Structural Identifiability : Tools and Applications

Applications

Torkel Glad, A. Sokolov

Division of AutomaticControl

Departmentof Electrical Engineering

Linkopings universitet, SE-581 83Linkoping, Sweden

WWW: http://www.control.isy.li u.se

E-mail: torkel@isy.liu.se, @isy.liu.se

2nd December2003

AUT

OMATIC CONTROL

COM

MUNICATION SYSTE

MS

LINKÖPING

Reportno.: LiTH-ISY-R-2557

Submitted toCDC'99

Technicalreportsfrom the Control &Communicationgroupin Linkoping are

Thepaperdealswiththeapplicationofidentiabilitycriteriato

mean-valuemodelsofturbochargedICengines. Awayofreducingsuchmodels

tolinearregressionsusingdierentialalgebraispresented. Theconditions

ofglobalidentiabilityandpersistentexcitationareformulatedinexplicit

formfor agivenset of sensors. It is accompaniedwith a techniquefor

reducing the set of sensors required for the engine identication. The

softwaretoolsrequiredare outlinedandtheircomplexityis discussed.

Keywords: Identiability,dierentialalgebra,internalcombustion

T. Glad

Divisionof AutomaticControl

Department of ElectricalEngineering

Linkoping University

A. Sokolov

Engine and Powertrain Electronics

Mecel AB

Box 73,SE-662 22



Amal, Sweden

1 Abstract

The paper deals with the application of the

identi-ability criteria to mean-value models of turbocharged

IC engines. A way of reducing such models to linear

regressionsusing dierentialalgebrais presented. The

conditionsoftheglobalidentiabilityandthepersistent

excitationareformulatedinexplicitformforagivenset

ofsensors. Itisaccompaniedwiththetechniquehowto

reducethesetofsensorsrequiredfortheengine

identi-cation. The softwaretoolsrequiredare outlined and

theircomplexityisdiscussed.

2 Parameterization ofnonlinear models

The models based on theoretical fundamentals such

asphysical,chemical, conservationand transportlaws

etc. arebenecial toimprovetheaccuracyin

descrip-tion ofphenomena. One theother hand,such models

canbeverycomplicatedandcontrolpurposesrequirea

semi-physical approach substituting somemodel parts

bysimpliedheuristicrelationshipsjustied

experimen-tally.

Whileconservationlawsofdynamicsarenormally

rep-resented by bilinear dierential-algebraic equations in

itsmostgeneralform,thestaticalphysicaland

chemi-calrelationshipsusuallycontainproductsofvariables.

The relationships justied physically have a natural

number of parameters which can be additionally

re-ducedbymathematicalmethodsusingtheengineering

knowledge. Theunknownstaticrelationshipsare

usu-allysoughtasformalmathematicalexpressions. Asfar

as thesetofparametersisspeciedthequestionofits

identiability[1] arises.

3 Structural identiability and choice of

sensors

The theoreticalaspects of structural identiabilityfor

dierential-algebraicmodelsaregivenin[1]. Structural

identiability depends on the sets of sensors,

parame-tersandinputs.

Thecommonproblemfacedbyindustries ishowto

re-ducethenumberofsensorsin aserialproductionunit

without signicant decrease in quality of the control

system. Sincemostmoderncontrolsystemsare

model-based,thequestionof identiability ofmodel

parame-tersisthequestionofourrelyinguponthegivenmodel

structureand, hence,uponthecontrolsystemin

prin-Intheory, ifweare unlimited in thechoice of sensors,

mostmodelstructureswillbeidentiable. Inpractice,

theset of measurements is very limited, especially on

serialproductionunits. Typically, variousparts ofthe

model are used asopen loopestimation algorithmsin

thecontrol strategy. Duringthe control strategy

cali-brationphase,thecalibratoris,indeed, limitedbythe

choice ofsensors used in production comparingto the

setused inthelab. Hence,theproblemwhat minimal

setofsensorsshould beusedtoidentify agivenmodel

structureisanimportantone.

Thus, if one is facedby the problem of gettingrid of

thesensormeasuringavariablex(t)neededtoestimate

animportantparametera, one hastond asourceof

additional information. It can be obtained from the

othermeasurementsandmodelequationslinking these

measurementsandtheir derivatives. An additional

in-formation source, common for all applications, is the

priorknowledge. It often happens that one can know

therangeforagivenparameteroritssigninadvance.

Based on these sources the test can be performed

whether theparametera is identiable. If theanswer

isnegativethen one hasto add onesensor, if positive

then one can try to get rid of one of them and run

the test again. In such an iterative way we will end

upwith theminimal set of sensors neededfor agiven

modelparameterization.

It turns out that this test can be carried out by

dif-ferential algebra tools. It was shown in [1] that for

themodelsdescribedbydierential-algebraicequations

(DAE)thereexistsanexplicitalgorithmtotestwhether

modelfreeparameterscanbeuniquelyrecoveredfrom

the data. The algorithm does not use numeric data

at all and, loosely speaking, reduces the

identiabil-ityproblem tothefollowing: whetherthegiven model

structurecanberearrangedasalinearregressionwith

respecttoeachparameter. Theextensionsofthe

algo-rithm have been developed in [2] taking into account

the prior knowledge about the system, which can be

written in theform ofinequalities. Thus, the

theoret-ical problem has been solved in an algorithmic form.

The question left is the computational complexity of

Ritt'salgorithmwhichisthebaseof thetechnique[9].

Inthepapergiven,thepracticalaspectsofapplication

of this algorithm to enginemean-valuemodels are

nonlinearmodels

It was shown in [4] that any system with elementary

functionscanberewritteninDAEform,i.e. consisting

of polynomialequations in variables and their

deriva-tives.

On the other hand, the rst principles, used for the

enginemodeling, are originallyofDAE type: the

con-servation of energy and mass, the gas dynamic

rela-tionships. Thus, the non-algebraic constructions are

apparently introduced by model developers trying to

rearrange equations in a simplerform eliminating the

variables. Intermsofidenticationsuchsimplication

is notdesirableand DAE form ofthe model is

prefer-able.

Dierential algebra deals with systems described by

polynomial equations f(x;y;u) = 0 in which

deriva-tivesofvariablesarealsoadmitted[9].

The dierential algebra allows to determine for each

systemof DAEacharacteristicset ofequationswhich

inheritsthemainpropertiesoftheoriginalsystemand

inmostcasescontainsinput-outputrelationship. There

existsanalgorithm,namedRitt'salgorithm,

construct-ing such characteristic set and obtaining an

input-outputrelationship. Itscomputerimplementationin[9]

isthemaintoolused inthispaper.

Letusconsideraninput-outputrelationA(u;y)

involv-inginputsuandoutputsy. Lettheparametersbe

in-cludedinthecoeÆcientsh k ;j

. Acanbewritteninthe

form A(u;y)=I(u;y)  y ( k ) k  fk + n k X j=1 h k ;j  k ;j (u;y) (1)

whereI(u;y)istheleaderofA k

[1]andf k

isthehighest

powerofthetermy ( k ) k .Theterms k ;j

(u;y)are

expres-sionsoftheform:

 k ;j (u;y)=y  k ;0 1 _ y  k ;1 1


 u ( k ;m ) m   k ; k ;m

,i.e. powerproductsoftheu i

,y i

andtheirderivatives.

Denition 4.1 A canonical parameterization of the

input-outputrelationshipisonewherethenew

param-eters i

arechosenasthecoeÆcientsh k ;j

in (1).

Proposition4.1Acanonicalparameterizationis

glob-ally identiable for generic solutions u, y. Generic

means\suÆcientlygeneral".

Proposition 4.2Thedeterminants

D k =           k ;1 :::  k ;n k  0 k ;1 :::  0 k ;n k . . . . . .  (n k 1) k ;1 :::  (n k 1) k ;nk          (2)

ticular solutionssatisedto such conditionsare called

persistentlyexcited. Seeproofsin [10].

5 Enginemodelparameterization

In the paper given, a basically congurated

tur-bochargedengineisconsidered. Forsimplicitythe

sub-models of fuel dynamics, combustion, intercooler and

EGR system are not considered. The main modeling

sourcesusedare[6,7]. Inthissection thelistof

mod-elingequationsisgiven. Themoredetaileddescription

ofthemodelisin[12].

5.1 Turbocharger

Thedynamicsoftheturbochargershaftisdescribedby

theenergyconservationlaw:

@ @t N tc (t)N tc (t)= m P t (t) P c (t) (3) where m

isthe turbine mechanicaleÆciency andN tc

isthe shaft speed. ThepowerP t

(t) generated by

tur-bineandthepowerP c

(t)consumedbycompressorcan

derivedfromthesteadyformofenergyconservation:

@ @t E c (t) = 0=P c (t)+(T amb T c (t))w c C p (4) @ @t E t (t) = 0= P t (t)+(T em (t) T amb )w t C p (5)

The outlet temperature and the pressure in the

tur-bochargerareassumedequaltotheambientconstants

T amb ; p amb ; T em ;p em

stand for the exhaust manifold

temperature and pressure; T c

;p c

are the compressor

outlet temperature and pressure; w c

;w t

are the mass

owratesthroughthecompressorandtheturbine,

re-spectively. The compressor and turbine isentropic

ef-ciencies  c

; t

are usually introduced by the

relation-ships:  c (t)( T c (t) T amb 1) = ( p c (t) p amb ) 1 1 (6) ( T amb T em (t) 1) =  t  p amb p em (t)  1 1 ! (7)

TheturbineeÆciencyisassumedconstantandthe

com-pressoreÆciencyisassumedthepolynomial:

 c =a 4 +a 5 N tc +a 6 N 2 tc +a 7 w c +a 8 w 2 c +a 9 w c N tc (8)

There are manyother functional forms studied in the

literature. Itistakenhereasademonstrationexample.

Denotebya c

theparameterset ofthispolynomial.

Theturbine massair owcanbemodeled with useof

theoriceequationin quazi-isentropicconditions[12]:

w t p RT em p em = A t pamb p em s 2t  1 ( p amb p em ) ( 1 )  1 1+  ( p amb p em ) ( 1 ) 1  (9)

t

assumed constant. No ow chocked orreversal is

ad-mittedduring theexperiments. Thegeneralset of

pa-rameterswhichis subjectto identication forthe

tur-bochargerisgivenasfollows:

P tc =fJ tc ; t ; m ;A t g[fa c g 5.2 AirThrottle

The air throttle mass ow rate is usually modeled as

theisentropic owthroughtheorice(=1).

w c p RT c p c =A at s 2 1  ( p im p c ) 2 ( p im p c ) +1  (10) wherep im ;T im

arethepressureandtemperatureinthe

intakemanifold. Againitisassumedthatreversaland

chocked owsarenotadmitted.

The eective ow areaA at

(t) can be modeled by the

followingequation[6,5]: A at (t)=a 0 ( 1 cos(a 1 (t)+a 2 ))+a 3 (11) The parametersa i

has technical meaning and, hence,

the priorknowledge can be formulatedas aset of

in-equalities: I at =fa 0  0;a 1 0;a 3 0gThe set of

throttleparametersissummarizedas:

P at =fa 0 ;a 1 ;a 2 ;a 3 g: 5.3 Manifolds

One of the simplest ways to model the manifolds

dy-namics is to apply the mass conservation principle in

conjunctionwiththeidealgaslaw[7]:

@ @t m im (t) = w c (t) w be (t) (12) p im (t) = m im (t) V im R T im (t) (13) @ @t m em (t) = w be (t)+w f (t) w t (t) (14) p em (t) = m em (t) V em R T em (t) (15)

The massair oww be

to the combustion chamberis

modeledbythefollowingequation:

w be (t)R T im (t)=  vol (t)N be (t)p im (t)V be 120 (16)

where the function  vol

(t) is called the volumetric

ef-ciency, V be

is thecylinder displacementvolume. The

volumetric eÆciency assumed to be a polynomial [6]

withtheparameterset a  v ol :  vol =a 27 +a 28 N be +a 29 N 2 be +a 30 N be p im : HereP im \P em =fV im ;a v ol g:

Thedynamicsoftheengineshaftcanbedescribedusing

theenergyconservationlaw:

@ @t N be (t)N be (t)=P be (t) P loss (t) T load (t)N be (t) (17) where N be

(t) is theengine shaftspeed. Here P be

(t) is

thepowergeneratedby the shaft, P loss

(t) is thetotal

losspower and T load

(t) is the load torque scaled with 30

 .

Themodelof P be

(t)is taken forthe spark-ignited

en-gine[7] assuming =1and ignoringtheterm forthe

sparkadvance: P be = w f P Pim (a 19 a 20 e Nbea21 ): (18) P Pim = (a 16 +a 17 p im +a 18 p 2 im ) (19) where P P im

is the partdependent on the intake

pres-sure. The set of parametersin this relationshipis

de-notedbya P be anda P P im ,respectively.

Thetotallosspower,referringtothesamesource[7],is

modeledbyaquadraticpolynomialintheenginespeed

N be

and the intake pressure p im

with the coeÆcients

a Ploss : P loss =a 22 N be +a 23 N 2 be +a 24 N 3 be +a 25 p im N be +a 26 p im N 2 be (20) P be =fJ be ;a P be ;a P loss ;a P P im g

The equation system described in this section will be

denotedshortlyEintherestofthepaper.

6 Nonlinear engine modelas a cascade of

linear regressions

Denition6.1 LetMbetheset ofmeasurements,P

be the set of parameters. The system of equations E

iscalled a linearregressionin P with respect to Mif

itappears asasystem ofpolynomial equationsof the

rst order in arguments from P with the coeÆcients

beingknownfunctions in theelementsofMandtheir

derivatives. ThesecoeÆcientsarecalledregressors.

Ifamodelisrearrangedasalinearregressionin

param-etersthentheidenticationoftheseparameterscanbe

performedbywell-developedandreliableleast-squares

techniques[3]. Itwillallowtobypassthelocalminima

problemusuallyarisinginthenon-convexoptimization.

Assume the idealsituation when wehavea full set of

measurements: M id =fN tc (t);N be (t);T load (t);p im (t);p em (t);p c (t); T im (t);T em (t);T c ;w f (t);(t)g

Thegeneralsetofparametersissummarizedas:

P =P tc [P at [P im [P em [P be

the newparameter set  =F(P) such that the

com-pleteenginemodelE can berearrangedasalinear

re-gressioninwithrespecttoM id

Generally,theproofofthispropositionfollowsfromthe

resultsof section4. Theproofgiven belowconsists of

four assertions . It is of constructive nature and its

main steps consist of sequential conversion of engine

subsystemstoDAEform whereitisnecessaryand

ap-plication of Ritt's algorithm to generate input-output

relationships.

Proposition 6.2 Theairthrottlemodel(10)-(11)can

berearranged to thefollowinglinearregressionin the

setofparameters  at =f at 1 ; at 2 ; at 3 ; at 4 g withrespectto M id : y at 1 (t) =  at 1  at 1 (t)+ at 1  at 2 (t)+ at 2  at 3 (t) y at 2 (t) =  at 4  at 4 (t) (21) where  at j ;y at i

arethe regressorswith respect to M id

,

 at j

arenewparametersandi=1


2;j=1


4:

 at 1 = a 0 2 a 1 2 a 0 2 a 3 2 a 1 2  at 2 = a 0 a 3 a 1 2  at 3 = a 1 2 (22)  at 4 = a 2

The explicit form of the regressors here and below is

givenin [12].

Remark6.1Theregressory at 2

dependsonthe

param-etersfromtherstregressionequation,i.e. the

regres-sionsform acascadeandthe dependentregressors

be-comeknownbyconsequentcomputations. Thus,itdoes

notcontradicttothedenitionoflinearregressiongiven

inthis section.

Remark 6.2 Thepractical implementation of the

re-gressorsgeneratedcanbedonebysamplingthe

deriva-tivesinvolved. Notice,that theregressorscontainonly

rstderivativesofthemeasurementsignalsfrom M id

.

Remark6.3Mostofthemathematicalmanipulations

needed in this derivation are automatized [1, 2] with

computeralgebrasystems. Below,wegiveanonly

man-ualprooftoelucidatesomestepsshowinghowitworks.

Proof of Proposition 6.2 By introducing new

variables z 1 (t) = cos(a 1 (t)+a 2 ) and z 1 (t) = sin(a 1 (t)+a 2

), dierentiating it and taking into

ac-count properties of trigonometric functions we obtain

that the eective area equation (11) is equivalent to

thefollowingdierential-algebraicsystem:

A at (t) a 0 z 1 (t) a 0 a 3 =0 d dt z 1 (t)+z 2 (t)a 1 d dt (t)=0 (23) (z 1 (t)) 2 +(z 2 (t)) 2 1=0 arccos( A at (t) a 3 a 0 )=a 1 (t)+a 2 (24)

Eliminating the variables z 1

;z 2

by dierentiation we

obtaintheequation:

 d dt A at (t)  2 =a 0 2 a 1 2  d dt (t)  2 a 1 2  d dt (t)  2 (A at (t)) 2 +2a 0 A at (t)a 1 2  d dt (t)  2 a 3 a 0 2 a 3 2 a 1 2  d dt (t)  2

Taking into account the notations used we

immedi-ately obtain the rst equation (21). The parameters

 at 0 ; at 1 ; at 3

canthenbedeterminedbyanyrelevant

es-timation procedure [3] if the excitation input (t) is

correctlychosen. Theanalysis ofthe algebraicsystem

(22)showsthatthissystemisuniquelyresolvablewith

respect to the parameters a at

if we utilize the prior

knowledgeformalizedin theinequalityset I at

.

Substi-tutingtheparametersobtainedin(24)werealizethat

this equationis alsoalinearregressionin a 2

. The

as-sertionfollows.

Proposition 6.3 Theturbochargermodel(3)-(9)can

be rearrangedasthe followinglinearregressionin the

setofparameters:  tc =f tc 1 ; tc 2 ; tc 3 ; tc 4 ; c ; pc g withrespecttoM id : y tc 1 (t) =  tc 1  tc 1 (t)+ tc 2  tc 2 (t) y tc 2 (t) =  tc 3  tc 3 (t) (25) y tc c (t) =  tc c T  tc c (t) where tc 1 =J tc ; tc 2 = m A t ; tc 3 = t ; tc c =a c .

The regressor vector  tc 

c

(t) is composed from the

re-gressorsofthecompressormap. Sincetheturbocharger

speedisknown,noconversiontoDAEformisrequired.

Theresultsfollowdirectlyrewritingthesystem(3)-(9).

Theparametersaretobecomputedrecursively.

Torepresentthebasicenginesubsystemincompact

re-gressionformwemaketheadditionalassumptionsthat

thelosspowerP loss

(t)and theintakepressure

depen-denttermdenotedbyP pim

(t)intheexpression(18)are

estimatedinadvanceand,hence,areknownfunctions.

Sincetheydepend onthe variablesfrom M id

onecan

attachthemtothisset asadditionalmeasurements.

~ P be =fa 19 ;a 20 ;a 21 ;J be g

Proposition 6.4 The basic engine model (17)-(20)

canbe rearrangedasthefollowinglinearregressionin

parameters  be =f be ; be ; be ; be ; be g

withrespecttoM id =M id [fP loss (t);P p im (t)g: y be 1 (t) =  be 1  be 1 (t)+ be 1  be 2 (t)+ be 3  be 3 (t)+ be 4  be 4 (t) y be 2 (t) =  be 5  be 5 (t) (26) where  be 1 = J be ;  be 2 = a 21 ; be 3 = a 19 a 21 ; be 4 = J be a 21 ; be 5 =a 20 :

Proof of Proposition 6.4 Convert thesystem

(17)-(20)toDAEform. Anewvariablez(t)anddierential

equation will appear. Apply Ritt's algorithm to this

DAEsystemspecifyingthenextinputsforelimination:

z(t);E be

(t);P be(t)

. The resultingrelationshipdoes not

contain theparameter a 20

. That means that this

pa-rameter is not identiable in this DAE form.

How-ever,itcanberecoveredfromtheoriginalnon-algebraic

equation. Again,similartoairthrottlemodel,wehave

thecascadeconstructionof linearregressions.

Proposition 6.5 The manifoldmodels (12,, 16) can

berearranged to thefollowinglinearregressionin

pa-rameters  im = f im 1 ; im 2 ; im 3 ; im 4 ; im 5 g and  em = f em 1 ; em 2 gwithrespecttoM id : y im (t)= im 1  im 1 (t)+ im 2  im 2 (t)+ im 3  im 3 (t)+  im 4  im 4 (t)+ im 5  im 5 (t) (27) y em (t)= em 1  em 1 (t)+ em 2  em 2 (t) where  im 1 = V im ; im 2 = a 27 ;  im 3 = a 28 ; im 4 = a 29 ;  im 5 =a 30 ; em 1 =V em ; em 2 =A t . 7 Identiability criteria

Thissectiondealswiththecriteriaofidentiabilityfor

themodelparameterization (3)-(20). It is shown that

this model is globally identiable with respect to the

measurementset ~ M

id

. Theconditionof global

identi-ability,whenthesensorofturbochargerspeedis

elim-inated,isgiven. Itisdiscussed howtochoosethe

per-sistentlyexcitingmodelinputstoperformengine

iden-tication.

Thedenitionofglobalidentiabilityisgivenaccording

to[1]in amodiedformasfollows:

Denition7.1 A model structure of equation,

mea-surement and parameter set E;M=fz(t)g;P=fg

withpriorknowledgeformalizedintheformof

inequal-itiesI is globallyidentiable iffor any feasible ?

the

followingimplicationholds:

z(t;)=z(t; ?

) ) = ?

:

Ifitholdsinanopenneighborhoodof ?

thenthemodel

structureiscalled locallyidentiablein ?

.

Proposition 7.1 Theenginemodel(3)-(20)withprior

knowledge I =I at

isglobally identiablewith respect

totheparametersetP and measurementset ~

M .

Proof of Proposition 7.1The linearregressions

de-rivedinthepropositions6.2-6.5arecanonically

param-eterizedbyconstruction. Accordingtotheresultsfrom

section4theyaregloballyidentiable. Hence,the

iden-tiability is reduced to the question whether the

sys-tems of algebraic equations dened in the assertions

of the section 6 and called here as identiability

ta-blesareuniquelyresolvablewithrespecttotheoriginal

physical parameters. The identiability table for the

manifolds is trivially uniquely resolvable with respect

toP im [P em [fA t

g. Asabonusweobtainthe

param-eterA t

usuallyreferredtotheturbochargersubsystem.

Withsuchaknowledgetheidentiabilitytablefor

tur-bocharger will be uniquely resolvable with respect to

P tc

. The identiability table for the air throttle. is

generally not resolvable uniquely with respect to P at

but withthe additionalknowledgeI tc

it is. Thebasic

enginesubsystemhasalsoauniquesolution. Theresult

follows.

Letusconsidersituationwhenasensorofturbocharger

speedisnotavailable. Assumethatwehavesomeprior

knowledge about thecompressor map (8) and it is as

limited as: I tc = fa 8 = a ? 8 ;a 9

= 0g. The rest of the

coeÆcientsofthecompressormap arenotxed.

Proposition 7.2 Theenginemodel(3)-(20)with the

priorknowledgeI tc

[I at

is globallyidentiableforthe

parametersetP andthemeasurementset ~ M id nfN tc g

ifthesignofparametera 5

in(8)is known.

ProofofProposition7.3Findtheregressors

depen-dent on turbocharger speed:  tc  c (t); tc pc (t); tc 1 (t). We

obtainthen fourequations involvingN tc and parame-ters: y tc 1 (t) c (t)=C p w c (t)y tc c (t) y tc 1 (t)= tc 1  tc 1 (t)+ tc 2  tc 2 (t)  c (t)=a 4 +a 5 N tc (t)+a 6 N tc (t) 2 +a 7 w c (t)+a 8 w c (t) 2  tc 1 (t)= N tc (t) @ @t N tc (t) Denote z 1 (t) = C p w c (t)y tc  c (t) and z 2 (t) = a 8 w c (t) 2

: Form the command to Ritt's algorithm

to generate input-output relationship eliminating

 c (t); tc 1 (t);N tc

(t):Theresultingrelationship, beinga

dierentialpolynomial,contains manyadditiveterms.

Introducingcanonicalre-parameterizationwegeta

sys-temofalgebraicequations. Thenumberofnew

canon-ical parameters is much bigger then original ones:

 1 ; 2 ;a 4 :a 5 ;a 6 ;a 7

. UsingGrobnerbasestools,the

sub-systemofequationswasfound

a 6  1 a 7 = b 1 a 6  1 = b 2 a 6  1  2 = b 3 a 6 a 2 7 = b 4 a 2 5 a 6 +4a 2 6 = b 5

whichisuniquelyresolvablewithrespecttotheoriginal

to a 5

withoutthis knowledge. Referringtothesection

4theresultfollows.

Inthetext abovetheidentiability wastreatedinthe

structuralsensewithoutregardingthedatatrajectories.

The formulation of persistent excitation conditions is

straightforwardaccordingtotheSection4. Wepresent

these conditions in explicit form for the air throttle

model.

Proposition 7.3 The persistent excitation condition

fortheairthrottlemodel(10)-(11)areasfollows:

@ @t (t)( @ @t A at (t)) 3 6=0

forgiveninputtrajectories(t);T load

(t);w f

(t).

It is clear that these conditionsare alwayssatised if

thethrottleanglechangesmonotonically.

8 Software issuesand computational

complexity

Intheworkpresentedthreeprogrammingpackagesfor

Maple computer algebra system were used. The

ba-sicprogrampackageisRitt'salgorithmimplementation

by[9]anditsmodication[4].

Themainprogramrittioworksin thefollowingway:

>rittio([E];[Y];[X])

whereEisthelistofmodelingequations,Y isthelistof

ordereddesirablevariables,usuallyinputs,outputsand

parameters,X isthelistofundesirablevariableswhich

are subjectto elimination. Ifonespecies parameters

as functions of time then the algorithm produces the

complete characteristicset of agiven DAE system. It

requires manycomputations andat this timeits

com-putationallimitisupto10time-dependentvariables. If

onespecies parametersastime-independentvariables

thenthealgorithmproducesjustinput-output

relation-ships. Thecomputational limitincreasessignicantly.

Inthiswayitwasappliedtotheenginemodelgiven.

The second packageis the convertor[4] from ODE to

DAEsystems. Itsmainprogramhastwoinputs:

>ode2ade(L;t)

thelistofarbitraryODEequationsLandindependent

variablet. Itproducesitsdierential-algebraic

equiva-lentofminimaldegree,automaticallyaugmentingnew

variablesifnecessary. Nocomputationalproblemshave

beennoticed withthisprogram.

The third package is the library of symbolic objects

for modeling of internal combustion engines. This

li-brary,presentedin [8],storesthebasicstructures

usu-canbeobtainbyoneline:

>orifice(inputs;outputs;parameters)

withthespecicationofvariablesand/orsubscripts

in-volved. These procedures are generated by the other

procedures, containing the derivation of this model

from the rst principles. The rst principles are also

considered asblack boxes in this library. Thus, some

changes in the underlying physical assumptions

auto-maticallyresultinthemodiedmodelofasubsystem.

Additionally, we used the Grobner bases package

in-cludedinthestandardMaplesetup.

Thecombination of these programsallows oneto

cre-ativelyautomatizetheenginemodelingprocess.

References

[1] L.Ljungand T. Glad. OnGlobal Identiability

for Arbitrary Model Parameterizations. Automatica,

Vol.30, No.2,pp.265-276,1994.

[2] S. T. Glad, and L. Ljung. Identiability with

constraints. report.LinkopingUniversity,1998.

[3] L. Ljung. System Identicataion. Prentice Hall,

EnglewoodClis,NewJersey,1987

[4] P. Lindskog. Methods, Algorithms and Tools

for System Identication Based on Prior Knowledge.

PhDthesis436,DepartmentofElectricalEngineering,

LinkopingUniversity,Sweden,May1996.

[5] M.NybergandL. Nielsen. ModelBased

Diagno-sis for the Air Intake System of the SI Engine. SAE

technical paper 970209.

[6] J.B.Heywood.InternalCombustionEngineF

un-damentals.1988.

[7] A. Hendriks and S. Sorenson. Mean value

mod-eling of spark ignition engines, SAE Technical paper

900616.

[8] A. Sokolov, A. Loria. An approach to

struc-turalmodelingofnonlinearsystemsfacilitatingrobust

design: application to IC engines. MTNS'98, July,

Padova,1998.

[9] S.T.Glad. ImplementingRitt'salgorithmof

dif-ferentialalgebra. InIFACSymposiumon Control

Sys-tems Design, NOLCOS'92, pages 610{614, Bordeaux,

France,June1992.

[10] S.T.Glad. Nonlinearinput-outputrelationsand

identiability. InProc. 31st CDC, Tucson, December

1992.

[11] A.SokolovGeneralizedmodelof owrestrictions

for IC engine control applications. Technical report

N2120,1999, ISY,Linkoping University.

[12] A. Sokolov, T. Glad Identiability of

Tur-bocharged IC Engine Models SAE Technical paper