Comparing Different Approaches to Model Error Modeling in Robust Identification

Identication

WolfgangReinelt #

,AndreaGarulli

∗

andLennartLjung #

#

Department ofElectrical Engineering

Linkoping University, 58183Linkoping, Sweden

WWW:

http://www.control.isy.liu.se/

E-mail:

{wolle,ljung}@isy.liu.se

∗

Dipartimentodi Ingegneriadell'Informazione

Universita di Siena,53100Siena,Italy

WWW:

http://www-dii.ing.unisi.it/~garulli/

E-mail:

garulli@ing.unisi.it

August2, 2000

REG

LERTEKNIK

AUTO

_{MATIC CONTR}

OL

LINKÖPING

Reportno.: LiTH-ISY-R-2293

PreparedforJournal Publication

Technical reports from the Automatic Control group in Linkoping are available by

anony-mous ftp at the address

ftp.control.isy.liu.se

. This report is contained inthe portable

in Robust Identication

∗

Wolfgang Reinelt #

,

†

,Andrea Garulli ##

andLennartLjung #

#

DepartmentofElectricalEngineering,

Linkoping University,58183 Linkoping,Sweden.

E-mail:

{wolle,ljung}@isy.liu.se

##

Dipartimentodi Ingegneriadell'Informazione,

Universita di Siena,53100Siena, Italy.

E-mail:

garulli@ing.unisi.it

Abstract

Identicationforrobustcontrolmustdelivernotonlyanominalmodel,butalsoa

re-liableestimateoftheuncertaintyassociatedwiththemodel. Thispaperaddressesrecent

approachestorobustidentication, thatexplicitly aimatseparatingcontributionsfrom

thetwomainuncertaintysources: unmodeled dynamicsandnoiseaectingthedata. In

particular,thefollowing methodsareconsidered: non-stationaryStochasticEmbedding,

Model ErrorModeling based onprediction error methods,Set Membership

Identica-tion. Modelvalidationissuesarealsoaddressedintheproposedframework. Moreover,

itisshownhowthecomputationoftheminimumnoiseboundforwhichanominalmodel

isnotfalsied byi/odata,canbeusedasarationale forselectinganappropriatemodel

class in thesetmembershipsetting. For allmethods,uncertainty isevaluatedin terms

ofthefrequencyresponse,sothatitcanbehandledby

H∞

controltechniques. An exam-ple,whereanontrivialundermodelingisensuredbythepresenceofanonlinearityinthe

systemgeneratingthedata,ispresentedtocomparethedierentmethods.

Keywords: identication for robust control, model error modeling, unmodeled dynamics,

modelvalidation,setmembershipestimation,stochasticembedding.

∗

Preliminary versions of thiswork have been presented atthe1999 Conference on Decisionand Control,

Phoenix,AZ,USA[19]andtheIFACSymposiumSystemIdenticationSYSID2000,SantaBarbara,CA,USA[6 ].

†

SupportedbytheEuropeanUnionwithintheEuropeanResearchNetworkinSystemIdentication(ERNSI)

Oneof themainobjectivesof control-orientedidenticationistoestimatemodelsthatare

suitable for robust control design techniques. For this purpose, the identication

proce-duremustdelivernotonly anominalmodel,butalsoareliableestimateoftheuncertainty

associated tothe model. Dierent paradigms forthe description ofuncertainty have been

addressedintheliterature(seee.g.thespecialissues[13,20]andtherecentbook[7]). Two

main philosophies are basically adopted. The rst one is based on statistical assumptions

andusuallyleadstoleastsquares estimationtechniquesand predictionerror methods,see

[16]. The second one relies on deterministic hypotheses, such as the identication error

being unknown-but-bounded (UBB), and has givenrise to a numberof techniques usually

addressedasboundederrororsetmembershipidentication(see[18,17]andthereferences

therein).

In standard identication problems, the error originates from two dierent sources: a

\variance"term,duetonoiseaectingthedata,anda\bias"term,duetosystemdynamics

whichis notcaptured by the estimatednominalmodel (oftenaddressed alsoas the model

error). Clearly,thenatureofthesetwoerrortermsisquitedierent: theformerisgenerally

uncorrelatedwiththeinputsignal(whenthedatais collectedinopenloop),whilethelatter

stronglydependsontheestimatednominalmodelandontheinputusedintheidentication

experiment. The model erroris notnegligible inmost practical situations,especiallythose

in which the order of the nominal model must be small (a typical requirement of robust

design techniques). Moreover, whileapriori informationon measurementnoise areoften

available,similarhypothesesontheunmodeleddynamicsseemtobelessrealistic.

This paperaddressesthreedierent approachestorobustidentication: Stochastic

Em-bedding,Model ErrorModelingandSetMembershipIdentication. Eachofthemexplicitly

separatecontributionsfrom theabovetwoerror sources,thusprovidingareliableestimate

ofmodeluncertainty. Thersttwoapproacheshavebeendevelopedinthestatistical

frame-work,whilethelatterrelieson UBBerror assumptions.

StochasticEmbedding[12,10]isafrequencydomainmethodwhichassumesthat

unmod-eled dynamicscanberepresentedadequatelyby anon-stationary stochasticprocesswhose

varianceincreaseswithfrequency. Thenominalmodelisobtainedvialeast-squares

estima-tionfromfrequencydomaindata; therefore, harmonicinputsarerequired. Theuncertainty

associatedtothemodelis evaluatedfrom statisticalpropertiesoftherandomwalkprocess

describingtheunmodeleddynamics.

Model Error Modeling [14, 15] employs standard prediction error methods to identify

a nominal model from input-output time domain data [16]. Then, one can estimate the

unmodeleddynamicsbylookingatthatpartoftheidenticationresidualthatoriginatesfrom

theinput. Identicationofresidualdynamics(thatcanbeperformedusingagainprediction

error methods) provides theso called model errormodel. The condenceregionof model

SetMembershipIdenticationprovideseÆcientalgorithmsforestimatingthesetof

fea-siblemodels,compatiblewiththeavailabledataandtheUBBerror assumption. Thechoice

ofthenominalmodelis usuallyperformedbyminimizingacostfunctionrelatedtothe

fea-sible set. The feasiblesetitselfgives the sizeof the uncertainty associatedto the nominal

model. Intherstworksonsetmembershipidentication(seethesurvey[25]andthe

ref-erencestherein)thecontributions fromunmodeleddynamics andnoisewerenotseparated

(which corresponds to assuming that the plant generating the data belongs to the

consid-eredmodelclass).Morerecently,thepresenceofmodelerrorshasbeenexplicitlyaccounted

for inseveral works and dierent settings (seee.g. [24, 9, 8]). In thispaper, modelerror

modelingconceptsareextendedtosetmembershipidentication,inordertoobtainafairly

generalstrategyforseparatingerrorcontributions.

Asitcanbeseenfromtheabovediscussion,quitedierentapproachestorobust

identi-cationareprovidedbytheconsideredtechniques,themaindierencesconcerningthetype

of data required, theselectedmodelclass, the assumptionson thenoise aectingthe data

andthenominalmodelestimationcriteria. Themaincontributionofthepaperistotestthe

abovethreeapproachesonthesamesimulationsetting,inordertohighlighttheadvantages

ofeachmethodandtoprovide,aslongasitispossible,afaircomparisonoftheresults. The

mostimportantfeaturesofeachtechniqueareemphasized. In ordertoensurethepresence

ofanontrivialundermodeling,anonlinearityispresentinthetruesystemconsideredinthe

test example, and identicationof a low order nominal model is required. Frequency

do-mainuncertaintyregionsareconsideredinordertoprovideadequatemodelsfor

H

∞

control design.

The secondcontributionofthepaperistoprovideafrequencydomainmodelvalidation

tool forsetmembershipidentication. The basicidea is toevaluatethe minimumvalue of

theerrorbound,whichallowsonetovalidatetheestimatednominalmodel. Itisalsoshown

thatthisprovidesausefulrationalefortheselectionofanadequatemodelclass.

Paper outline: Section 2 brie yreviews the technique ofnon-stationary stochastic

embed-ding, while in Section3, modelerror modeling ideas aresummarized from aquite general

point of view. Section 4 introduces the concept of set membership identication,

embed-ded in a model error modeling setup, and discusses identication of both nominal model

and model error, modelvalidation and model classselection. Section 5reports a

compar-ative simulation example, which illustrates the main features of non-stationary stochastic

embedding and model error modeling, the latter using eitherprediction error methods or

theproposedsetmembershipidenticationapproach. Benets andadvantagesofthethree

techniques are discussed. Finally, Section 6 gives some concluding remarks and suggests

ThebasicideabehindStochasticEmbeddingcanbedescribedasfollows. Thetruesystem

G

canbegivenas

G(iω) = ^

G

0

(iω) + ∆G(iω)

(1)

where

G

^

0

is a \nominal system" that can be exactly represented within a parameterized

family,and

∆G(iω)

isarandomvariableindependentof

^

G

0

. Noticethatthisisaturn-around oftheconventionalinterpretation,wherethenominal,estimatedmodel

^

G

0

(iω) = G(iω) + ∆G(iω)

is seen as the true system plus a model error

∆G

, that is independent of

G

. In contrast,

thetruesystemin(1)isarandomvariableas well,whichistherootoftheterm\stochastic

embedding". Now,supposethatwehavenoisyobservationsofthetruesystem

G

atcertain

frequencies:

^

^

G

k

= G(iω

k

) + ν

k

(2)

wherethenoiseterm

ν

k

isindependentof

G

and

∆G

. Then,combining(1)and(2),onehas

^

^

G

k

= ^

G

0

(iω

k

) + ν

k

+ ∆G(iω

k

).

(3)

Thisis thebasicestimationequation,thatneedstobecomplementedwitha

parameteriza-tion ofthe model

G

^

0

and assumptions about the variance of

ν

k

and

∆G(iω

k

)

. In [10]it is

suggestedtousealinearregressionparameterizationofthenominalmodelintermsofsome

orthonormal basis functions,i.e.

G

^

0

(θ) =

P

n

i=1

θ

i

B

i

, where

B := [B

1

, . . . , B

n

]

represents the selectedbasis and

θ

is thevectorofparameters. Moreover, themodelerror

∆G

is

parame-terized according tothe samebasis

B

and itis assumedthat the relativemodel error hasa variancethatincreaseslinearlyorquadraticallywith

ω

. Thismeansthat

∆G

canbewritten

as

B



θΛ

where

Λ

is a random walk process over

ω

(this argument being suppressed) and

θ

comesfrompriorknowledge. Allthisleadsto

G

=

G

^

0

(θ) + G

0

(

θ)Λ



(4)

=

Bθ + B

θΛ.



(5)

Theidenticationprocedurethenconsistsofthreemainsteps:

1. Pointwiseleastsquaresestimationofthetransferfunctionforcertainfrequencies:

there-fore,theinput

u

hastobeasumofsinusoids. Thisstepdeliversthevalues

^

^

G

k

,for

cer-tainfrequencies. Additionally, statisticalproperties ofthe noise

v

are calculated(i.e.

anunbiasedestimateofitsvariance),assumingGaussiannoise.

3. Estimation ofthe parameter

θ

and therandomwalk process

Λ

in eqn.(5):

^

_^

G

k

=

Bθ +

B

θΛ+ν



k

,basedonthefrequencyfunctionpointestimates

^

^

G

k

. Thisisusuallyperformed accordingtothefollowingprocedure:

(a) Estimationof

θ

(called

θ

^

),basedontheknowledgeof

^

_^

G

k

. Thus,thenominalmodel

^

G

0

=

B^θ

ineqn.(4) is the leastsquares estimateapproximationof

^

^

G

k

inthe

sub-spacespannedbybasisfunctions

B

.

(b) Asmodelerrorparameterization 

θ

,theestimate

θ

^

ischosen(thisisatypicalchoice, whennoaprioriinformationonunmodeleddynamicsis available). Thus,eqn.(5)

becomes:

^

^

G

k

=

B^θ(1 + Λ) + ν

k

.

(c) Computationofanunbiasedestimateofthevarianceoftherandomwalk

Λ

,

basi-callyusing anunbiased estimate ofthe varianceofthe noise, see[10] forclosed

formexpressions and technicaldetails. Here,the randomwalkis amodel whose

varianceofthefrequencyresponseincreaseslinearlywithfrequencyuptoacertain

rollo.

(d) Quantication of the model error

B



θΛ

for any frequency, i.e. calculation of its statisticalproperties.

Avariantoftherandomwalkforresonant systems(integratedrandomwalk)isdescribedin

[2]: here, the error increases quadraticallywith the frequency, in order to produce tighter

error boundsat lowfrequenciesinthepresenceofresonant poles.

Step2containsthecrucialchoiceofthesetof(orthonormal)basisfunctions. Inthesimple

caseofaLaguerreexpansion,onecanemploytheminimizationoftheaveragemodelingerror,

as suggestedin[1, 2]. Atanyfrequency

ω

j

, the total modeling error is givenby

G

e

(ω

j

) :=

B(ω

j

)^

θ−G(ω

j

)

. Foraclosedformexpressionof

G

e

(ω

j

)

anditscovariance

Σ

e

(ω

j

)

,thereaderis

referredto[11 ]. Wenotethat,fordierentchoicesofthepoleintheLaguerreexpansion,the

covariancematrixisafunctionoftheLaguerrepoleaswell:

Σ

e

(ω

j

) = Σ

e

(ω

j

, p

i

)

. Therfore,we

denetheaveragemodelingerror(overacertainnitefrequencygrid

Ω

)foraxedLaguerre

poleas

X

ω

j

∈Ω

trace

(Σ

e

(ω

j

, p

i

))

(6)

The\optimal"Laguerrepole

p

∗

i

istheonethatminimizesthemeanerror(6).

3 Model Error Modeling

Inthissection,thebasicideasofmodelerrormodelingaredescribed. Athoroughtreatment

of this approach is given in [15] , where prediction error methods are used to calculate

nominal model as well as error model. The idea described there, however, is muchmore

Let

(u

m

, y

m

)

beacollectionof i/o measurements and assumethata nominalmodel

G

n

ofthe systemthat generatedthedatahas beenestimated, accordingtosome identication

procedure. Then,themodelerror modelingstrategycanbesummarizedasfollows.

1. Computetheresidual

 = y

m

− G

n

u

m

.

2. Consider the \error" system, with input

u

m

and output



, and identify a model

G

e

for thissystem. This is an estimateof the error due toundermodeling, the so-called

modelerrormodel. Standardstatisticalpropertiesoftheestimatedmodelleadtothe

denitionoftheuncertaintyregionofthemodelerror(e.g.,givenbythe99%condence

regionoftheestimatedmodelerror

G

e

).

3. The uncertainty region of the nominal model is obtained by adding up the nominal

modelandtheuncertaintyregionofthemodelerror.

4. Model validation: the nominal model is not falsied if and only if it lies inside its

ownuncertaintyregionor,equivalently,ifand onlyif zerois anelementofthemodel

error uncertainty region. This can beeasily checked by lookingat the corresponding

uncertaintybandsintheBodeorNyquistplots.

Identicationofthemodelerrorfromresidualdataprovidesaseparationbetweennoise

andunmodeleddynamics. Infact,

G

e

canbeseenasanestimateofthedynamicsystem

∆(

·)

(possiblynonlinear,time-varying,etc.), suchthat

 = ∆(u) + v

(7)

where

v

isthenoiseterm,whichis assumedtobeuncorrelatedwiththeinput

u

.

Afewobservationscanbemade.

•

The modelerror

G

e

must beselectedamong apre-speciedclassofmodels. Unfortu-nately,thereisnotastandardprocedureforselectingthestructureofthemodelerror.

Thisisratherbasedonaprioriknowledgeonthesystemgeneratingthedata,andonthe

purposesforwhichthemodeluncertaintyisestimated. Forexample,atypical

require-mentis thatunmodeled dynamicsis capturedat thosefrequencies thataresignicant

forcontroldesignpurposes.

•

Thesizeoftheuncertaintyregionclearlydependsontheselectednominalmodel. When a severe undermodeling occurs (due for example to the fact that we are required to

delivera loworder nominalmodeltothe robust controldesigner), one cannotexpect

themodelerroruncertaintyregiontobesmallatallfrequencies,i.e. thenominalmodel

willbefalsied,whateverthestructureofthemodelerroris.

•

Afalsiednominalmodelmaystillbeaccepted, provided thattherelateduncertainty bandissmallatfrequenciesofinterest.

In thispaper, wewill adopt the general classof linearmodels proposed in[16] and apply

4.1 BasicIdeasand connectiontoModel ErrorModeling

ThetermSetMembershipIdenticationhasbeenusedinrecentyearstoindicateawide

va-rietyofrobustidenticationtechniques,thatareabletohandlehardboundsonthe

identi-cationerror. Theapproachconsideredinthispaperistheonetakenin[8]. Inthefollowing,

wesummarizeits mainfeatures.

Assumethat themeasuredi/odataaregeneratedby asystem

S

0

,accordingto

y

m

= S

0

(u

m

) + v

where

S

0

belongs to a set

K

(the a priori information on the system) and the noise

v

is bounded in somenorm

Y

, i.e.

kvk

Y

≤ δ

, for given

δ > 0

. Then itis possible todene the feasiblesystemset

FSS =

{S ∈ K : ky

m

− S(u

m

)

k

Y

≤ δ}

(8)

which is the set of systems that are compatible with the measured data and the a priori

assumptions. The set

FSS

can be very complicated, depending on the structure of

K

and the norm

k · k

Y

. For example, if

K

is the classof LTI systems whose impulse response has anassignedexponentialdecayand noiseis

`

∞

-bounded,then

FSS

isaninnitedimensional

polytope. Obviously,forreal-worldsystems,includingnonlinearitiesortime-varyingdrifts,

FSSmaybemuchmorecomplex.

Sincetheset

FSS

containsalltheinformationprovidedbydataandaprioriassumptions,

itisnaturaltoevaluatethequalityofanominalmodel

G

n

according toitsworst-caseerror

withrespecttoelementsof

FSS

. In otherwords,theidenticationerrorassociatedto

G

n

is

givenby

E(G

n

) =

sup

S

∈FSS

kS − G

n

k

S

(9)

where

k · k

S

isasuitablenorminthesystemspace.

Inorder toidentifyanominalmodel,amodelclassmustbechosen.Acommon

require-ment is that the nominal modelmust be simple(low dimensional,linearly parameterized,

etc.). Hence,atypicalstructurefor

G

n

is

G

n

(q; θ) =

n

X

i=1

θ

i

B

i

(q)

(10)

wherethe

B

i

(q)

areuserdenedbasisfunctions,suchasFIRlters,LaguerreorKautzlters

[23],generalizedorthonormalbasisfunctions[22],etc. Ifwedenoteby

M

thesetof nom-inal models parameterized as in (10), the problem of selecting a model in

M

according to

optimalnominalmodelis givenbytheconditionalcentralestimate

G

∗

_n

=

arg inf

G

∈M

sup

S

∈FSS

kS − Gk

S

.

(11)

Since in most practical situation nding an exact solution of the min-maxoptimization

problem(11) isaprohibitivetask,suboptimal estimators areconsidered[5]. Inthispaper,

we will assume that noise is

`

∞

-bounded (i.e.,

k · k

Y

= `

∞

) and select as nominal model

G

n

(q, θ

r

)

,where

θ

r

=

arg inf

θ

∈

IR

n

ky

m

−

n

X

i=1

θ

i

B

i

(q)u

m

k

∞

.

(12)

This is known as restricted projection estimateand enjoys someniceproperties, including

thefactthatitdoesnotdependontheactualvalueofthenoisebound

δ

. Herethischoiceis

motivatedonly by computationalreasons,as

θ

r

canbeeasilycomputedby linear

program-ming;other moresophisticatedestimationalgorithmscanbechosenwithoutmodifyingthe

wholerobustidenticationstrategy.

Whatever nominalmodel hasbeen identied, the actualsize of theunmodeled

dynam-ics can be evaluated from available data by exploiting the Model Error Modeling concept

described in Section3. The identicationof the error system(7) canbeperformed viaset

membership identicationalgorithms, exploiting the noisebound on

v

. If the structure of

themodelerrormodelischosenas

G

e

(q;

θ) =

 

n

X

i=1



θ

i

B



i

(q)

(13)

thefeasiblesetforthemodelerrorparametersisgivenby

FES =

{

θ



∈

IR 

n

_:

_{k −}



n

X

i=1



θ

i

B



i

(q)u

m

k

Y

≤ δ}.

(14)

Then,onemayselecttheworst-caseoptimalerrormodelbycomputingtheChebishevcenter

of

FES



θ

∗

=

arg inf 

θ

∈

IR 

n

sup ~

θ

∈FES

k

θ −

~

θ



k

(15)

where

k · k

denotestheEuclideannorm(ofcourse,othernorms canbechosen).

As for (11), problem (15) maybe computationally infeasible, and suboptimal solutions

are sought. For example, since for

`

∞

-bounded noise

FES

is a polytope in IR 

n

, it can be

recursivelyapproximatedbysimplerregions,likeellipsoidsorparallelotopes[4,3],andthe

centeroftheseapproximatingsetsmaybechosenasanestimateof 

θ

∗

.

If robust identicationis orientedto

H

∞

control,anuncertainty bandassociatedtothe nominalmodelfrequencyresponsemustbedeliveredtothecontroldesigner. Inthis

theset

FES

ontothecomplexplaneforeachfrequencyofinterest. Thisleadstothefrequency

domainuncertaintyset

V(FES) =

[

ω

V

ω

(FES)

where

V

ω

(FES) =

{z ∈

Cl

: z = A

ω



θ,

θ



∈ FES}

(16) and

A

ω

= [



B

1

(e

jω

) . . .

B

 

n

(e

jω

)]

∈

Cl

1

×

n



. Once again, the computation of

V(FES)

may be a formidabletaskiftheexact

FES

denedby(14)isconsidered. Hence,setapproximationsare

usefulalsointhisrespect. Exampleoftheresultingplotsfornominalmodelandmodelerror

willbeshown inSection5.

The overallsetmembershipidenticationstrategycanbesummarizedasfollows.

1. Identify the nominal model

G

n

(q; θ)

, using conditional estimators (12) based on the

feasiblesystemset(8).

2. Computetheresidual

 = y

m

− G

n

(q; θ)u

m

.

3. Selectamodelerrorstructure,choosethenoisebound

δ

andcompute(orapproximate)

thefeasiblemodelerror set(14).

4. Identifyanominalmodelerror

G

e

(q;



θ)

,usingoptimalorsuboptimalestimators based on

FES

.

5. Mapthe nominalmodelplusthe modelerror and its uncertaintyregiononto the

fre-quencydomain.

Remark. The use of residual data in set membership identication for evaluating the

worst-case

H

∞

norm ofthe unmodeled dynamics has beenintroduced in[9], for standard least-squares nominal models. The above set membership model error modeling strategy

canbeseenasageneralframeworkinwhichthestructuresofthenominalanderrormodels,

andthecorrespondingidenticationalgorithmsmustbechosenbytheuseraccordingtothe

specicproblem(a prioriknowledge,noisebound,error norm,etc.).

Inthenextsubsection,itwillbeshownhowitispossibletoexploittheaboveframework

toobtainareliable measureofthe sizeof theunmodeled dynamics. This will alsoprovide

usefulinformationabout theselectionofthenominalmodelclass.

4.2 Model Validationand ModelClass Selection

Lift this to a seperate section (WR+AG)?

In the Set Membership procedure previously described, a key parameter is the noise

Hence,

δ

canbeusedasatuningparameter,inordertoevaluatethe\distancetovalidation"

oftheselectednominalmodel.

Letusconsidertheset

V

ω

(FES)

in(16)anddene

d(ω) =

min

z

∈V

ω

(FES)

|z|.

(17) Ifweset

d =

sup

ω

d(ω)

(18)

thenitisclearthatthenominalmodelcanbedeemedtobeunfalsiedif

d = 0

. Ifellipsoidal

approximationsof

FES

areconsidered,thesets

V

ω

(FES)

areellipsesinthecomplexplaneand theoptimizationproblem(17) canbeeasilysolved. Then,

d

in(18) canbeapproximatedby

takingthemaximumoveranitenumberoffrequenciesintherangeofinterest.

Itisworthobservingthat

d

isafunctionof

δ

,since

FES

(anditsfrequencyimage

V

ω

(FES)

) dependon the noisebound. Then,one canlook forthe minimumvalue of

δ

for whichthe

nominalmodelisstillnotfalsiedbydata,i.e.

δ

∗

=

min

{δ: d=0}

δ.

(19)

Since

d

is anincreasingfunctionof

δ

, thecomputationof

δ

∗

canbeeasilyperformed within

thedesiredprecision, usingastandardbisection on

δ

. Obviously,

δ

∗

willnotbelargerthan

thevalueofthenormgivenineqn.(12),whichistheminimumnoisebound forthenominal

modelwhenusingnomodelerror model.

Itisbelievedthatthequantity

δ

∗

isimportantinseveralrespects. Inparticular,itallows

one to deliver the smallest frequency domain uncertainty region associated to the model

error model, which does not falsify the nominal model. This meets a typical requirement

ofthecontroldesigner: the\tightest"uncertaintybandaroundthenominalmodel,without

contradictingtheinformationgivenbydata. Clearly,

δ

∗

dependsonthemodelclassselected

for identicationof the model error model: the\richer" thisclass, thesmaller

δ

∗

. On the

otherhand,atoosimplemodelclassmayresultinalargevalueof

δ

∗

,whichwouldcontradict

the a priori assumptionon the noise size. Hence,

δ

∗

can be usedto \validate" either the

noiseboundor themodelerrormodelclass,accordingtotheavailableaprioriinformation.

Ifareliableaprioriestimate

^

δ

ofthenoisesizeis available,themodelclassformodelerror

identicationshouldbechosensothat

δ

∗

turnouttobeascloseaspossibleto

^

δ

. Conversely,

ifthemodelerrormodelclassissuggestedbyaprioriinformationaboutthesystemdynamics,

δ

∗

providesavalidationtoolforthenoisebound(forexample,if

δ

∗

ismuchsmallerthanthe

aprioribound,one mayconcludethatthelatterwasoverestimated).

Anotherusefulpropertyof

δ

∗

isthatitgivesarationaleforselectinganappropriatemodel

classfornominalmodelidentication. Letusdenoteby

p

avectorofparametersthatdene

a model class for

G

n

(for example, the numberand/or the pole locations of the family of

modelandmodelerrormodelassuggestedinSection4.1,andcompute

δ

∗

_(p)

. Then,atleast

inprinciple,onecanchoose theoptimal modelstructurebysolving

inf

p

δ

∗

_(p).

(20)

In practice, the optimization in (20) canbe performed over a nite set

P = {p

1

. . . p

m

}

(see theexampleinthenextsection,where

p

isthepoleofaLaguerreexpansionofxedorder).

5 Simulation Example 5.1 ExperimentalSetup

Gnom

u

zg

zsat

zf

F

y

noise

Figure1: Theexperimentalsetup: Alinearsystem

G

nom

inparallelwithalinearFilter

F

and

asaturationnonlinearity. Theoutputsignal

y

iscorruptedbynoise.

ConsidertheexperimentalsetupasshowninFig.1. Itscoreconsistsofthefollowingfth

orderlinearsystem

G

nom

(s) =

24.15(s + 1)

0.012s

5

_{+ 0.25s}

4

_{+ 1.80s}

3

_{+ 5.86s}

2

_{+ 7.33s + 1}

.

Its dynamics are perturbed by a nonlinearity in the medium frequency range. This is due

toa parallelconnectionwith alinear secondorder lter

F(s) =

2s

0.1s

2

_+s+0.09

and a saturation nonlinearity, that saturates at level

l = 1.2

. According to theamplitude curve ofthe lter,

whichisgiveninFig.2,thenonlinearitywillbe\active"inthefrequencyrange

ω = [0.1; 10]

,

dependentontheamplitudeoftheinputsignal. Thesystemwillbehavelinearly,whenthe

outputofthelter

z

f

issmallerthan

1.2

inamplitude;then

y = z

g

+ z

sat

= (G

nom

+ F)u

holds.

In order to be able to apply Stochastic Embedding, the input signal we use is a linear

combination of

53

sinusoids(thecontainedfrequencies aremarkedas crossesinFig. 4). We

use a sampletime of

T

s

= 0.04s

and

3000

samples. We will use four dierent data sets to

examinethebehaviorofthelinearidenticationmethodstothisnonlinear plant,whichare

10

−2

10

−1

10

0

10

1

10

2

10

−1

10

0

10

1

Amplitude

Amplitude, Filter F

Figure2: Amplitudeplotofthesecondorder linearlter

F

.

0

20

40

60

80

100

120

−10

−5

0

5

10

input u

0

20

40

60

80

100

120

−40

−20

0

20

40

noisy output y

0

20

40

60

80

100

120

−2

−1

0

1

2

noise

time/s

0

20

40

60

80

100

120

−15

−10

−5

0

5

10

15

filter output z

f

0

20

40

60

80

100

120

−1

−0.5

0

0.5

1

sat output z

sat

time/s

Figure3: Selectedsignalsfromdataset3. Left: Testinput

u

,noisyoutput

y

andoutputerror

noise(top-down). Right: Outputofthelter

F

,

z

f

,and ofthesaturationnonlinearity,

z

sat

.

Dataset excitationofsaturation inputlevel(rsthalf) inputlevel(sec.half)

1 none(linearbehavior)

1

1

2 low

1

2

3 medium

1

6

4 high

10

10

Experiments1 and4excitethesystemwith aninputsignal asdescribed abovewithan

am-plitude of about

1

and

10

respectively. These input signalswill lead toan either linear or

completelynonlinearbehavior. Experiments2and3switchfromalowamplitudeinput

sig-naltoahigherone. Wenote thattheresultsobtainedlateron willnotchangesignicantly

when switching form a higher to a lower amplitude. For experiment 3, these signals are

reportedinFig.3.

corre-10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

Amplitude

Spectral analysis of Noiseless Data: D1 (k−), D2 (b:), D3 (g−.), D4 (r−−)

Frequency, Excited frequencies (kx)

10

−2

10

−1

10

0

10

1

10

2

10

−1

10

0

10

1

10

2

Amplitude

Spectral Analysis of Noiseless Data (E1, −); and linear model (−−)

Frequency, Excited frequencies (kx)

Figure4: Left: Spectralanalysis(employingasmallsmoothingwindow)ofthefournoiseless

datasets: dataset1(solid),dataset2(dotted),dataset3(dash-dotted)anddataset4(dashed).

Thecrossesonthefrequencyaxismarkthose53frequencies,containedinthemulti-sinusoid

inputsignal. Right: Bodeplotofthelinear\mode"

G

nom

+ F

(dashed)incomparisontothe

spectralanalysisofdataset1(solid).

sponding inputsignal

u

. The resultis depictedinFig. 4 (left), togetherwith theBode plot

of the linear system

G + F

, which will be \the true plant" for low input level (right). All

frequency proles are quite similarfor low frequencies, while they are signicantly

dier-ent at higher frequencies. Clearly,the frequency measurements obtained inthis case may

changefor dierent inputsignals, especiallyforaninputsignal with acompletely dierent

powerspectrum. Nevertheless,wekeeptheobtainedfrequencyprolesasareferenceinthe

comparisonofthedierentmethods,forthespecicinputsselected.

Add something about 2nd moment equivalent (LL)

Additionally, theoutput is corruptedby noise, whichis alsoshown in Fig.3 (left). The

noiseisanormallydistributedonewithvariance

0.0102

,addedtoascaledversion(toa

max-imum amplitudeof about

1.4

) ofthe

200Hz

signal registered during theLomaPrieta

earth-quakeintheSantaCruzMountainsinOctober1989(whichhasbeenmeasuredattheCharles

F.RichterSeismologicalLaboratoryandmadebeenavailablebyTheMathWorksInc.). This

noisesignal hasbeen choseninorder toavoid usageof standardstationary stochastic

pro-cessesor boundaryvisitingsignalsasnoisemodels.

Inthefollowing,datasets1-4willbeusedforidenticationpurposes. Dierentdatasets

(obtainedbyaddingadierentoutputnoise)willbeemployedinthemodelvalidationstep,

for those methods that allow an explicit validation procedure (i.e. Model Error Modeling

andSetMembershipIdentication).

Thegoaloftheidenticationexperimentsistoobtainanominalmodelofordernotlarger

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

pole location p

average modeling error SigmaGbar

Figure 5: NSSE: Laguerre pole vs. mean error over all frequencies, in order to choose the

\optimal"Laguerrepole,thatguaranteesminimumerror. Result isbasedondataset4.

sizeforrobustcontrollerdesignusing standard

H

∞

methods.

5.2 Identication via Non-Stationary StochasticEmbedding

The rst stepwithin non-stationarystochasticembedding is the estimation ofthe transfer

functionatthosefrequenciescontainedintheinputsignal. Thesecondstepisthenthe

esti-mationofthenominalmodel,basedonthesetransferfunctionpointestimates. Byproblem

denitionwearerestrictedtomodels oforder

4

,thus wechoose suchaLaguerreexpansion

forouridentication. Inordertoobtaincomparableresults,weuseacondencelevelof

99

%

hereaswellasforthepredictionerrormethodsintheModelErrorModelingapproach. We

remark, however, that theshape of the uncertainty band does not change paramountly at

lowandmediumfrequencies,whendecreasingthecondencelevel. Moreover,preliminary

testrunsshowthattheintegratedrandomwalkwillproduceuncertaintybandsthatarequite

tightforlowerfrequenciesanduselesslargeforhigherfrequencies,whereastherandomwalk

(increasing linearly with frequency)produces reasonableuncertainty bands. Therefore, we

conclude that thesystemdoes not containimportant resonances, and continuewith

error-propagationbyrandomwalkstrategy. WeareleftwiththechoiceofthepoleintheLaguerre

basis. Therefore, weemploythe minimizationofthe average modeling error,as suggested

inSec.2. A typicalplotof meanerror vs. polelocationforxedorder is reportedinFig. 5

(dataset 4).

Choosing pole for the Laguerre expansion to

p = −0.2895

(dataset1,2) and

p = −0.5737

(dataset 3,4)respectively, leastsquares estimation ofthe nominalmodel and error

propa-gationleadstotheresults asdepictedinFig. 6. Note, thatchoiceoferror propagation and

condenceleveldonotin uencethenominalmodel. Inallfourcases,theuncertaintyband

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

For Comparison: Spectral analysis of Dataset (b−−)

Frequency

Magnitude

Estimated model

Point Estimate

99.99% conf. cloud, random walk

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

For Comparison: Spectral analysis of Dataset (b−−)

Frequency

Magnitude

Estimated model

Point Estimate

99.99% conf. cloud, random walk

Figure 6: NSSE: Identication based on datasets 3 (left) and 4 (right) respectively. Both

approachesusea

4th

orderLaguerreexpansionwithpolelocatedat

p = −0.5737

andrandom

walkaserrorpropagation. The

99

%condencecloudsaredrawn(shaded),aswellastransfer

functionpointestimates(crosses), estimatednominalmodel(solid)andspectralanalysisof

noiselessdata(dashed).

5.3 Identication via Prediction ErrorMethods

WeestimateanOutputErrormodeloforder

4

asnominalmodelusingpredictionerror

meth-ods. The result fordataset1,togetherwiththestandardresidualtest,isreportedinFig.7.

The realplantis clearlynotinthe condenceregion,whichis rathersmalldue tothelarge

amountofdata(3000samples) usedhere. Moreover,the modeldoesnot seemtopass the

classicalresidualtest. A similarstatementholds for the remaining threedatasets. Hence,

weareinneed ofmore detailed informationabout the accuracyof the identied

4rth

oder

OE model. Basedon theobtained nominalmodel, wethereforeproceed withestimationof

anerrormodelbasedontheresidualdata(calculatedfromthevalidationdataset). Acrucial

stepistochoosethestructureoftheerror model,sothatthenominalmodelisnotfalsied

byitsownerrormodel,i.e.zerohastobeinsidethe,say99%,condenceregionoftheerror

model.

Wechooseaerror modeloftheform

A(q)(t) =

B(q)

F(q)

u(t − n

k

) +

C(q)

D(q)

v(t)

withpolynomialorders

n

A

= 1, n

B

= 9, n

C

= n

D

= 5, n

F

= 10

anddelay

n

k

= 1

forvalidation

dataset1. TheresultisreportedinFig.8(left). Themodelerrormodelvalidatesthenominal

model,atthepriceofalargercondenceregion(comparedtotheoneinFig.7).

Wefollowthesamereasoningforvalidationdataset4. Aswefacenon-trivial

straight-10

−2

10

−1

10

0

10

1

10

2

10

−1

10

0

10

1

10

2

Amplitude

Nominal plant (b−) and Real plant (r−−)

0

5

10

15

20

25

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Correlation function of residuals. Output validation noise data

lag

−25

−20

−15

−10

−5

0

5

10

15

20

25

−0.2

−0.1

0

0.1

0.2

0.3

Cross corr. function between input Sum of 53 sinusoids and residuals from output validation noise data

lag

Figure7: PEM: Left:

4rth

order outputerror model, using dataset1for identication. The

99

% condenceregionis givenas well(shaded). Right: Correlation functionoftheresiduals (upperplot) andcross-correlationfunction(lowerplot) usingthevalidation dataset.

forward how to choose order and structure of the model error model in order to validate

the nominal model. After several trials, we used an error model with polynomial orders

n

A

= 4, n

B

= 9, n

C

= n

D

= 5, = n

F

= 10

and delay

n

k

= 1

, whichdoes not verify the

nomi-nalmodel,butgives betterinformationonthefrequencyrangewheretheerror occurs. The

result isreportedinFig.8(right).

5.4 Identication via SetMembershipEstimation

Toobtaina

4rth

ordernominalmodelweusea

4rth

orderLaguerreexpansion. Thepolesof

theLaguerreexpansionarechosenaccordingtothestrategysuggestedinSection4.2. Hence

we will use the hard bound on the noise as additional tuning parameter for the selection

ofthe modelclass. A typicalpictureof Laguerrepole locationvs.minimum, non-falsifying

noisebound isreportedinFig.9.

Using dataset 1,3 and 2,4, weend up with a Laguerre pole at

p = 0.97

and

p = 0.98

re-spectively. Duetotheincreasingamountofnonlinearity,theminimum,non-falsifyingerror

bounds increaseby

δ = 0.70, 0.83, 1.29, 2.63

fordatasets 1-4 respectively. These \identied"

noiseboundsfordatasets1-3coincidequitewellwiththesizeoftheactualnoiseofabout

1

,

asgiveninFig.3. Clearly,thehighervaluefordataset4isdue toasignicantcontribution

fromthenonlinearity.

In order toobtainafrequencydomainuncertaintyrepresentation,we followthe

proce-duresuggestedinSection4.2. Toreducethecomputationalburden,weapproximatetheFES

in(14) via ellipsoids[4]. Fig. 10 reports the nal result for datasets1 and 4. Weobserve,

10

0

10

1

10

2

10

−1

10

0

10

1

Nominal plant (b−) and Real plant (r−−)

Magnitude

10

0

10

1

10

2

10

−4

10

−2

10

0

10

2

Model Error Model With Uncertainty Region

Frequency

Magnitude

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

Nominal plant (b−) and Spectral analysis (r−.)

Magnitude

10

0

10

1

10

2

10

−4

10

−2

10

0

10

2

Model Error Model With Uncertainty Region

Frequency

Magnitude

Figure8: PEM:Left:

4rth

orderoutputerrormodel,usingdataset1foridentication. Model

errormodelingwitha

(1,9,5,5,10,1)

-errormodelusingvalidationdataset1. Therealplant

is shown as well(dashed). Right:

4rth

order output error model, using dataset4 for

iden-tication. Model error modeling with a

(4,9,1,1,11,2)

-errormodel using validation data

set4. Upperplots: nominalmodel(solid)alongwithuncertaintyregion(shaded) and

spec-tralanalysis ofnoiseless data(dashed). Lowerplots: modelerror modelswith uncertainty

regions.

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

0.6

0.8

1

1.2

1.4

1.6

Laguerre pole

minimum noise bound for model error

Figure9: SMI:LocationofLaguerrepoleversusminimum,non-falsifyingnoisebound ofthe

10

−2

10

−1

10

0

10

1

10

2

10

−4

10

−2

10

0

10

2

Frequency

Magnitude

Nominal plant (b−) and Real plant (r−−)

10

−2

10

−1

10

0

10

1

10

2

10

−4

10

−2

10

0

10

2

Frequency

Magnitude

Model error model (b−) with uncertainty region

10

−2

10

−1

10

0

10

1

10

2

10

−4

10

−2

10

0

10

2

Frequency

Magnitude

Nominal plant (b−) and Spectral analysis (r−.)

10

−2

10

−1

10

0

10

1

10

2

10

−4

10

−2

10

0

10

2

Frequency

Magnitude

Model error model (b−) with uncertainty region

Figure10: SMI:Fordatasets1(left)and4(right): Theupperplotshowsrestrictedprojection

estimate (solid) insidethe unfalsieduncertainty region(shaded) compared to linearplant

(left plot) and spectral analysis of noiseless data (right plot) respectively. The lower plots

showthemodelerror modelinsideits unfalsieduncertaintyregion.

ranges. Moreover,theuncertaintybandisreasonablysmallandincludesthefrequency

pro-lesprovided byspectralanalysis.

5.5

Why the comparison in the next subsection?

(LL)

5.6 Comparison

Thesystemgeneratingthedata(seeFig.1)containsasaturation,whichisanoddstatic

non-linearitywithagainvaryingbetween

0

and

1

. Wethereforecomparetheuncertaintyregions,

deliveredby identication with the magnitudeenvelope given by

G

nom

and

(G

nom

+ F)

as

lowerand upperbound respectively. For easeofnotation,wewillrefertothelatterregion

as the \true uncertainty region". Figures 11and 12report the results fordatasets 2 and 4

respectively. The identied uncertainty regions should cover parts of the true uncertainty

region,obviouslydependedoftheexcitationofthenonlinearitybytheinputsignal.

Weobserve,thatforbothdatasetstheuncertaintysetproducedbypredictionerror

meth-odsis misleadinglytight,becauseof alarge amountof data(

1500

samples). Moreover, the

nominalmodelisfalsiedforthemediumfrequencyrangefordataset4,seeFig.8.

Moreover, we note that for dataset 4, Fig. 12, preferably the \lower part" of the true

uncertainty region is captured by all methods. This is clear due to the lower gain of the

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

Bode Magnitude: nominal model (r−) and G

nom

, G

nom

+F (b−−)

Frequency

Magnitude

Estimated model

Point Estimate

99.99% conf. cloud, random walk

10

−2

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

Bode Magnitude: nominal model (b−) and G

nom

,G

nom

+F (b−−)

Magnitude

Frequency

10

−2

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

Frequency

Magnitude

Bode Magnitude: nominal model (b−) and G

nom

,G

nom

+F (b−−)

Figure11: Dataset 2: Nominalmodel(solid)anduncertaintyregion(shaded) incomparison

totheamplitudesof

(G

nom

+

1

2

F)

±

1

2

F

(dashed). FromlefttorightforNSSE(continuoustime), MEMandSMI(discretetime).

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

Bode Magnitude: nominal model (r−) and G

nom

,G

nom

+F (b−−)

Frequency

Magnitude

Estimated model

Point Estimate

99.99% conf. cloud, random walk

10

−2

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

Bode Magnitude: nominal model (b−) and G

nom

,G

nom

+F (b−−)

Magnitude

Frequency

10

−2

10

−1

10

0

10

1

10

2

10

−2

10

−1

10

0

10

1

10

2

Frequency

Magnitude

Bode Magnitude: nominal model (b−) and G

nom

,G

nom

+F (b−−)

Figure12: Dataset 4: Nominalmodel(solid)anduncertaintyregion(shaded) incomparison

totheamplitudesof

(G

nom

+

1

2

F)

±

1

2

F

(dashed). FromlefttorightforNSSE(continuoustime), MEMandSMI(discretetime).

Our comparison addresses three dierent approaches to robust identication: Stochastic

Embedding, Model Error Modeling and Set Membership Identication. The rst two

ap-proaches have been developed in the statisticalframework, while the latter relies on UBB

errorassumptions. Duetothedierentnatureofthethreemethods,anoverallcomparison

mightbecomeunfairat acertainstage. Duetorestrictionsonharmonicinputsignal for

ex-ample,onemight notalwaysbeabletoapplyStochasticEmbeddingtoagivendataset. We

thereforeconcentrateourselves onhighlighting themostimportant propertiesand benets

ofeachmethod.

Model Error Modeling and Set Membership Identication allow an explicit validation

step (on freshvalidation data). Asno furtherknowledge than the nominali/o behavior is

needed,theseframeworksalsoallowvalidationofanalreadyexistingmodelfortheprocess.

Evenwhen invalidated, valuable information on the underlying error canbe obtained, for

instanceinwhichfrequencyrangethemodelissuÆcientlyreliable.

Incontrast,StochasticEmbeddingallowsthechoicebetweendierentrandomwalk

mod-els(thevariancemayincreaseindierentwayswithfrequency)toperformtheerror

propa-gation. Althoughthisisnoexplicitvalidationstep,itisaquitereasonableconcepttoproduce

reliablecondenceregions.

The Model Error Modeling setup enjoys exibility in the choiceof the structure of the

nominal model, to end up with for instance OE or ARX models. In contrast, Stochastic

Embeddingand the presentedSetMembership Identicationare(thelatteratleastinthis

context)tiedtoaparameterizationoforthonormalbasisfunctions. Bothschemes,however,

comealongwitharationaleforchoiceofthepoleinaLaguerreexpansion. Itisworthtonote

that both end up with basicallythe samerecommendation(in our caseat

s = −0.29, −0.57

,

correspondingto

z = 0.98, 0.97

indiscretetime).

Aremainingprobleminthemodelerrormodelingapproachisthecorrectorderand

struc-tureselectionofthemodelerrormodel,whichisingeneralnotstraightforward. This

prob-lemhasalsobeenreportedin[21],wheretheauthorssuggestanadaptiveandnonparametric

frequency-domainmethodthatestimatesthefrequencyresponseofthemodelerrorandalso

allowslocaltuningindierentfrequencybands.

We observe, that all methods deliver an estimated nominal model, along with an

un-certaintyregion,whichis certainlysuitedforrobustcontroller design,forinstanceusingan

H

∞

mixedsensitivity approach, based on the estimateduncertainty band in thefrequency domain.

6 Conclusions and Future Works

Wecomparedidenticationmethods,workinginthetimeandfrequencydomainand

dependon the apriori knowledge (for example,if harmonicinput signalsare possibleor a

prioriinformationonthe noiseamplitudeor statistics areavailable). However,weshowed

themain features ofthemethods andobtainedreliablenominalmodels andacceptable

re-lateduncertaintiesinallthreecases.

Some ‘‘nonlinear’’ comments (LL)

Moreover, theideasof modelerror modeling andtheirapplicationinthecontextofset

membership identication have been analyzed. In particular, it has been shown that the

separationofnoiseandunmodeleddynamicsisquitenaturalinthisframework,andthatthe

minimumnoisebound forwhicha nominalmodel is notfalsied by the data canbeeasily

computedand usedasatoolformodelclassselection.

Complementaryworkhasstilltobedone,inseveraldirections. Forexample,more

com-plicatedmodelerrormodelstructuresareneeded,tocopewithnonlinearitiesortime-varying

drifts. Obviously,thisrequiresmoresophisticatedidenticationalgorithmsandsmarter

ap-proximationsinthecomputationoftheuncertaintyregions. Moreover,accuratecriteriafor

the a priori selectionof the structureof the model error model, depending on the specic

control-orientedidenticationproblem,must stillbeinvestigated.

References

[1] J. H. Braslavsky. Model error quantication via non-stationary stochastic

embed-ding: Hairdryer example. CIDAC, Dept of Electrical Engineering, Univ of Newcastle,

Callaghan,Australia.ManualforNSSE-package,Mar.1999.

[2] J.H. Braslavskyand G. C.Goodwin. A noteon non-stationarystochasticembedding

for modelling error quantication in the estimation of resonant systems. Technical

Report EE99014,CIDAC,DeptofElectricalEngineering,UnivofNewcastle,Callaghan,

Australia,Jan.1999.

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