Proceedings of the 40th IEEE

Guaranteed numerical alternatives to structural

identiability testing

I. Braems 1

, L.Jaulin 2

,M. Kieer 1

and E.Walter 1

1

LaboratoiredesSignauxetSystemes,CNRS-Supelec-UniversiteParis-Sud

91192Gif-sur-Yvette,France

fbraems,kieer,walterg@lss.supelec.fr

2

Laboratoired'IngenieriedesSystemesAutomatises,Universited'Angers

2bldLavoisier,49045Angers,France

luc.jaulin@univ-angers.fr

Abstract

Testing models for structural identiability is particularly

important forknowledge-basedmodels. Ifseveralvaluesof

theparametervectorleadtothesameobservedbehaviorof

the model, then one may try to modify the experimental

setup to eliminate this ambiguity (qualitative experiment

design). Thetediousnessofthealgebraic manipulationsin-

volvedmakescomputeralgebraparticularlyattractive. The

purposeofthispaperistoexploreanalternativeroutebased

onguaranteednumericalcomputation. Anewdenitionof

identiabilityinadomainallowstestingtobecastintothe

frameworkofconstraint-satisfactionproblems,andmakesit

possibletousethetoolsofintervalanalysisandintervalcon-

straint propagation to get guaranteed answers. When the

data have already been collected, the notion of structural

identiabilitymaynotbethemostpertinentconcept. This

paper shows how interval analysis and interval constraint

propagationcanagain beused tobypassthe identiability

studyandestimateevenparametersthatarenotidentiable

uniquely.

Keywords: constraintsatisfactionproblems,compart-

mental models, experiment design, guaranteed numer-

icalcomputation, identiability, identication, interval

analysis,parameterestimation.

1 Introduction

Underlyingthenotionofidentiabilityisthequestionof

whetheronecanhopeuniquelyto estimatetheparam-

eters of a model from the experimental data that can

becollected. This question is particularly relevantfor

knowledge-basedmodels, where theseparameters have

aconcrete meaning, and whenever decisionsare to be

takenonthebasisoftheirnumericalvalues. Theimpor-

tance of thenotion has been recognized morethan 50

yearsago[1],butmuchremainstobedoneto convince

potential usersand to provide them with tools to test

theirmodelsforidentiability. Asthealgebraicmanip-

ulations involved in identiability testing can become

really tedious, the use of computer algebra and elimi-

nation theory to obtain reliable results is particularly

attractive,see,e.g.,[2]-[7].

The purpose of the present paper is to explore an al-

ternative route, mainly based on numerical computa-

tion,anddiscussitsadvantagesanddisadvantages. The

paperis organizedasfollows.Thenotion ofstructural

identiabilityisbrieyrecalledinSection2,wheresome

limits of aformal approach are alsomentioned. These

limits are the rationale for the development of an al-

ternativenumericalapproachbasedonintervalanalysis

and interval constraint propagation in Section 3. Sec-

tion 4 shows that these toolsmay make it possibleto

bypassthestructuralidentiabilitystudyaltogetherby

allowing oneto characterize, in a guaranteed way, the

set ofall valuesof the parametervectorthat are con-

sistentwiththeexperimentaldatacollectedonthesys-

temtobemodeled.Itthusbecomespossibletoidentify

unidentiablemodels.

2 Formal approach to structural identiability

Structuralidentiabilityis studiedin anidealizedcon-

text where thedata are assumedto begenerated bya

model with the same structure M(:) as the model to

beidentied.Theunknowntruevalueoftheparameter

vector is denoted by p

, and is assumed to belong to

somepriorfeasiblesetPR dimp

. Fromnoise-freedata

generatedbyM(p

),onewantstoestimatetheparam-

eters of amodel M(bp). In these idealized conditions,

it isalwayspossibleto tunepb so asto ensurethat the

outputs generated by the \model"M(bp) are identical

to those generated by the\process" M(p

) forall in-

putsandtimes.Theidentityoftheseexternalbehaviors

will thenbeexpressedconciselyby

M(b

p)=M(p

): (1)

Theidentiabilityof bponlydepends onthenumberof

solutionsof(1)forbpin P.Iftheset Sofallthesesolu-

tionsreducestothesingletonfp

g,thenpbisglobally(or

ThM06-6

Proceedings of the 40th IEEE

Conference on Decision and Control

Orlando, Florida USA, December 2001

uniquely) identiable at p . If Sisdenumerable (most

oftennite),thenbpislocally (orcountably)identiable

at p

.If Sis notdenumerable, thenbpisunidentiable

at p

. Ofcourse, agiven entrypb

i

of bpmay be locally

orglobally identiableevenwhentheentireparameter

vectorisunidentiable.

Nowthisidentiabilitystudymaytakeplacebeforedata

collection. This is actuallyadvisable, asthe resultsof

theidentiabilitystudymayimpactontheexperiments

tobeconducted[8].Nonumericalvalueisthenavailable

for p

, and we would like our conclusions to be valid

whateverthevalueofp

maybe. Unfortunately,thisis

notalwayspossible,whichledtothenotionofstructural

(orgeneric)identiability.Aparameterp

i

isstructurally

globally identiable (s.g.i.) if, foralmost any p

in P,

the set Sof the solutions of (1) for pb

i

reduces to the

singletonp

i

.Itisstructurally locallyidentiable (s.l.i.)

if,foralmostanyp

in P,Sisdenumerableornite.If

Sisnotdenumerable,foralmostanyp

inP,thenp

i is

structurallyunidentiable. Ofcourse,piss.g.i. ors.l.i.

ifallofitscomponentsares.g.i. ors.l.i.

When testing model structures for identiability, the

standardapproachisintwosteps. Duringtherststep,

the equations to besatised bypb for (1) to hold true

areestablished, andthis is usuallymuch facilitatedby

theuseofcomputeralgebra. Veryoften,(1) translates

intoasetofequations:

r(bp )=r(p

): (2)

During the second step, the set of all solutions of (2)

for pb is sought for. When each component of r(p) is

polynomialin the entries of p; elimination theory can

beused to put this set of polynomial equationsinto a

triangularform

t(bp)=t(p

); (3)

whereagaineachcomponentoft(p)ispolynomialinthe

entriesofp. It thenbecomespossible,at leastinprin-

ciple,tosolve(3)bysolvingasuccessionofpolynomial

equationsinasingleunknown,inawaysimilartothat

usedtosolvelinearsystemsofequationsbytriangular-

ization,andthustoobtainthesetSofalltheparameter

vectors b

pthatsatisfytheseequations. Themethodused

tobuild(3)from(2)guaranteesthatnosolutioncanbe

lost,soifthedegreeofeachofthepolynomialequations

tobesolvedintheprocessisgenericallyone,thenithas

beenproven that pis s.g.i. Evenwhen this is notso,

itis sometimespossibleto generatetheset of allsolu-

tionsof(2)forpbasanexplicitfunctionofp

,whathas

beencalledexhaustive modeling [9]. Providedthat the

cardinalofthissetkeepsthesamevalueforalmostany

valueofp

,thisformalapproachallowsonetoconclude

about thestructural identiability of themodel under

study.

Testingmodelsforstructuralidentiabilityisaveryin-

teresting domain of application for computer algebra,

becauseonelooksforasimpleanswertothesimplequal-

itativequestionofwhetherthemodeliss.g.i. Thereare,

howeverseveralreasonswhytheapproachmayturnout

nottobesatisfactory.

{ (1) may not translate into polynomialequations, or

thetranslationmaymaketheproblemexceedinglycom-

plicated tosolve.

{Reachingaconclusionmayrequireformal manipula-

tionsthat aremuchmorecomplicatedthanallowedby

present-daycomputers.

{ Thedegreeof someof theunivariatepolynomialsto

besolvedmaybetoolargeforananalyticexpressionto

exist,whichthenimposestheuseofnumericalmethods

and thelossoftheformalnature ofthesolution.

{Thenumberofreal solutionsforthebp

i

's(theonlyones

in whichweareinterested) maydependonthevalueof

p

insuchawaythatnoconclusionofastructuralna-

ture can be reached. Thefollowing examplewill serve

to illustrate this point. It will be treated in Section 3

withthealternativeapproachadvocatedin thispaper.

´ ( ) p ^¤

p ^¤

−2

−2 2

−4 4

−6 6

−a a

−1 1 2

Figure 1:OutputofthepolynomialmodelofExample1;

thegreyzonecorrespondstothevaluesofp

for

whichthemodelisnotuniquelyidentiable

Example1 Consider the model

(p)=p(p 1)(p+1);

with one parameterpin P=[ 2;2]and noinput. For

any (bp;p

) inPP,M(bp)=M(p

) ifandonly if

p

(p

1)(p

+1) p(bbp 1)(bp+1)=0:

The set of feasible values for pbis a singleton for p

2

] 2; a[[]a;2[, a pair for p

= a or p

= a, and

a triple for p

in ] a;a[, with a = 2

p

3

(see Figure 1).

So unique identiability isnot a structural property in

P.

3 Alternative guaranteed numericalapproaches

Theroute tobefollowedwilldependonwhether anu-

mericalvaluecanbegiventop

.

3.1 Anumericalvalue can be given to p

This willbethecase,for instance, whenasatisfactory

model with structureM(:) hasalreadybeenidentied

fromexperimentaldataandoneisinterestedinestimat-

ingallthemodelswiththesamestructureandthesame

externalbehavior.ItthensuÆcestotakeforp

thenu-

mericalvalueofthevectoroftheestimatedparameters,

andtogetareliableestimateofthenumberofsolutions

of (2)for bpbyusing aguaranteedsolver. This canbe

donebymeansofintervalanalysis[10],[11]. Solving(2)

amountsto nding the zeros of thefunction f dened

inPby

f(bp)=r(bp ) r(p

): (4)

When dimp = dimf(p), under the hypothesis that f

iscontinuouslydierentiable,anintervalNewtonsolver

can be employed to enclose all these zeros in a union

L of interval vectors (or boxes)[bp] [12]. The width of

these boxesdependson atuning parameter witha di-

rectimpactontheamountofcomputationrequired.In

this solverwe used the Newton-Gauss-Seidel operator

N

GS

([p]) [12],which hasthefollowingproperties:

(i) each zero b

p of (4) in any box [p] satises b

p 2

N

GS ([p]);

(ii)ifN

GS

([p])=;,thenthereisnozeroof(4)in [p];

(iii) ifN

GS

([p]) isstrictly inside [p], then [p]contains

exactlyonezerooff.

Properties(i)and(ii)indicatethattheresultisreliable

in thesense that nozero canbelost. Property(iii) is

usedto evaluatetheactualnumberof zeros. IfL isre-

duced to a single box [bp] and (iii) is satised for [bp],

then themodel isglobally identiable at p

. Else, [bp]

may contain one ormorezeros. A solution is then to

split [bp] into two subboxes[p

1

] and [p

2

] and to apply

N

GS

againon[p

1

]and [p

2

]. Ifall boxesin L aresuch

that (iii) issatised,thenthemodelislocally identi-

ableat p

.

3.2 Aprior domain can be given to p

One of the interest of structural identiability is that

itcanbetestedbeforeexperimentation andthus serve

asatool forqualitativeexperimentdesign [8]. Onthe

other hand, the experimenter may feel uncomfortable

withthefact that globalidentiability isnotnecessar-

ily astructuralproperty(as shown in Example1) and

that, evenif the model is s.g.i., there maybe atypical

regionswheretheconclusionreachedmaybefalse.This

motivatesthefollowingnewdenition.

Denition1 The parameterp

i

is globallyidentiable

in P(g.i.i.P)if

8(p

;p)b 2PP; M(bp)=M(p

))pb

i

=p

i

; (5)

andtheparametervectorpisg.i.i.Pifallitscomponents

are g.i.i.P.

Denition 1 no longer allows the existence of atypical

regionsinPandunderlinestheimportanceofthesearch

domainP. Iftheproblem

M(bp)=M(p

); kbp p

k

1

>0 (6)

hasnosolutionfor(bp;p

)inPP,thenpisg.i.i.P. This

makesiteasytoexpressthetestofglobalidentiability

asaconstraintsatisfactionproblem (CSP).ACSPcon-

sistsofasetofvariables(here,fbp

1

;:::;pb

n

;p

1

;:::;p

n g),a

setofdomainsassumedtocontainthesevariables(here

we shall consider intervals, and P will be taken as a

box, but more sophisticated search domains, such as

unions of boxes, could also be considered), and a set

of constraints to be satised (here, those in (6)). In-

tervalconstraintpropagation(ICP)hasbeendeveloped

to deliverguaranteedouterapproximationsofthesolu-

tionsofCSP's(see,e.g.,[13]andthereferencestherein).

ICPcontractstheinitialdomainsintosmallerones.Ifa

deadlockisreached,bisection ofthedomainofinterest

canonceagainbeapplied, atthecostofanincreasein

complexity.

Inpractice,(6)isreplacedby

M(bp)=M(p

);kbp p

k

1

>"; (7)

where " is some positive coeÆcient to be set by the

user to expressa distancebelowwhich parametervec-

tors willnotbedistinguished. Weshallthen talkof "-

g.i.i.Pparametersormodels. For"-g.i.i.Pmodels,ICP

may allow a conclusion to be reached without a sin-

gle bisection,which isparticularly attractiveforlarge-

dimensional problems.

2

−2 2

−2 L

D ₁ £D ₁ D ₁ D ₂ £D ₂

D ₂

Figure 2:L,ascomputedinthe(bpp

)spaceforExam-

ple2,andthedomainsD1 andD

2

;sinceD

2 D2

intersects L,Mmaynotbe"-g.i.i.D

2

,whereas

itis"-g.i.i.D

1

Example2 Consider again the model of Example 1,

and the domains D

1

=[0:8;2]and D

2

=[0:3;2]. With

"=0:02,in1msonaPentium350,ICPprovedthatM

is"-g.i.i.D

1

;butcouldnotconcludeaboutwhetherMis

"-g.i.i.D

2

. Figure 2presents an outer approximation L

of allsolutions of (7)for (bp;p

)in [ 2;2][ 2;2], as

2 2

itmay exist(bp;p

)in D

2 D

2

satisfying(7). Itcanbe

shown on this simple example that the actual solution

set Sof (7) is included in the ellipse dened by (bp+

p

) 2

b pp

=1. Theconstraint kb

p p

k

1

>"in(7)

implies that a part of this ellipse does not belong toS.

Thisanalysis conrms thatM isnot"-g.i.i.D

2

.

Example3 Consider the following compartmental

model, usedto describe the behavior of adrug suchas

Glafenine administeredorally:

M(p): 8

>

>

>

>

<

>

>

>

>

: _ q

1

= (p

1 +p

2 )q

1 +u;

_ q

2

= p

1 q

1 (p

3 +p

5 )q

2

;

_ q

3

= p

2 q

1 +p

3 q

2 p

4 q

3

;

1

= p

6 q

2

;

2

= p

7 q

3 :

(8)

The initial conditions on the state variables q

i are all

taken to be zero. M(bp) = M(p

) translates into (2)

wherethe components ofr(p) arethecoeÆcientsof the

transfermatrixassociatedwith(8).Forthistobevalid,

the entries of this matrix should be expressed in some

canonical form. For P=[0:6;1]

7

and "=10 9

; ICP

ndsin 0:01s that the model is "-g.i.i.P. On P=R 7

;

the modelis not s.g.i. [8].

4 Bypassing the identiability study

When the data have already been collected, it is no

longersoimportanttoobtainconclusionsthatarevalid

forall(oralmostall)possiblevaluesofp

.Instead,one

wouldliketocharacterizethesetSofallvaluesof b

pthat

areconsistentwiththesedata,inasensetobespecied.

Intervalanalysis can also be used forthis purpose, as

illustratedbythenexttwosections.Theidentiability

analysis is thus bypassed, to address the actual prob-

lem of interest directly, namely nding all optimal or

acceptable models based on the data. The resultsare

guaranteed,irrespectiveofwhetherthemodelstructure

is globally identiable, contrary to those provided by

traditionallocalnumericalmethods.

4.1 Globaloptimization

Assumethat thesetSofinterestisthesetofallglobal

minimizersof asuitablecostfunction J(:;:),obtained,

e.g., by amaximum-likelihood approach based on hy-

pothesesaboutthenoisecorruptingthedata

S=fbp2P jJ(bp ;y )6J(p;y) 8p2Pg; (9)

where y is the (numerically known) vector of all the

datacontaininginformationaboutp.

OneofthemainnumericaldiÆcultiesattachedwiththis

approach isthe fact that J(p;y ) isusually notconvex

withrespecttop,sothere maybeseveralglobal mini-

mizersp,b aswellasparasiticlocalminimizersthatmay

searchonly.Evenwithmoreglobaltoolssuch assimu-

lated annealing,adaptiverandom search orgenetical-

gorithms, one will never be sure in a nite time that

all globalminimizers havebeenlocated. Deterministic

globaloptimizationbasedonintervalanalysis,exampli-

edbyHansen'salgorithm[11],doesnotsuerfromthe

samelimitations.Itallowsonetoeliminatepartsofthe

priorsearchspacePthatcannotcontainanyglobalmin-

imizer ofthe costfunction,and to reduce theboxesof

parameterspacethatcannotbeeliminated.Theresults

obtainedareguaranteed,in thesensethat anouterap-

proximationofSisobtained.Itmayevenbepossibleto

estimate parameters that are notglobally identiable,

asevidencedbythefollowingexample.

p ₁ p ₂ p ₃ u

1 2

Figure3: Two-compartmentmodelofExample4

Example4 As in [14], consider the system described

by Figure 3. The evolution of the vector q =(q

1

;q

2 )

T

ofthe quantitiesof materialinthetwocompartmentsis

describedby thelineartime-invariant stateequation

_ q

1

= (p

1 +p

3 )q

1 +p

2 q

2 +u;

_ q

2

=p

3 q

1 p

2 q

2 :

(10)

Take the system in zero initial condition (q(0 )= 0),

and assumethata Dirac inputu(t)=Æ(t)is appliedto

Compartment1,soq

1 (0

+

)=1andq

2 (0

+

)=0. Assume

also that the content of Compartment 2 is observed at

16 instants oftime, according to

y

i

=q

2 (t

i )+n

i

; i=1;:::;16; (11)

wheren

i

issomemeasurementnoise.Itistrivialtoshow

that the corresponding model outputssatisfy

i

(p)=( exp(

1 t

i

) exp(

2 t

i

)); i=1;:::;16;

(12)

where ,

1 and

2

are simple functions of p

1

;p

2 and

p

3

. The parameter vector to be estimated is p =

(p

1

;p

2

;p

3 )

T

.Theexperimentaldatay

i

,togetherwiththe

times t

i

at which theyhave been collected are given in

Table1. Nopriorinformationisavailableaboutp,and

the measurementsy

i

areall deemedequally reliable, so

the costfunction ischosen as

J(p;y )= 16

X

i=1 (y

i

i (p))

2

: (13)

For P = [0:01;2:0][0:05;3:0][0:05;3:0], and with

its precision parameters "

p and "

J

both equal to 10 9

,

inP areinthe boxes

[1:925402;1:925404][0:232717;0:232719]

[0:145075;0:145077]

and

[0:232717;0:232719][1:925402;1:925404]

[0:145075;0:145077]:

Notethat these boxescan bededucedfrom one another

byexchangingtheirintervalvaluesforp

1 andp

2

.Thisis

consistentwiththeconclusionofanidentiabilitystudy.

It seems important to stress that the conclusion of the

presentestimationwasnotbasedonsuchastudy,which

canthereforebedispensedwith. Optimizingrstwithre-

spectto;

1 and

2

andthenestimatingpwithaguar-

anteed interval-based method, takes about one minute

[14 ].

Table1: Experimentaldata

t

i y

i

t

i y

i

1 0:0532 9 0:0099

2 0:0478 10 0:0081

3 0:0410 11 0:0065

4 0:0328 12 0:0043

5 0:0323 13 0:0013

6 0:0148 14 0:0015

7 0:0216 15 0:0060

8 0:0127 16 0:0126

Insteadof looking for allglobal minimizersof thecost

function, onemaylook forthe set Sofall values of p

such that thevalue ofthe costfunction is below some

threshold. Thismaybeimportantinpractice,asthere

doesnotseemtobeanyseriousreasontoeliminateval-

ues of p that would correspond to values of the cost

function almost equal to the absolute minimum. Sis

then dened by an inequality, and the methods to be

usedarethoseofthenextsection. Notethatsuchvalues

ofpcouldnotbeobtainedbyastructuralidentiability

studybasedonpurely algebraicmanipulations.

4.2 Setsdened by inequalities

Assumenowthat

S=fbp2Pjg (bp ;y )60g; (14)

where g (b

p;y )60is tobeunderstood componentwise.

Smay be a condence region (at some level of con-

dencetobespeciedbytheuser).The(usuallyscalar)

inequality g(bp;y ) 6 0 dening Sis then deduced by

probabilisticconsiderationsfrom hypothesesaboutthe

statisticaldistributionofthenoisecorruptingthedata.

Theinequalitiesin(14)mayalsoexpressthattheerrors

i

correspondingmodel outputs

i

(p) lie betweenknown

bounds.Thiscorrespondstoparameterbounding orset-

membershipestimation [15].

In both cases, characterizing S can be cast into the

frameworkofset inversionandperformedusingtheal-

gorithm SIVIA [16]. In this algorithm, a positiveac-

curacy coeÆcient"must alsobetuned by theuser,to

choosethewidth belowwhichuncertainboxeswill not

bebisected.

0 5

0 5

p ₁ p ₂

Figure 4: ProjectionofSontotheplane(p

1 ,p

2 )

Example5 Consider againtheproblem ofExample4.

Take P = [0;5]

3

. Assume now that pb is accept-

able if the absolute deviations between the data y

i and

corresponding model outputs

i

satisfy jy

i

i (bp )j 6

0:002; i =1;:::;16, which can of course be reformu-

lated as g (bp;y ) 6 0. For " = 0:0025, in 2 mn on a

Pentium233, SIVIAcomputesanouterapproximation

Sof the setSdenedby(14):Sconsistsof twodiscon-

nectedsubsetscontainedinthe boxes

[1:5046;2:9546][0:22245;0:25954][0:12061;0:22069]

and

[0:22245;0:25954][1:5046;2:9546][0:12061;0:22069]:

The projection of Sonto the plane (p

1

;p

2

) is given in

Figure4. Thefactthatp

1 andp

2

canbeexchangedwith-

outmodifyingthemodelbehaviorexplainsthesymmetry

of the gure.

5 Conclusions

Testingmodelsforidentiabilityisimportantinatleast

three cases:

{whentheparametersto beidentied haveaphysical

meaningandonewantstoestimatetheiractualvalues,

{whenstatevariablesthatcannotbemeasureddirectly

are to be estimated from the available measurements

and themodelbeingbuilt,

{whenadecisionistobetakenbasedonthenumerical

valuesoftheparametersorstatevariables.

Whenthedatahavenotbeencollectedyet,oneisinter-

estedin aconclusionthat isasindependentaspossible

from the numerical value of p

. This conclusion may

leadoneto modifythe locationand natureof thesen-

sorsandactuatorsinaneorttoimproveidentiability

[8]. We have shown here that it may depend on the

value of p

in such a way that no generic conclusion

canbereached. Thisledustoproposinganalternative

approachbasedonanewdenition ofidentiability.A

parametervectorpissaidto begloballyidentiablein

Pifitisgloballyidentiableforallvaluesofp

inP. We

havebriey explainedhowinterval analysis and inter-

valconstraintpropagationcan beusedtotest whether

amodelstructureisgloballyidentiableinPforagiven

setP.

Whenthedatahavealreadybeencollected,theactual

question of interest is: what is the set of all values of

the parametervectorthatareacceptable, given thedata

andwhatweknowaboutthe noise? Identiabilitythen

becomesawaytogetapartialanswertothisquestion,

in an idealized context. We have shown that interval

analysis may make it possible to reach a guaranteed

conclusion in a more realistic context. There are, of

course, limitations to the complexity of the problems

thatcanbehandled,andoneofthechallengesofinter-

valanalysisistoenlargetheclassoftheproblemsthat

can be treated.Thenotionofintervalconstraintprop-

agation, for instance, allows one to limit the number

of bisections needed, which is anextremely important

contributiontorepellingthecurseofdimensionality.

We have only considered the notion of identiability,

where the structures of the model and process are as-

sumedtobeidentical.Thesametypeofstudycouldbe

conducted in thecontext ofdistinguishability, where it

is assumed that the structure of themodel maydier

from that oftheprocess [17]. The notionof structural

distinguishability could similarly bereplaced by ano-

tionofdistinguishabilityinP, andthetoolsofinterval

analysis could again be used to test whether a given

model structure isdistinguishable from another onein

thisnewsense.

References

[1] T.C. Koopmans and O. Reiersol. The identi-

cation of structural characteristics. Ann. Math. Stat.,

21(2):165{181,1950.

[2] E. Walter and Y. Lecourtier. Global approaches

to identiability testing for linear and nonlinear state

spacemodels. Mathematics andComputers in Simula-

tion,24:472{482,1982.

[3] A. Raksanyi, Y. Lecourtier, E. Walter, and

E. Venot. Identiability and distinguishability testing

1985.

[4] S.Vajda,K.R.Godfrey, andH.Rabitz. Similar-

ity transformation approach to identiability analysis

of nonlinearcomportemental models. Math. Biosc., 93

(2):217{248,1989.

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