◊ Perhapsmostimportantofall,theGauss–Newtonfiltersarenotvulnerabletothe
instabilityproblemsthatplaguetheextendedKalmanfilters–inwhich,unpre-
dictably, the estimation errors become inconsistent with the filter’s covariance
matrix.
The Gauss–Newton filters therefore do not require what is known in Kalman
parlanceas‘tuning’–searchingforandincludingaQmatrixthatonehopeswill
preventsomethingthatisbothunpredictableandalmostinexplicable,andinthe
processdegradingtheaccuracyoftheKalmanestimate.
ThisinturnmeansthatinpracticetheGauss–Newtonfiltersaremoreaccurate,
andthattheyareeasiertoimplement,requireshorterprojecttime,areextremely
robustand,bestofall,unconditionallystable.
Andwhatcanonesayaboutexecutiontime?
Using Gauss–Newton on a 1958-era machine, the extraction of a satellite’s orbital
parameters from a set of radar observations would have added about one hour of
processingtimeaftertheobservationshadbeenobtained.
However, on today’s readily available desk-top or lap-top machines, that hour has
shrunktoafewhundredmilliseconds,andascomputerperformanceandothertech-
nologiessuchasFPGAcontinuetoadvance,itissafetosaythatinthenottoodistant
futureitwillhaveshrunktotensofmillisecondsorevenless.
(cid:1)(cid:2)
As envisaged here, tracking filters are computer programs that extract estimates of
the states of dynamic processes in near real-time from sequences of discrete real
observations.
Trackingfilterengineeringisthecollectionoftechniques,knowledgeandactivities
thatweemploywhenwecreateandoperatesuchfilters.
Thefollowingisapartiallistoffieldsinwhichtrackingfiltersareused:
Air-traffic control, artificial intelligence, astronomy, atmospheric re-entry,
ballistic missile defence, chemical engineering, civil engineering, control
engineering,econometrics,electricalengineering,GPSandWAAS,industrial
engineering,inertialnavigation,mechanicalengineering,missileengineering,
neural networks, physics, pilotless-aircraft engineering, radar and tracking,
robotics,satelliteengineering,spacenavigation,statistics,telecommunications,
telescopeengineeringandwind-powerengineering.
Preface xix
Thisbookwillbeofinteresttothreegroupsofpeople:
◊ Practitionersworkingintheaforementionedorsimilarfields.
◊ Graduate-levelnewcomerswishingtolearnaboutGauss–Newtonandpolynomial
filters,andhowtheycanbeusedinfilterengineering.
◊ UniversitylecturerswhomightwishtoincludematerialonGauss–Newtonand
polynomialfiltersingraduate-levelcoursesontrackingfilterengineering.
(cid:1)(cid:2)
Tracking filter engineering came into existence around 1958 at the start of the
SatelliteEra.2
Engineersandscientistshadlongbeenfittingcurvestodatabyleastsquares,butthis
wasdifferent.Nowtherewerehugetrackingradarsmakingobservationsonartificial
satellites,fromwhichestimatesoforbitalparameters–knownasKeplerians–were
tobeextracted,andthefilteringalgorithmsfordoingsowerefarmoresophisticated
thansimpleleastsquares.
Astronomers had been estimating Keplerians of objects in orbit since the time of
Gauss, usingwhatwasknownastheGauss–Newtonalgorithm. However, suchcal-
culationsweredonebyhandandittookmanymonthstoobtainresults,whereaswith
artificial satellites the need was for the observations to be processed by computers
andfortheestimatestobeavailableinnearreal-time.
RunningGauss–Newtonona1958-eramachinewouldhavetakenroughlyanhour
to extract a satellite’s Keplerians from a set of radar observations after the latter
hadbeenobtained.Suchadelaywasclearlytoofarremovedfromreal-time,andso
Gauss–Newtoncouldnotbeused.
Initsplacetwonewfiltersweredevised–bothrelatedtoGauss–Newtonbutcom-
putationallydifferent.ThefirstofthesewaspublishedbySwerlingin19583andthe
secondbyKalman4andBucy5in1960/61,bothofwhichcouldextractKepleriansin
nearreal-timeontheexistingmachines.6
Starting from the extraction of Keplerians, use of the Kalman filter spread rapidly
tothemanyotherfieldslistedabovewheretodayitoccupiesthedominantplacein
2Sputnik-1wasplacedinorbitbytheSovietUniononOctober4th,1957.SeeReference40.
3PeterSwerling(Americanmathematicianandradartheoretician), 1929–2000. SeeReferences7, 8,
9,91.
4RudolfEmilKalman(Hungarian-bornAmericanscientist),b.1930.SeeReferences1,2,92.
5RichardS.Bucy(Americanmathematician),b.1935.SeeReference93.
6SeeChapter11forwhatwemeanby‘Swerlingfilter’and‘Kalmanfilter’.
xx Trackingfilterengineering:theGauss–Newtonandpolynomialfilters
filterengineering.TheKalmanfilterhasalsocometooccupythedominantplacein
filter-engineeringcurriculathroughouttheacademicworld.
(cid:1)(cid:2)
Butwhileallofthiswastakingplace,somethingelsewasalsohappening.
Since1958,computertechnologyhasmadeadvancesthataredifficulttocomprehend,
and the end is still nowhere in sight. Consider for example the following item that
appearedontheIntelcorporatewebsite:
‘Intel remains at the forefront of Moore’s Law. Our 22 nanometer technology-
basedIntelmicroprocessorswillenablenever-before-seenlevelsofperformance,
capability,andenergy-efficiencyinarangeofcomputingdevices.’
Andso,whileitmayhavebeenappropriatetorejectGauss–Newtonin1958,itisno
longerappropriatetodosotoday.Inthisbookweaccordinglydosomethingdifferent.
Many books have been written about the Kalman filter – the author is aware of at
least fifteen. However, to our knowledge nobody has yet written a book about the
Gauss–Newton filters and their remarkable fit to certain important areas of filter
engineering.
This then is our attempt to do so, and to present those filters in a readable and
self-containedway.
(cid:1)(cid:2)
TheGauss–Newtonfilterspossessattributesthatmakethemparticularlywell-suited
foruseintrackingfilterengineering,amongthemthefollowing:
◊ Idealfortrackingbothmanoeuvringandnon-manoeuvringtargets.
◊ Canbeusedwhentheobservationsarestage-wisecorrelated.
◊ ArenotvulnerabletotheinstabilityphenomenathatplagueboththeKalmanand
Swerlingextendedfilters,andhencerequirenotuning.
◊ Requirenoinitializationintheall-linearcaseandverylittleinthethreenonlinear
cases.
◊ Possesstotalflexibilitywithregardto
◊ Memories that can be configured as fixed length, variable length or
expandinglength–andifneedbe,reconfiguredcyclebycycle.
◊ Filtermodelsthatcanalsobereconfigured,ifneedbe,fromcyclebycycle.
Preface xxi
◊ Offerimmediateaccesstotheresidualswhichcanthenbeusedtorungoodness-
of-fittests.7
◊ Unconditionally stable, meaning that they are always error/covariance-matrix
consistent.8
◊ Cramér–Rao consistent, meaning that their accuracy is the best that it can
possiblybe.
◊ Canemploynonlinearmodelsofalmostanycomplexity.
◊ Nonrecursive,andhencedevoidofinternaldynamics.
◊ Easytoimplement.
◊ Extremelyrobust.
Thetechniquesproposedinthisbookhaveallbeentestedinsimulations,andmany
ofthemhavebeenusedinoperationalhardwarebuiltbyindustrialorganizationswith
whichtheauthorhasbeenassociated.
Theengineerswhocreatedthefiltersforthoseapplicationshavemadeitknownhow
easyitwastodothat,andhowrobustweretheresultingfilters.
(cid:1)(cid:2)
RegardingGauss–Newtonexecutiontimeontoday’smachines:
Wehavecarriedoutsimulator-basedtimingstudiesinwhichtheGauss–Newtonfilters
were used to track a mix of manoeuvring targets such as high-performance mili-
tary jets, cruise missiles and helicopters. We found that readily available modern
computers–eitherdesktoporlaptop–canreadilyhandletheloadimposedbythese
filterswhiletrackingmanysuchtargets, includingalloftheoverheadcalculations,
namelytrackformation,trackdeletionandplot-to-trackallocationusingtheauction
algorithm.
Indeed,dependingonthelevelofradarclutter,ourstudieshavesuggestedthatcon-
currenttrackingofamixofpossiblyasmanyasfiftymanoeuvringtargetsofthiskind
canbeaccomplishedonsuchcomputersusingthesefilters,andasprocessingpower
continuestoincrease,theconservativefigureoffiftywillcontinuetorise.
7Goodness-of-fittestsareusedbytheMasterControlAlgorithms(MCAs)tocontrolaGaussfilter’s
memorylengthandfiltermodelwhentrackingmanoeuvringtargets,insuchawaythatitsperformanceis
alwaysoptimal.TheMCAsarediscussedinChapter10.
8Error/covariance-matrix(ECM)consistencyisdiscussedinChapter1.WhenaKalmanorSwerlingfilter
becomesunstablethenitisECMconsistencythatisabsent,aconditionwhichisfatal.
