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1Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
Chapter 5 – Seakeeping Theory
5.1HydrodynamicConceptsandPotentialTheory5.2SeakeepingandManeuveringKinematics5.3TheClassicalFrequency-DomainModel5.4Time-DomainModelsincludingFluidMemoryEffects
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2Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
Chapter 5 - Seakeeping TheoryEquationsofMotionSeakeepingtheoryisformulatedinequilibrium(SEAKEEPING)axes{s}butitcanbe
transformedtoBODYaxes{b}byincludingfluidmemoryeffectsrepresentedbyimpulseresponsefunctions.
Thetransformationisisdonewithinalinearframeworksuchthatadditionalnonlinearviscousdampingmustbeaddedinthetime-domainundertheassumptionoflinearsuperposition.
μisanadditionaltermrepresentingthefluidmemoryeffects.
Inertia forces: !MRB !MA" "! ! CRB#!$! ! CA#!r$!r
Damping forces: !#Dp ! DV$!r ! Dn#!r$!r ! "
Restoring forces: !g##$ ! goWind and wave forces: # $wind ! $wave
Propulsion forces: !$
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3Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
StripTheory(2-DPotentialTheory)Forslenderbodies,themotionofthefluidcanbeformulatedasa2-Dproblem.Anaccurateestimateofthehydrodynamicforcescanbeobtainedbyapplyingstriptheory(Newman,1977;Faltinsen,1990;Journee andMassie,2001).
The2-Dtheorytakesintoaccountthatvariationoftheflowinthecross-directionalplaneismuchlargerthanthevariationinthelongitudinaldirectionoftheship.
Theprincipleofstriptheoryinvolvesdividingthesubmergedpartofthecraftintoafinitenumberofstrips.Hence,2-Dhydrodynamiccoefficientsforaddedmasscanbecomputedforeachstripandthensummedoverthelengthofthebodytoyieldthe3-Dcoefficients.
CommercialCodes:MARINTEK(ShipX-Veres)andAmarcon (OctopusOffice)
5.1 Hydrodynamic Concepts and Potential Theory
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4Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
ShipX (VERES) by MARINTEK
VERES- VEssel RESponse program isaStripTheoryProgram whichcalculateswave-inducedloadsonandmotionsofmono-hullsandbargesindeeptoveryshallowwater.TheprogramisbasedonthefamouspaperbySalvesen,TuckandFaltinsen (1970).ShipMotionsandSeaLoads.Trans.SNAME.
MARINTEK- theNorwegianMarineTechnologyResearchInstitute- doesresearchanddevelopmentinthemaritimesectorforindustryandthepublicsector.TheInstitutedevelopsandverifiestechnologicalsolutionsfortheshippingandmaritimeequipmentindustriesandforoffshorepetroleumproduction.
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5Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
ShipX (Veres)
ShipX (VERES) by MARINTEK
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6Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
OCTOPUS SEAWAY by Amarcon
andAMARCONcooperateinfurtherdevelopmentofSEAWAY
SEAWAY isdevelopedbyProfessorJ.M.J.Journée attheDelftUniviversity ofTechnology
SEAWAY isaStripTheoryProgram tocalculatewave-inducedloadsonandmotionsofmono-hullsandbargesindeeptoveryshallowwater.Whennotaccountingforinteractioneffectsbetweenthehulls,alsocatamaranscanbeanalyzed.Workofveryacknowledgedhydromechanicscientists(suchas Ursell,Tasai,Frank,Keil,Newman,Faltinsen,Ikeda,etc.)hasbeenused,whendevelopingthiscode.
SEAWAY hasextensivelybeenverifiedandvalidatedusingothercomputercodesandexperimentaldata.
TheMaritimeResearchInstituteNetherlands(MARIN)andAMARCONagreetocooperateinfurtherdevelopmentofSEAWAY.MARIN isaninternationallyrecognizedauthorityonhydrodynamics,involvedinfrontierbreakingresearchprogramsforthemaritimeandoffshoreindustriesandnavies.
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7Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) Copyright © 2005 Marine Cybernetics AS7
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8Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.1 Hydrodynamic Concepts and Potential TheoryPanelMethods(3-DPotentialTheory)Forpotentialflows,theintegralsoverthefluiddomaincanbetransformedtointegralsovertheboundariesofthefluiddomain.Thisallowstheapplicationofpanelorboundaryelementmethodstosolvethe3-Dpotentialtheoryproblem.
Panelmethodsdividethesurfaceoftheshipandthesurroundingwaterintodiscreteelements(panels).Oneachoftheseelements,adistributionofsourcesandsinksisdefinedwhichfulfilltheLaplaceequation.
Commercialcode:WAMIT(www.wamit.com)
-40
-30
-20
-10
0
10
20
30
40 -30
-20
-10
0
10
20
30
-12
-10
-8
-6
-4
-2
0
2
4
Y-axis (m)
3D Visualization of the Wamit file: supply.gdf
X-axis (m)
Z-ax
is (m
)
3D Panelization ofa Supply Vessel
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9Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
WAMIT
WAMIT® is the most advanced set of tools available for analyzing wave interactions with offshore platforms and other structures or vessels. WAMIT® was developed by Professor Newman and coworkers at MIT in 1987, and it has gained widespread recognition for its ability to analyze the complex structures with a high degree of accuracy and efficiency.
Over the past 20 years WAMIT has been licensed to more than80 industrial and research organizations worldwide.
Panelization of semi-submersible using WAMIT user supplied tools
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10Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.1 Hydrodynamic Concepts and Potential TheoryPotentialtheoryprogramstypicallycompute:
• Frequency-dependentaddedmass,A(w)• Potentialdampingcoefficients,B(w)• Restoringterms,C• 1st- and2nd-orderwave-inducedforcesandmotions
(amplitudesandphases)forgivenwavedirectionsandfrequencies• …andmuchmore
OnespecialfeatureofWAMITisthattheprogramsolvesaboundaryvalueproblemforzeroandinfiniteaddedmass.Theseboundaryvaluesareparticularusefulwhencomputingtheretardationfunctionsdescribingthefluidmemoryeffects.
ProcessingofHydrodynamicDatausingMSSHYDRO– www.marinecontrol.orgThetoolboxreadsoutputdatafilesgeneratedbythehydrodynamicprograms:
• ShipX (Veres)byMARINTEKAS• WAMITbyWAMITInc.
andprocessesthedataforuseinMatlab/Simulink.
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11Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.2 Seakeeping and Maneuvering KinematicsSeakeepingTheory(PerturbationCoordinates)TheSEAKEEPINGreferenceframe{s}isnotfixedtothecraft;itisfixedtotheequilibriumstate:
e1 ! !1,0,0,0,0,0"!
! ! !"
L :!
0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 !1 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
!" ! !#" ! !$
! ! !!1,!2,!3,!4,!5,!6"T #
!! ! ! !U!L!" " e1"
!!# ! !# !UL! # #
Transformationbetween{b}and{s}
!
"
#
!
00#"
#
$!
$"
$#
#
vnsn ! !Ucos!, Usin!, 0"!
!nsn ! !0,0,0"!
"ns ! !0,0,!" "!
# # #
-Intheabsenceofwaveexcitation,{s}coincideswith{b}.- Undertheactionofthewaves,thehullisdisturbedfromitsequilibriumand{s}oscillates,withrespecttoitsequilibriumposition.
!sb ! !!4,!5,!6"! ! !"#,"#,"$"! #
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12Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
SeakeepingAnalysisTheseakeepingequationsofmotionareconsideredtobeinertial:
5.3 The Classical Frequency-Domain Model
! ! !" ! !!x,!y,!z,!",!#,!$"! #
EquationsofMotion
MRB!" ! #hyd " #hs " #exc #
Cummins(1962)showedthattheradiation-inducedhydrodynamicforcesinanidealfluidcanberelatedtofrequency-dependentaddedmassA(ω) andpotentialdampingB(ω)accordingto:
!hyd ! !Ā"# ! "0
tK$ !t ! !""%!!"d! #
K!t" ! 2! !0
"B!""cos!"t"d" #
Ā ! A!!"
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13Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.3 The Classical Frequency-Domain Model
Ā ! A!!"
Frequency-dependentaddedmassA22(ω)andpotentialdampingB22(ω)insway
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14Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
CumminsModel
5.3 The Classical Frequency-Domain Model
K!t" ! 2! !0
"B!""cos!"t"d" #
!MRB ! A!!""!" ! "0
tK# !t # !"!$!!"d! ! C! " %exc #
Iflinearrestoringforcesτhs = -Cξ areincludedinthemodel,thisresultsinthetime-domainmodel:
Matrixofretardationfunctionsgivenby
!hyd ! !Ā"# ! "0
tK$ !t ! !""#!!"d! #
Thefluidmemoryeffectscanbereplacedbyastate-spacemodeltoavoidtheintegral
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15Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.3 The Classical Frequency-Domain Model
Longitudinaladdedmasscoefficientsasafunctionoffrequency.
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16Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.3 The Classical Frequency-Domain Model
Lateraladdedmasscoefficientsasafunctionoffrequency.
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17Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.3 The Classical Frequency-Domain Model
Longitudinalpotentialdampingcoefficientsasafunctionoffrequency.ExponentialdecayingviscousdampingisincludedforB11.
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18Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.3 The Classical Frequency-Domain Model
Lateralpotentialdampingcoefficientsasafunctionoffrequency.ExponentialdecayingviscousdampingisincludedforB22 andB66 whileviscousIKEDAdampingisincludedinB44
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19Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.3.1 Potential Coefficients and the Concept of Forced Oscillations
foreachfrequencyω.
Thematrices A(ω),B(ω)andC representsa"hydrodynamicmass-damper-springsystem"whichvarieswiththefrequencyoftheforcedoscillation.
Thismodelisrooteddeeplyintheliteratureofhydrodynamicsandtheabuseofnotationofthisfalsetime-domainmodelhasbeendiscussedeloquentlyintheliterature(incorrectmixtureoftimeandfrequencyinanODE).Consequently,wewilluseCumminstime-domainmodelandtransformthismodeltothefrequencydomain– nomixtureoftimeandfrequency!
Inanexperimentalsetupwitharestrainedscalemodel,itispossibletovarythewaveexcitationfrequencyω andtheamplitudesfi oftheexcitationforce.Hence,bymeasuringthepositionandattitudevectorη,theresponseofthe2nd-orderordersystemcanbefittedtoalinearmodel:
MRB!" ! #hyd " #hs " fcos!!t" #
!MRB ! A"!#$!" ! B"!#!# ! C! " fcos"!t# #
harmonicexcitation
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20Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
!MRB ! A!!""!" ! "0
tK# !t # !"!$!!"d! ! C! " %exc #
5.3.2 Frequency-Domain SeakeepingModels
Cumminsequationcanbetransformedtothefrequencydomain(Newman,1977;Faltinsen 1990)byassumingthatthevesselcarriesoutharmonicoscillationsin6DOF(seeSection5.4.1)::
ThepotentialcoefficientsA(ω)andB(ω)areusuallycomputedusingaseakeepingprogrambutthefrequencyresponsewillnot beaccurateunlessviscousdampingisincluded.
TheoptionalviscousdampingmatrixBV(ω) canbeusedtomodelviscousdampingsuchasskinfriction,surgeresistanceandviscousrolldamping(forinstanceIKEDArolldamping).
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21Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
BV!!" !
"1e!#!"NITTC!A1" 0 0 0 0 0
0 "2e!#! 0 0 0 0
0 0 0 0 0 0
0 0 0 "IKEDA!!" 0 0
0 0 0 0 0 0
0 0 0 0 0 "6e!#!
#
5.3.2 Frequency-Domain SeakeepingModels
B total!!" ! B!!" " BV!!" #
u ! Asin!!t" #
y ! c1x " c2x|x|"c3x33 #
N!A" ! c1 " 8A3! c2 " 3A2
4 c3 # y ! N!A"u #
X ! !X |u|u|u|u" NITTC!A1"u #
Viscousfrequency-dependentdamping:
Quadraticdampingisapproximatedusingdescribingfunctions(similartotheequivalentlinearizationmethod):
!ie!"#
QuadraticITTCdrag:
Viscousskinfriction:
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22Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.4 Time-Domain Models including Fluid Memory Effects
Unifiedmaneuveringandseakeeping model(nonlinearviscousdamping/maneuveringcoefficients
areaddedmanually)
Linearseakeeping equationsinBODYcoordinates(fluidmemoryeffectsareapproximatedasstate-spacemodels)
TransformfromSEAKEEPINGtoBODYcoordinates(linearizedkinematictransformation)
CumminsequationinSEAKEEPINGcoordinates(lineartheorywhichincludesfluidmemoryeffects)
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23Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
!MRB ! A"!#$!" ! B total"!#!# ! "0
tK"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ #
Fromanumericalpointofviewisitbettertointegratethedifference:
ThiscanbedonbyrewritingCumminsequationas:
5.4.1 Cummins Equation in SEAKEEPING Coordinates
!MRB ! Ā"!" ! !"#
tK# !t " !"!$!!"d! ! C# ! " %wind ! %wave ! "% #
Ā ! A!!" #
K! !t" ! 2! !0
"B total!""cos!"t"d" #
Cummins(1962)Equation
TheOgilvie(1964)Transformationgives
K!t" ! 2! !0
"#B total!"" # B total!""$ cos!"t"d" #
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24Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
Itispossibletotransformthetime-domainrepresentationofCumminsequationfrom{s}to{b}usingthekinematicrelationships:
Thisgives:
Thesteady-statecontrolforceτ neededtoobtaintheforwardspeedU whenτwind =τwave=0 andδη =0 is:
Hence,
5.4.2 Linear Time-Domain SeakeepingEquations in BODY Coordinates!MRB ! A"!#$!" ! B total"!#!# ! "
0
tK"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ #
!MRB ! A"!#$!!" !UL!$ ! B total"!#!! !U"L!# " e1#$ ! #0
tK"t " "#!!""#d" ! C!# " $wind ! $wave ! "$ " $%# #
!! ! ! !U!L!" " e1"
!!# ! !# !UL! # #
!" ! B total!!"Ue1 #
! ! !"
!" ! !#
!MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! "0
tK"t # "#!!""#d" ! C!# " $wind ! $wave ! $ #
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25Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
Whencomputingthedampingandretardationfunctions,itiscommontoneglecttheinfluenceofδη ontheforwardspeedsuchthat:
Finally,letusereplaceν bytherelativevelocityνr toincludeoceancurrentsanddefine:M =MRB+MA suchthat:
where
5.4.2 Linear Time-Domain SeakeepingEquations in BODY Coordinates!MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! "
0
tK"t # "#!!""#d" ! C!# " $wind ! $wave ! $ #
!! ! v !U!L!" " e1" ! v "Ue1 #
M!" ! CRB! ! ! CA!!r ! D!r ! "0
tK!t # !"#!!!"#Ue1$d! ! G# " $wind ! $wave ! $ #
MA ! A!!"CA" ! UA!!"LCRB" ! UMRBLD ! B total!!"
G ! C
LinearCoriolis andcentripetalforcesduetoarotationof{b}about{s}
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26Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.4.2 Linear Time-Domain SeakeepingEquations in BODY Coordinates
! :! !0
tK!t " !"#"!!""Ue1$
""d! #
FluidMemoryEffectsTheintegralinthefollowingequationrepresentsthefluidmemoryeffects:
! ! H!s"#" !Ue1$ #
!x " Arx # B r!!" " Crx
#
Approximatedbyastate-spacemodel
K!t" ! 2! !0
"#B!"" # B!""$ cos!"t"d" #
Impulseresponsefunction
0 5 10 15 20 25-1
-0.5
0
0.5
1
1.5
2
2.5x 107
time (s)
K22(t)
M!" ! CRB! ! ! CA!!r ! D!r ! "0
tK!t # !"#!!!"#Ue1$d! ! G# " $wind ! $wave ! $ #
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27Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory EffectsLinearSeakeeping Equations(BODYcoordinates)
UnifiedNonlinearSeakeeping andManeuveringModel
• Usenonlinearkinematics• ReplacelinearCoriolis andcentripetalforceswiththeirnonlinearcounterparts• Includemaneuveringcoefficientsinanonlineardampingmatrix(linearsuperposition)
!" ! J"!!"#M#" r # CRB!#"# # CA!#r"#r # D!#r"#r # $ # G! ! %wind # %wave # %
# #
M!" ! CRB! ! ! CA!!r ! D!r ! # ! G$ " %wind ! %wave ! % # Copyright © Bjarne Stenberg/NTNU
Copyright © The US Navy
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28Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.5 Case Study: Identification of Fluid Memory EffectsThefluidmemoryeffectscanbeapproximatedusingfrequency-domainidentification.ThemaintoolforthisistheMSSFDItoolbox(PerezandFossen2009)- www.marinecontrol.org
Whenusingthefrequency-domainapproach,thepropertythatthemapping:hasrelativedegreeoneisexploited.Hence,thefluidmemoryeffectsμ canbeapproximatedbyamatrixH(s)containingrelativedegreeonetransferfunctions:
!! ! " #
! ! H!s"!" #
H!s" ! Cr!sI ! Ar"!1B r #
!x " Arx # B r!!" " Crx
#
hij!s" ! prsr"pr!1sr!1"..."p0sn"qn!1sn!1"..."q0
r ! n ! 1, n " 2
State-spacemodel:
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29Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.5.1 Frequency-Domain Identification using the MSS FDI ToolboxConsidertheFPSOdatasetintheMSStoolbox(FDItool)andassumesthattheinfiniteaddedmassmatrixisunknown.Hence,wecanestimatethefluidtransferfunctionh33(s)byusingthefollowingMatlab code:
load fpsoDof = [3,3]; %Use coupling 3-3 heave-heaveNf = length(vessel.freqs);W = vessel.freqs(1:Nf-1)';Ainf = vessel.A(Dof(1),Dof(2),Nf); % Ainf computed by WAMIT
A = reshape(vessel.A(Dof(1),Dof(2),1:Nf-1),1,length(W))';B = reshape(vessel.B(Dof(1),Dof(2),1:Nf-1),1,length(W))';
FDIopt.OrdMax = 20;FDIopt.AinfFlag = 0;FDIopt.Method = 2;FDIopt.Iterations = 20;FDIopt.PlotFlag = 0;FDIopt.LogLin = 1;FDIopt.wsFactor = 0.1;FDIopt.wminFactor = 0.1;FDIopt.wmaxFactor = 5;
[KradNum,KradDen,Ainf] = FDIRadMod(W,A,0,B,FDIopt,Dof)
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30Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox
FPSOidentificationresultsforh₃₃(s)withoutusingtheinfiniteaddedmassA₃₃(∞).Theleft-hand-sideplotsshowthecomplexcoefficientanditsestimatewhileaddedmassanddampingareplottedontheright-hand-side.
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31Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)
5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox
h33!s" !1.672e007 s3 " 2.286e007 s2 " 2.06e006 s
s4 " 1.233 s3 " 0.7295 s2 " 0.1955 s " 0.01639
Ar !
!1.2335 !0.7295 !0.1955 !0.01641 0 0 00 1 0 00 0 1 0
B r !
1000
Cr ! 1.672e007 2.286e007 2.06e006 0
Dr ! 0
! ! H!s"#" !Ue1$ #
!x " Arx # B r!!" " Crx
#