系统辨识第四章

SYSTEMIDENTIFICATIONTheoryfortheUserByLENNARTLJUNGLinkopingUniversity,SwedenC40COLLEGEOFELECTRICALENGINEERINGPartISystemsandModelsC41COLLEGEOFELECTRICALENGINEERINGCHAPTER4ModelsofLinearTime-InvariantSystemsC42COLLEGEOFELECTRICALENGINEERINGAmodelofasystemisadescriptionof(someof)itsproperties,suitableforacertainpurpose.Inordertoserveitspurpose,themodel

neednotbeatrueand

accuratedescriptionofthesystem.Systemidentificationisthesubjectofconstructingorselectingmodelsofdynamicalsystemstoservecertainpurposes.Thefirststepistodetermineaclassofmodelswithinwhichthesearchforthemostsuitablemodelistobeconducted.C43COLLEGEOFELECTRICALENGINEERING4.1LinearModelsandSetsofLinearModelsC44COLLEGEOFELECTRICALENGINEERINGInChapter2,alineartime-invariantmodelisspecifiedbytheimpulseresponse{g(k)}1,∞,thespectrumFv(w)=l|H(eiw)|2

ofthe

additivedisturbance,andtheprobabilitydensityfunction

(PDF)

ofthedisturbancee(t):

y(t)=G(q)u(t)+H(q)e(t)

fe(.),thePDFofe

(4.1)

with

G(q)=Sk=1,∞g(k)q-k,H(q)=

1

+Sk=1,∞h(k)q-k

(4.2)Inmostcases,itisimpracticaltomakethisspecificationbyenumeratingtheinfinitesequences{g(k)},{h(k)}andtogetherwiththefunctionfe(x).ThespecificationofGandHintermsofafinitenumberofnumericalvaluesisrequired.Typicalexamplesare:Rationaltransferfunctionandfinite-dimensionalstatespacedescriptionsC45COLLEGEOFELECTRICALENGINEERINGAlso,thePDFfeisoftennotspecifiedasafunction,butdescribedtypicallybythefirstandsecondmoments:

(4.3)Ife(t)isassumedtobeGaussian,thePDFisentirelyspecifiedby(4.3).

Thefinitenumberofcoefficientsin(4.1)areoftennotpossibletodetermineapriorifromknowledgeofthephysicalmechanism.Thedeterminationofallorsomeofthemmustbelefttoestimationprocedures.Denotingsuchparametersbythevectorq,thuswehavethefollowingdescription:

y(t)=G(q,q)u(t)+H(q,q)e(t)

(4.4a)

fe(x,q),thePDFofe(t);{e(t)}whitenoise

(4.4b)C46COLLEGEOFELECTRICALENGINEERINGUsing(3.20),theone-step-aheadpredictionfor(4.4)canbecomputedas

y^(t|q)=H-1(q,q)G(q,q)u(t)+[1-H-1(q,q)]y(t)

(4.6)Thepredictionisdenotedbyy^(t|q)toemphasizeitsdependenceonq.

Thispredictorformdoesnotdependonfe(x,q).ThetermpredictormodelisusedformodelsthatjustspecifyGandH

in(4.4)orintheform(4.6).Probabilisticmodelswillsignifydescriptions(4.4)thatgiveacompletecharacterizationoftheprobabilisticproperties.Inthefollowing,differentwaysofdescribing(4.4)intermsofqwillbediscussed.C47COLLEGEOFELECTRICALENGINEERING4.2AFamilyofTransfer-FunctionModelsC48COLLEGEOFELECTRICALENGINEERINGEquationError

ModelStructureC49COLLEGEOFELECTRICALENGINEERINGARXModelStructureThemostsimpleinput-outputrelationshipisobtainedbydescribingitasalineardifferenceequation:y(t)+a1y(t-1)+…+anay(t-na)=b1u(t-1)+…+bnbu(t-nb)+e(t)

(4.7)Sincethewhite-noiseterme(t)entersasadirecterror(Why?UseEq.4.6),themodelisoftencalledanequationerrormodel.Theadjustableparametersareq

=

[a1a2…anab1…bnb](4.8)IntroducingA(q)=1+a1q-1+…+anaq-naandB(q)=b1q-1+…bnbq-nb,(4.7)willcorrespondto(4.4)y=G

u+H

ewith G(q,q)=B(q)/A(q),H(q,q)=1/A(q)

(4.9)Themodel(4.7)isalsocalledanARXmodel,whereARreferstotheAutoRegressivepartA(q)y(t)andXtotheeXtrainputB(q)u(t).Whenna=

0,y(t)ismodeledasafiniteimpulseresponse(FIR).C410COLLEGEOFELECTRICALENGINEERINGTheequationerrormodelsethasaveryimportantproperty:

Thepredictordefinesalinearregression.

Thatis:y^(t|q)=B(q)u(t)+[1-A(q)]y(t)Thispropertymakesitaprimechoiceinmanyapplications.C411COLLEGEOFELECTRICALENGINEERINGLinearRegressionsWithoutastochasticframework,thepredictorforA

y

=

B

u

+

ecanbecomputedbyinserting(4.9)into(4.6)as: y^(t|q)=B(q)u(t)+[1-A(q)]y(t)

(4.10)(4.10)isalsoanaturalchoiceiftheterme(t)isconsideredtobe“insignificant”or“difficulttoguess”ignoringe.Thusitisperfectfor“deterministic”models.Introducingthevector

j(t)=[-

y(t

-1)…-

y(t

-

na)u(t

-1)…u(t

-

nb)]T

(4.11)

Then(4.10)canberewrittenas y^(t|q)=qTj(t)=jT(t)

q

(4.12)Thepredictorisascalarproductbetweenaknowndatavectorj(t)andtheparametervector

qtobeestimated.C412COLLEGEOFELECTRICALENGINEERINGSuchamodeliscalledalinearregressioninstatistics,andthevectorj(t)isknownastheregressionvector.IncasessomecoefficientsofthepolynomialsAandBareknown,wearriveatlinearregressionsoftheform y^(t|q)=jT(t)

q+m(t)

(4.13)wherem(t)isaknownterm.ImportantAdvantage:SomepowerfulandsimpleestimationmethodssuchasLScanbeappliedforthedeterminationofq.BasicDisadvantage:Itlackstheadequatefreedomindescribingthepropertiesofthedisturbanceterm.Thewhitenoisee(t)isassumedtogothroughthedenominatordynamicsofthesystem

(y=B/Au+1/Ae).C413COLLEGEOFELECTRICALENGINEERINGARMAXModelStructureTodescribethepropertiesofthedisturbance

term

moreadequately,weconsidertheequationerrorasamovingaverage(MA)ofwhitenoise: A(q)y(t)=B(q)u(t)+C(q)e(t)

(4.15)withC(q)=1+c1q-1+…+cncq-nc.TheARMAX

modelhasbecomeastandardtoolincontrolandeconometricsforbothsystemdescriptionandcontroldesign.ComparedwithlinearregressionsfortheARXmodel,theregressionfortheARMAXmodelispseudolinear.C414COLLEGEOFELECTRICALENGINEERINGPeudolinearRegressionsThepredictorHy^=Gu+[H-1]yforARMAXmodelAy=Bu+Ceis C(q)y^(t|q)=B(q)u(t)+[C(q)–A(q)]y(t)

(4.18)Adding[1–C(q)]y^(t|q)tobothsideof(4.18)gives y^(t|q)=B(q)u(t)+[1-A(q)]y(t)+[C(q)-1][y(t)-y^(t|q)](4.19)Introducethepredictionerror

e(t,q)=y(t)–y^(t|q) andthevectorj(t,q)=[-y(t-1)…-y(t-na)u(t-1)… u(t-nb)

e(t-1,

q)…e(t-nc,

q)]T(4.20)Then(4.19)canbewrittenasy^(t|q)=jT(t,q)

q

(4.21)Theequation(4.21)itselfisnotalinearregression,duetothenonlineareffectofqinthevectorj(t,

q),andisthereforecalledapseudolinearregreesion.C415COLLEGEOFELECTRICALENGINEERINGOutputError

ModelStructureC416COLLEGEOFELECTRICALENGINEERINGInequationerrormodel,thetransferfunctionsG(=B/A)andH(=1/AorC/A)havethepolynomialAasacommonfactorinthedenominators.Fromaphysicalpointofview,itismorenaturaltoparametrizethesetransferfunctionindependently.Supposingthattherelationshipbetweeninputandundisturbedoutputwcanbewrittenasalineardifferenceequation,andthatthedisturbancesconsistofwhitemeasurementnoise,thenwehave:

w(t)+f1w(t-1)+…+fnfw(t-nf)=b1u(t-1)+…+bnbu(t-nb)

(4.24a) y(t)=w(t)+e(t)

(4.24b)or y(t)=B(q)/F(q)*u(t)+e(t)

(4.25)(4.25)iscalledanoutputerror

(OE)

model.C417COLLEGEOFELECTRICALENGINEERINGe

uwy

Fig.4.3TheoutputerrormodelstructureTheparametervectortobedeterminedis

q=[b1b2…bnbf1f2…fnf]T

(4.26)Sincew(t)

is

neverobservedandisconstructedfromuusing(4.24a)w=B/F*u,itshouldcarryanindexq:

w(t,q)+f1w(t-1,q)+…+fnfw(t-nf,q)=b1u(t-1)+…+bnbu(t-nb)

(4.27)Comparingy

=

B/F*u

+

ewith(4.4),H(q,q)=1,thepredictoris y^(t|q)=B(q)/F(q)*u(t)=w(t,q)

(4.28)Notethaty^(t|q)isconstructedfrompastinputsonly.B/F+C418COLLEGEOFELECTRICALENGINEERINGWiththeaidofthevector

j(t,q)=[u(t

-1)…u(t

-

nb)-

w(t

-1,

q)…-

w(t

-

nf,

q)]T

(4.29)

wehave

y^(t|q)=jT(t,q)

q

(4.30)Note

thatin(4.29)the

w

(t

-1,

q)arenotobserved,but,using

(4.28),

theycanbe

computed: w(t

-

k,

q)=y^(t

-

k|q)=B/F

*

u(t

-

k),k=1,2,…,nf.C419COLLEGEOFELECTRICALENGINEERINGBox-JenkinsModelStructureThismodelissuggestedbyBoxandJenkinsin1970.Itisanaturaldevelopmentoftheoutputerrormodeltofurthermodel

thepropertiesoftheoutputerror: y(t)=B(q)/F(q)*u(t)+C(q)/D(q)*e(t)

(4.31)Thisisthemostnatural

finite-dimensionalparameterization:GandHareindependentlyparameterizedasrationalfunctions.Accordingto(4.6):y^(t|q)=H-1(q,q)G(q,q)u(t)+[1-H-1(q,q)]y(t),thepredictorfor(4.31)is

y^(t|q)=D(q)B(q)/(C(q)F(q))*u(t)+(C(q)

-

D(q))/C(q)*y(t)

(4.32)C420COLLEGEOFELECTRICALENGINEERINGAGeneralFamilyofModelStructuresC421COLLEGEOFELECTRICALENGINEERINGSelectingfromthefivepolynomialsused:A,B,C,D,andF,wecouldactuallyconstruct32differentmodelsets.Thesixpossibilitiesexplicitlydisplayedinthissectionarethemostcommonlyusedonesinpractice.Fortheconvenienceofexplicitalgorithmsandanalyticresults,ageneralizedmodelstructureshouldbeused: A(q)

y(t)=B(q)/F(q)*u(t)+C(q)/D(q)*e(t)

(4.33)From(4.6):y^(t|q)=H-1(q,q)

G(q,

q)

u(t)+[1

-

H-1(q,

q)]

y(t),thepredictorfor(4.33)is

y^(t|q)=D(q)B(q)/(C(q)F(q))*u(t)+[1

-

D(q)A(q)/C(q)]*y(t)

(4.35)C422COLLEGEOFELECTRICALENGINEERINGTable4.1SomecommonBlack-boxSISOModelsasSpecialCasesof(4.33)A

y=B/F

*

u+C/D

*

e

---------------------------------------------------------------------------PolynomialsUsedin(4.33)NameofModelStructure

---------------------------------------------------------------------------

B FIR

AB ARX

ABC ARMAX

AC ARMA

ABD ARARX

ABCD ARARMAX

BF OE(outputerror)

BFCD BJ(Box-Jenkins)

---------------------------------------------------------------------------C423COLLEGEOFELECTRICALENGINEERINGOtherModelExpansionsTheFIRmodelstructureG(q,

q)

=

Sk=1,n

bk

q-khastwoimportantadvantages:Itisalinearregression

(aspecialcaseofARX,(EE)).Itisanoutputerrormodel

(aspecialcaseofOE).Thismeansthatthemodelcanbeefficientlyestimatedandthatitisrobustagainstnoise.Thebasicdisadvantageisthatmanyparametersmaybeneeded.Apoleclosetotheunitcirclemakestheimpulseresponsedecayslowly,sonhastobelargetoapproximatethesystemwell.C424COLLEGEOFELECTRICALENGINEERINGAmodelwhichretainsthelinearregressionandoutputerrorfeaturewhileoffersbetterpossibilitiestotreatslowlydecayingimpulseresponsewouldlooklike

G(q,q)=Sk=1,nqkLk(q,a)

(4.47)

whereLk(q,a)representsafunctionexpansioninthedecayoperatorwithaauser-chosenparameter.Theparameteraistreatedasfixedtomake(4.47)alinearregression.Asimplechoiceis Lk(q,a)=q-k/(q

-

a)whereaisanestimateofthesystempole

closesttotheunitcircle.Asamoresophisticatedchoiceintermsoforthonormalbasisexpansion,Laguerrepolynomialshavebeenused: Lk(q,a)=1/(q

-

a)*[(1

–

a

q)/(q

-

a)]k-1

(4.48)withaanestimateofthedominatingpole

(timeconstant).C425COLLEGEOFELECTRICALENGINEERING4.3State-SpaceModelsC426COLLEGEOFELECTRICALENGINEERINGInthestate-space,therelationshipamongtheinput,noiseandoutputiswrittenasasystemoffirst-orderdifferentialor

differenceequationsusinganauxiliarystatevectorx(t).ThisdescriptionbecameanincreasinglydominatingapproachafterKalman’s(1960)workonpredictionandlinearquadraticcontrol.Theinsightsintophysicalmechanismsofthesystemcanmoreeasilybeincorporatedintostate-spacemodels.C427COLLEGEOFELECTRICALENGINEERINGContinuous-timeModelsBasedonPhysicalInsightWhythemodelisconstructedincontinuoustimeratherthanindiscretetime?Mostlawsofphysics

(Newton’slawofmotion,relationshipsinelectricalcircuits,etc.)areexpressedincontinuoustime.State-spacemodelingnormallyleadstoarepresentationx’(t)=F(q)

x(t)+G(q)

u(t)or[p

I-F(q)]

x(t)=G(q)

u(t)

(4.62)Herepisthedifferentiationoperator,FandGarematricesofn

x

nandn

x

m,andqisavectorofparametersthattypicallycorrespondtounknownvaluesofphysicalcoefficients,materialconstantsandthelike.Thestatevariablesxusuallyhavephysicalsignificance

(position,velocities,etc.)andthemeasuredoutputyistheknowncombinationofthestates.C428COLLEGEOFELECTRICALENGINEERINGLet

h(t)

bethe

measurements

obtainedwith

ideal,noise-free

sensors:

h(t)=H

x(t)

(4.63)

Thenthe

transferoperator

from

utohin

(4.63)is

h(t)=Gc(p,

q)u(t)withGc(p,

q)=H

[p

I-F(q)]-1G(q)

(4.64)Thisisthecontinuous-timetransfer-functionmodelofthesystemparameterizedintermsofphysicalcoefficients.Consideringthemeasurementimperfections

(affectingtheoutput)anddisturbancesactingon(4.62)

(affectingthestatevariables),somenoise-corruptedversionofh(t)isobtained.Thereareseveraldifferentpossibilitiestodescribethesenoiseanddisturbanceeffects.ThesimplestapproachwithmeasurementimperfectionsvT(kT)is y(kT)=H

x(kT)+vT(kT)=Gc(p,

q)u(t)+vT(kT)

(4.65)C429COLLEGEOFELECTRICALENGINEERINGSamplingtheTransferFunctionOnewaytotransportGc(p,q)toarepresentationinexplicitlydiscretetime:SupposetheinputisconstantoverthesamplingintervalT: u(t)=uk=u(kT),kT≤t≤(k+1)T(4.66)Thenx’(t)=F(q)

x(t)+G(q)

u(t)caneasilybesolvedfromt=kTtot=kT

+

Tas x(kT

+

T)=AT(q)

x(kT)+BT(q)

u(kT)

(4.67)whereAT(q)=eF(q)T,(4.68a,b)IntroducingqfortheforwardshiftofTtimeunits,wecanrewrite(4.67)as[q

I-AT(q)]x(kT)=BT(q)u(kT)or

h(kT)=GT(q,

q)u(kT)

(4.70) GT(q,

q)=H[q

I-AT(q)]-1BT(q)

(4.71)C430COLLEGEOFELECTRICALENGINEERINGHence(4.65)can

equivalently

begiveninthe

sampled-dataform

y(t)=GT(q,

q)u(t)+vT(t),t=T,2T,…(4.72)Note:Whenu(t)=uk

=u(kT)holds,noapproximationisinvolvedinthisrepresentation.InviewofAT(q)andBT(q),GT(q,

q)couldbequiteacomplicatedfunctionofq.ReadExample4.1withthefollowingquestions(p.95):Howmanydisturbancescanbeconsideredinastate-spacemodel?ComparingwithARX(EE)orOEmodel,whatisthemainadvantageanddisadvantageofusingastate-spacemodeltodescribethesystem?C431COLLEGEOFELECTRICALENGINEERINGEquations(4.67):x(kT

+

T)=AT(q)x(kT)+BT(q)u(kT)

and(4.65):

y(kT)=H

x(kT)+vT(kT)=Gc(p,

q)u(t)+vT(kT)

constitute

a

standarddiscrete-time

state-spacemodel.Forsimplicity,wetakeT

=

1anddropthecorrespondingindex,andintroduceamatrixrelatingxtoh:H=C(q),thuswehave

x(t

+1)=A(q)x(t)+B(q)u(t)

(4.80a)

y(t)=C(q)x(t)+v(t)

(4.80b)Correspondingto

y(t)=G(q,

q)u(t)+v(t)

(4.81) G(q,

q)=C(q)[q

I-A(q)]-1B(q)

(4.82)C432COLLEGEOFELECTRICALENGINEERINGNoiseRepresentationandtheTime-invariableKalmanFilterAstraightforwardbutentirelyvalidapproachtomodelthenoiseterm{v(t)}in(4.80)and(4.81)istopostulateitas v(t)=H(q,q)e(t)

(4.83)with{e(t)}beingwhitenoisewithvariancel.Forstate-spacedescription,itismorecommontosplitv(t)intomeasurementnoisev

(t)actingontheoutputandprocessnoisew(t)actingonthestates:x(t+1)

=

A(q)

x(t)

+

B(q)

u(t)

+

w

(t)y(t)

=

C(q)

x(t)

+

v

(t)

(4.84)withw

(t)andv

(t)beingindependentrandomvariableswithzeromeanvaluesandcovariancesE

w(t)

wT(t)

=

R1(q)E

v(t)

vT(t)=R2(q)E

w(t)

vT(t)

=

R12(q)

(4.85)Whenwandvarenotwhitenoises,extramodelingandextensionofthestatevectorarerequiredin(4.84)and(4.85).C433COLLEGEOFELECTRICALENGINEERINGApplythecelebratedKalmanfilter,theconditionalexpectationofy(t),givendatay(s)andu(s),s

≤

t-1,is,providedwandvareGaussianprocess,givenbyx^(t+1,q)=A(q)x^(t,q)+B(q)u(t)+K(q)[y(t)-C(q)x^(t,q)] y^(t|q)=C(q)x^(t,q)

(4.86)

HereK(q)isgivenas

whereP(q)isthepositivesemidefinitesolutionofthestationaryRiccatiequation:andisthecovariancematrixofthestateestimateerror:

Howtopredicty(t)in(4.84)?C434COLLEGEOFELECTRICALENGINEERINGTowritethepredictor

intermofinput,wehave:

y^(t|q)

=C(q)[q

I-A(q)+K(q)C(q)]-1B(q)u(t) +C(q)[q

I-A(q)+K(q)C(q)]-1K(q)y(t)

(4.88)C435COLLEGEOFELECTRICALENGINEERINGInnovationsRepresentationThepredicterrorof(4.86)isthepartofy(t)thatcannotbepredictedfrompastdata:“theinnovation”denotedby

e(t):y(t)-C(q)x^(t,q)=C(q)[x(t)-x^(t,q)]+v(t)=e(t)

(4.90)Then(4.86)canberewrittenas x^(t+1,q)=A(q)x^(t,q)+B(q)u(t)+K(q)e(t) y(t)=C(q)x^(t,q)+e(t)

(4.91a)Thecovarianceofe(t)canbefoundfrom(4.90)and(4.89): Ee(t)eT(t)=L(q)=C(q)P(q)CT(q)+R2(q)

(4.91b)Sincee(t)appearsexplicitly,thisrepresentationisknownastheinnovationsformofthestate-spacedescription.C436COLLEGEOFELECTRICALENGINEERINGDirectlyParameterizedInnovationsFormIn(4.91)theKalmangainK(q)iscomputedfromA,C,R1,R12andR2inthefairlycomplicatedmannergivenby(4.87)forK,P.ItisanattractiveideatoparameterizeK(q)

intermsofq

directly.Theimportantadvantageofthisisthatthepredictor

(4.88)becomesamuchsimplerfunctionofq.Suchamodelstructureiscalledadirectlyparameterizedinnovationsform.IfwehavenopriorknowledgeabouttheR-matrices,whichmeansthatmanyparametersareneededtodescribethem,thedirectparameterizationofK(q)isabetteralternative.Ifthephysicalinsightinto(4.84)entailsknowing,forexample,thattheprocessnoise

affectsonlyonestateandisindependentofmeasurementnoise,thecalculationofK(q)

via(4.85)and(4.87)isdoneeasilycomparedwiththedirectparameterizationofK(q).C437COLLEGEOFELECTRICALENGINEERINGGuidanceofthechoiceofparameterizedmodelsetsTwodifferentphilosophies

guidethechoiceofthemodel:Black-boxmodelstructure:Theyareflexiblemodelsetsthatcanaccommodateavarietyofsystems,withoutlookingintotheirinternalstructure.Theinput-outputmodelstructuresaswellascanonicallyparameterizedstate-spacemodelsareofthischaracter.Modelstructureswithphysicalparameters:Thephysicalinsightisincorporatedintothemodelsetsoastobringthenumberofadjustableparameters

downtowhatisactuallyunknownaboutthesystem.Continuous-timestate-spacemodelsaretypicalrepresentationsforthisapproach.C438COLLEGEOFELECTRICALENGINEERING4.6IdentifiabilityofSomeModelStructuresC439COLLEGEOFELECTRICALENGINEERINGIdentifiabilityConceptDefinition4.6.AmodelstructureMisgloballyidentifiableat q*ifM(q)=M(q*),q

eDM

q=q*(4.130)Theidentifiabilityconceptconcernstheuniquerepresentationofagivensystem.AgivensystemS: y(t)=G0(q)u(t)+H0(q)e(t)

(4.132)LetMbeamodelstructurebasedonone-step-aheadpredictorsfor y(t)=G(q,q)u(t)+H(q,q)e(t)

(4.133)DefinethesetDT(S,M)asthoseq-valuesinDMforwhichS=M(q): DT(S,M)={qeDM|G0(z)=G(z,q),H0(z)=H(z,q)almostallz} (4.134)ThissetisemptyincaseSnote

M.C440COLLEGEOFELECTRICALENGINEERINGNowsupposethatS

e

MsothatS=M(q0)forsomevalueq0.SupposethatMisgloballyidentifiableatq0.Than DT(S,M)={q0} (4.135)Note:

Mshouldbeselectedsothat(4.135)holdsforagivenS.SinceSisunknown,severaldifferentstructuresMshouldbetested.TheidentifiabilityconceptwillprovideusefulguidanceinfindinganMsuchthat(4.135)holds.C441COLLEGEOFELECTRICALENGINEERINGAmodelstructureisgloballyidentifiableatq*ifandonlyif G(z,q)=G(z,q*)andH(z,q)=H(z,q*) foralmostallz

q=q*

(4.136)Forlocalidentifiability,qwillbeconsideredtobeconfinedtoasufficientlysmallneighborhoodofq*.Globalidentifiabilityismoredifficulttodealwithingeneralterms.Weshallonlybrieflydiscussidentifiabilityofphysicalparametersandgivesomeresultsforgeneralblack-boxSISOmodels.C442COLLEGEOFELECTRICALENGINEERINGParametrizationsinTermsofPhysicalParametersForamodel y(t)=Gc(p,

q)u(t)+v(t)

(4.137)Asimpleridentifiabilitytesttoapplyis Gc(s,q)=Gc(s,q*)almostalls

q=q*

?

(4.138)Thisequationisnotsufficientfor(4.136)withbothGandHtoholdbutisareasonabletestforglobalidentifiabilityofthemodelatq*.But(4.138)isstilladifficultenoughproblemexceptforspecialstructures.C443COLLEGEOFELECTRICALENGINEERINGSISOTransfer-functionModelStructuresConsidertheARXmodelstructure G

(z,q)=B(z)/A(z),H(z,q)=1/A(z)

(4.139)with q=[a1…anab1…

bnb]TEqualityforH(z,

q)=H(z,

q*)in(4.136)impliesthattheA-polynomialscoincide,aa*,whichinturnimpliesthattheB-polynomialsmustcoincidefortheGtobeequal,bb*.Thatisq

q*.Itverifiesthat(4.136)holdsforallq*inthemodelstructure(4.139)---Thestructureisstrictlygloballyidentifiable.C444COLLEGEOFELECTRICALENGINEERINGFortheOEmodelstructure

(4.25)withordersnbandnf,atq=q*wehaveLetB~*(z)=znbB*(z)andqbeanarbitraryparametervalue,thenequation(4.136)

znf-nb

B~*

/

F~*

=

G(z,q*)

=

G(z,q)

=

B(z)

/

F(z)=

znf-nb

B~(z)

/

F~(z)canbewrittenas F~(z)B~*(z)–F~*(z)B~(z)=0(4.141)C445COLLEGEOFELECTRICALENGINEERING F~(z)B~*(z)–F~*(z)B~(z)=0(4.141)SinceF~*(z)isapolynomialofdegreenf,ithasnfzeros: F~*(ai)=0,i=1,…,nfSupposethatB~*andF~*arecoprime

B~*(ai)≠0.Then(4.141)impliesthat F~(ai)=0,i=1,…,nfConsequently,wehaveF~(z)

F~*(z),whichintu