Solution Manual For Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, 1st Edition

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||CONTENTS-—Introductionv1Linearsystemstheory12Probabilitytheory153Leastsquaresestimation294Propagationofstatesandcovariances415Thediscrete-timeKalmanfilter516AltemateKalmanfilterformulations657Kalmanfiltergeneralizations798Thecontinuous-timeKalmanfilter939Optimalsmoothing103iii|=n

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||ivCONTENTS10AdditionaltopicsinKalmanfiltering11311TheHyfilter12712AdditionaltopicsinHo,filtering14113NonlinearKalmanfiltering14914TheunscentedKalmanfilter16915Theparticlefilter179Sourcecodeforcomputerproblems193ll

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-_—OOOOO_oThissolutionmanualis&companiontothetextOptimalStateEstimation;Kalman,Hoo,andNonlinearApproaches,byDanSimon(JohnWiley&Sons,2006).TheMATLAB!sourcecodeforthecomputerexercisesolutionsisgivenattheendofthissolutionmanual.ThereferencesinthissolutionmanualrefertothereferencessectioninthetextOptimalStateEstimation.TheequationnumbersinthissolutionmanualrefertotheequationsinthebookOptimalStateEstimation.AlthoughtheMATLABcodeforthesolutionsisnotavailableogtheInternet,MATLAB-basedsourcecodefortheexamplesinthetextisavailableattheauthor'sWebsite?Theauthor'se-mailaddressisalsoavailableontheWebsite,andIeagerlyinvitefeedback,comments,suggestionsforimprovements,andcorrections.AnoteonnotationThreedotsbetweendelimiters(parenthesis,brackets,orbraces)meansthattheduantitybetweenthedelimitersisthesameasthequantitybetweentheprevioussetofidenticaldelimitersinthesameequation.Forexample,U+BEDY+(97=(A+BCD)+(A+BoD)™A+IBC-DIEL]=A+BDIE[B(C+D)]MATLABisaregisteredtrademarkofTheMathWorks,Inc,http:/facademic.csuohio,edu/simond/estimation—iftheWebsiteaddresschanges,itshouidbeeasytofindwithanInternetsearch.v

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CHAPTER1ee————————————————————————LinearsystemstheoryJProblemsWrittenexercises;foo1.1Findsherankofthematrix|o |-SolutionTherankofamatrixAcanbedefinedasthedimensionofthelargestsubmarrixconsistingofrowsandcolumnsofAwhosedeterminantisnonzero.Withthisdefinitionweseethattherankofthezeromatrixiszero.1.2Findtwo2x2matricesAandBsuchthatA#B,neitherAnorBarediagonal,4#cBforanyscalar¢,andAB=BA.Findtheeigenvectorsof4andB.Notethattheyshareaneigenvector,Interestingly,everypairofcommutingmatricessharesutleastoneeigenvector[Hor83,p.51].OptimalStateEstimation.SolutionManual,FirstEdition.ByDanJ.Simon1(©2006JohnWiley&Sons.Inc.—|—+StudyXY

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2LINEARSYSTEMSTHECRYSolutionSuppose4andBaregivenas_Jaarela]_hobB=[bzbyThenweseethatap—|@bitamhashy+ass-azby+aghy axby+ashyBA=arb+ashyashy+ashy.-@1by+ashyazby+ashyWeseethat48=BAifa;by+azbs=aby+aghy.Thiswillbetrue,forexample,Hay=10y=2a3=1,b=1,by=3andby=1.Thisgives12ao[st][13=0s?ForthesematricesAhastheeigenvalues—1and3,Bhastheeigenvalues—2and4,andboth4andBhovetheeigenvectors[~117and[11".1.3ProvethethreeidentitiesofEquation(1.26).Solutiona).SupposeAisannxmmatrix,andBisanmxpmatrix.ThenAnoim1]BaoBy1\T“py”=CoPoeAnorAnBri+BpTTAB;CAB,LABooYCAyBy|YAByooYAuBa|AGB3AwByyBu+BmAnnAmBTAT=FE:JE:Bip+BmpAimAgm|YBuAycoYLBud|Bp;oYBipdy,

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PROBLEMS3QEDb).Supposethat(AB)!=C.ThenCAB=I.PostmultiplyingbothsidesofthisequationbyB~!givesC4=B~!.PostmultiplyingbothsidesofthisequationbyA=!givesC=B~'A~1.Henceweseethat(AB)~>=B~1A~1,QEDc).SupposeAisannxmmatrix,andBisanmxnmatrix.ThenAnocAimBix+oBin(4B)=||:AEAuAumBry+BonSAB;crYoAyBie=Tr:.,:YCAnBycoTAnBinnom=33AB==(Bui+rBigAnooAim]Tr(BA}=Tr::FE:{Bry+++BunAnAumJBudcoLUBidim=Tr:iS:2BrjAjrcr2Bmidjmmn-Ys11QED1.4FindthepartialderivativeofthetraceofABwithrespecttoA.SolutionSuppose4isan7x7matrix,andBisanmxnmatrix.ThenTr(AB)8“arTAluni=15m1|ShrTimDimAubwhLiEiAu|ohSiShABwhUi0kABs|By+++BmBin+Bm=BT

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4LINEARSYSTEMSTHEORY1.5Considerthematrixa_[a®“lboeRecallthattheeigenvaluesofAarefoundbyfindtherootsofthepolynomialP(3)=|\=A].ShowthatP(A)=0.(ThisisanillustrationoftheCayley—Hamiltontheorem[Bay99,Che99,Kai00).)SolutionPy=A-4_lA-aTl=oa-e=M_(a+tcr+ac~b?P(A)=A*—(a+c)A+(ac—b)Ia?+6ab+beabac—0=[G18aIPhlEE_[ooloo1.6SupposethatAisinvertibleand4A174]JoBalciT1FindBandCintermsofA[Lie67].SolutionMultiplyingoutthematrixequationgivesthefollowingtwoequations.ALAC=0BA+AC=1SolvingforBandCintermsof4givesB=A+a!C=-AL.7ShowthatABmaynotbesymmetriceventhoughbothAandBaresym-metric.SolutionSupposethesymmetricmatricesAandBaregivenas4=aeyagazJEE

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PROBLEMS5ThenweseethatAp—|Gbitashyaby+ashyashy+aghy azbs+ashyABisnotsymmetricifaybz+a2bs#azby+aba.1.8ConsiderthematrixaceeTleoewherea,b,and¢arereal,andaand¢arenonnegative.a)ComputethesolutionsofthecharacteristicpolynomialofAtoprovethattheeigenvaluesofAarereal.b)ForwhatvaluesofbisApositivesemidefinite?Solutiona).ThecharacteristicpolynomialofAisPQ)=I-A]_r-a—b-|—bA-c=M-(a+r+ac—bFindingLherootsofthisgives15A=3[aesviasc)?+47Thediscriminantisnon-negativesoAisreal.QEDb).InorderforAtobepositivesemidefinite,itseigenvaluesmustbepositive.TheeigenvaluesareH[averNicerA=3faseNicerThefirsteigenvalueisalwaysnon-negative.Thesecondeigenvaluejsnon-negativeifa+¢>\/(e—cj?+46%.Solvingthisequdtiongives|b]<v/acastheconditionofpositivesemidefiniteness.1.9DerivethepropertiesofthestatetransitionmatrixgiveninEquation{1.72)Solutiond=(At)EEOitf|Vv|TTStudyXY

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6LINEARSYSTEMSTHEORY_d(41)?(4?=5[roar2FtAg?=AranACL2At)?=AlvaBEL]2!=Ae™Thisprovesthefirstequality.Afterwritingthethirdexpressionoftheabovesequenceofequations,wecanbringthecommonfactorAouttotherighttoobtaindat(At)?“4,=At2207FrI+At+5+A=etyQED1.10SupposethatthematrixAhaseigenvaluesA;andeigenvectorsv;(i=1,.-+,n).Whataretheeigenvaluesandeigenvectorsof—A?SolutionAv;=Ay,therefore—Av;=—Mw;Fromthiswescethat—4haseigenvalues—A;andeigenvectorsv;.1.11Showthat|e**|=eMforanysquarematrixASolution:A?ett=Toate=00.A%?led=rjAEA?=inpapsAPE=elkQED1.12ShowthatifA=BA.then414]——=Tr(B)|4=TimaSolution:TheequationA=BAcanbesolvedasA=€8*.4(0).Takingthedeterminantofthisequationgives[Al=[Aw

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PROBLEMS7=[e™lA@)]=FA)Fromthisweseethat4484«|B11)4¢0)Fr[Blel?F|A(0)]=TrB)A|QED1.13ThelinearpositionpofanobjectunderconstantaccelerationisLl,P=potpttGitwherepoistheinitielpositionoftheobject.a)Defineastatevectorasz=[ppp|"andwritethestatespaceequa-tion&=Azforthissystem.b)UseallthreeexpressionsinEquation(1.71)tofindthestatetransitionmatrixeA?forthesystem.©)Proveforthestatetransitionmatrixfoundabovethat¢4%=I.Solutiona).ar010plp=joo01pB000sb).FromthefirstexpressioninEquation(1.72)weobtainA_esaPa=o(antar(An?(an=othtmtramtoot000#2=1+|00t+000|+0640000001t2/2=lo:¢001FromthesecondexpressioninEquation{1.72)weobtaineMo=L7(sI-A)7Ys-1017"=cto5s1000os

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8LINEARSYSTEMSTHEORY1s1/s*1/8?=£700ls1s?001s122=[01¢00FromthethirdexpressioninEquation(1.72)weobtainAtQettgtTheeigendataof4arefoundtobeA={00,0}10[iv=of,lt],l0001ActuallywecannotethatAisalreadyinJordanform,whichmeansthatitseigenvaluesareonthedisgonal,anditseigenvectorsformtheidentitymatrixwhenaugmentedtogether.RecallforathirdorderJordanblockthatSophLoneo)0etfed00ertInourcaseA=0so1ot$82ef=101001©).Fromtheaboveexpressionfor4,ifwesubstitute¢=0weseethatct—I.QED1.14Considerthefollowingsystemmatrix.10r=[o5]Showthatthematrixsme]0To2satisfiestherelation$2)=A(t),butS(¢)isnotthestatetransitionmatrixofthesystem.Solutionsip=[dOEE02

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PROBLEMS.9As—|100e&0@=021]]oze_[et0“lo-2tWeseethat$(¢)=AS(t).However,thestatetransitionmatrixisfoundtobeet0t=[0et|QED1.15Giveanexampleofadiscrete-timesystemthatismarginallystablebutnotasymptoticallystable.SolutionThesystem7341=Tjismarginallystable,becausethestateisboundedforanyinitialboundedstate,butitisnotasymptoticallystable,becauseitisnottruethatthestateapproacheszeroforallinitialstates.1.16Show(H,Fisanobservablematrixpairifandonlyif(H,F~*)isobservable(assumingthatFisnonsingular).SolutionIf(H.F)isobservable,thenQx#0forallnonzerox,whereoHrQ=.HE!SinceFisnonsingularF~®=1zspanstheentiren-dimensionalspace.(Thatis,anyn-element.vectorcanbewrittenasF~"~9zforsomen-elementvectorz.)Sotheobservabilityof(H,F)isequivalenttoQF"=z20forallnonzeroz.ThisisequivalenttoQ'z#0forallnonzerox,whereHF=n=UHF=(n=2)Q=.Hwhichistheobservabilitymatrixof(H,1).QEDComputerexercises1.17ThedynamicsofaDCmotorcanbedescribedasJ§+F§=T

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10LINEARSYSTEMSTHEORY.where6istheangularpositionofthemotor,Jisthemomentofinertia,Fisthecoefficient.ofviscousfriction,ardTisthetorqueappliedtothemotora)Generateatwo-statelinearsystemequationforthismotorintheformi=Az+Bub)Simulatethesystemfor5sandplottheangularpositionandvelocity.UseJ=10kgw?,F=100kgm*/s,2(@)={00%andT=10Nm.Userectangularintegrationwithastepsizeof0.05s.Dotheoutputplotslookcorrect?WhathappenswhenyouchangethestepsizeAtto0.287Whathappenswhenyouchangethestepsizeto0.5s?WhataretheeigenvaluesoftheAmatrix,andhowcanyourelatetheirmagnitudestothestepsizethatisrequiredforacorrectsimulation?Solutiona).Let2;=0and74=6.Then[010[5Laerb).OutputplotsforvarioussimulationstepsizesareshowninFigures1.1-1.3,WithAt=0.05thesimulationworksfine.WithAt=0.2thesimulationresultsareobviouslyincorrect,althoughthesimulationissillstable.WithAt=0.5thesimulationblowsup.TheeigenvaluesofAare0and—10.Thesimuistionstepsizeshouldbeappreciablysmallerthan1/(Alugs,whichimpliesthatthestepsizeshouldbesmallerthan1/10.whichisconsistentwithourexperimentalresults.dt=0.050.5position0.45velocity0.4]0.350.3]0.25)0.20.15c10.05%12345Seconds.Figure1.1Problem1,17simulationwithAf—0.05.Goodsimulation1.18ThedynamicequationsforaseriesRICcircuitcanbewrittenasw=IR+LI+V,

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PROBLEMS11ai=020.5——positionc4025]030250.20.15)0.10050o12345SecondsFigure1.2Problem1.17simulationwithAf=0.2.Poorsimulation(x10&=05——positon2|velocityOf2]By|6h|_8-|~10rjial2%12345SecondsFigure1.3Problem1.17simulationwithAt=0.5.UnstablesimulationI=ov,whereuistheappliedvoltage,1isthecurrentthroughthecircuit,andV,isthevoltageacrossthecapacitor.a)Writeastate-spaceequationinmatrixformforthissystemwithryasthecapacitorvoltageand25asthecurrent,b)SupposethatR=3,I=1,andC=0.5.Pindananalyticalexpressionforthecapacitorvoltagefor20,assumingthattheinitialstateiszero,andtheinputvoltageisu(t)=¢~2,c)Simulatethesystemusingrectangular,trapezoidal,andfourth-orderRunge—Kuttaintegrationtoobtainanumericalsolutionforthecapacitorvoltage.

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