Identification of physical parameters of a pneumatic servosystem

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Identification of physical parameters of a pneumatic servosystem

Mohamed Smaoui, Minh Tu Pham, Xavier Brun, Sylvie Sesmat

To cite this version:

Mohamed Smaoui, Minh Tu Pham, Xavier Brun, Sylvie Sesmat. Identification of physical param- eters of a pneumatic servosystem. 16th IFAC World Congress, Jul 2005, Prague, Czech Republic.

�10.3182/20050703-6-CZ-1902.00061�. �hal-02064093�

PARAMETERS OFA PNEUMATIC

SERVOSYSTEM

M. Smaoui,M. T. Pham,X. Brun, S. Sesmat

Laboratoired'AutomatiqueIndustrielle

INSAde Lyon, 20avenueAlbertEinstein

69621Villeurbanne Cedex,FRANCE

Abstrat: This paper deals with the estimation of physial parameters of a

pneumati servopositioning system. It onsists of two servodistributors and an

atuator.This work onernsnotonlytheidentiationofthemass,visous and

Coulombnonsymmetrifritionsparametersofthemehanialpartbut alsothe

polytropi oeÆient of the gas onsidered as a perfet one. At rst the mass

ow rate harateristi of the servodistributor is determined indiretly by an

approximationusingseveralpolynomialfuntions.Next,adynamimodelwhih

is linear in relation to a set of dynami parameters is proposed. The dynami

parameters are estimated using the weighted least squares solution of an over

determinedlinearsystemobtainedfromthesamplingofthedynamimodelalonga

losedlooptrakingtrajetory.Anexperimentalstudyexhibitsgoodidentiation

results.Copyright

2005 IFAC

Keywords:losed loopidentiation, pneumatisystem,physialparameters,

polynomialinterpolation,weightedleastsquares

1. INTRODUCTION

Beause of inreasing performanes of pneu-

mati servo positioning systems, aurate mod-

els are needed to hek their performanes (a-

uray and rapidity) by simulation and to im-

prove their design and their ontrol (Isermann,

1996)(Janiszowski, 2004). Some investigations in

modellingofpneumatiatuatorsuselinearmod-

elsoramulti-modelstruture(ShulteandHahn,

2001). The authors in (Liu and Bobrow, 1988)

have worked on a linearized state spae model

to implement an optimal regulator for a xed

operating point. These models are well adapted

if eorts are made in developing robust ontrol

strategy to address the diÆulties in modelling

The main diÆulties in modelling pneumati

atuators are their highly nonlinear behaviors

(M Cloy,1968)(Shearer,1956)(Blakburnet al.,

1960). These ones are assoiated with the non-

linear dynami properties of pneumati systems

suh as servodistributor ow harateristis, the

thermodynami propertiesofairompressedin a

ylinder and the nonlinear frition between the

ontating surfaes of the slider-piston system.

The majority of papers in the eld of researh

on identiation of physial parameters of pneu-

matiservopositioningsystemsfouses oneither

the identiation of mehanial parameterssuh

as fritions or the mass to be moved (Uebing

andVaughan,2001)(Wangetal.,2004)eitherthe

study of model of the ow stage delivered by a

servodistributor (Belgharbi et al., 1999)(Rihard

and Savarda, 1996). In(Nouri et al., 2000),the

tioningsystemhavebeenmodelled/identiedsep-

aratelywithoutonstrutinganoverallsimulation

model inorder toestablishonformitybehaviour

with pratie. An identiation sheme of pneu-

matiandmehanialpartsisproposedin(Zorlu

etal.,2003).Butthisapproahisbasedonagood

ompensationofdryandstatifritionstoobtain

alinearized model.

This paper dealswith the estimation ofphysial

parametersofapneumatiservopositioningsys-

tem. This work onernsnot only theidentia-

tionofmehanialparametersbutalsopneumati

parameters. At rst the mass ow rate hara-

teristi of the servodistributor is determined in-

diretlyby anapproximationusing several poly-

nomialfuntions.Next,adynamimodelwhihis

linearinrelationtoasetofdynamiparametersis

proposed.Thedynamiparametersareestimated

using the weighted least squares solution of an

overdeterminedlinearsystemobtainedfrom the

sampling of the dynami model along a losed

loop traking trajetory. An experimental study

exhibitsidentiationresults.

Thispaperisorganizedasfollows:Setion2deals

with the modelling of the pneumati servosys-

tem. Setion 3 is devoted to the approximation

of theservodistributor ow stage harateristis.

Setion4presentsthemethodofidentiationby

weighted least squares and the pratial aspets

ofthemethodintermofdataaquisitionandl-

tering.Setion5isdediatedtotheexperimental

resultsofalosedloopidentiation.

2. MODELLING

2.1 Dynamimodel

The eletropneumati system (gure 1) uses the

following struture: two three-way proportional

servodistributors/atuator/mass in translation.

The atuator under onsideration is an in-line

eletropneumatiylinderusingasimplerod.The

eletropneumati system model an be obtained

using three physial laws: the mass ow rate

through arestrition, thepressure behaviorin a

hamberwithvariablevolumeandthefundamen-

tal mehanialequation.

The two servodistributors are supposed to be

idential. This omponent is a pneumati ow

valve andonsists of amathing spool-sleeveas-

sembly and a proportional magnet diretly on-

trolling the movement of the spool against a

spring. The spool is ontrolled in position by

means of a position sensor. On the ontrary of

many other valve designs used in automotive or

railwayappliationsorinpneumatiiruits,the

Fig.1.Theeletropneumatisystem

poppet tehnology. This meansthat pressurea-

uray around zero opening has been set to the

detriment ofleakage.Sothis tehnology leadsto

harateristis without dead zone. In our ase,

thebandwidthoftheServotroniJouomatiser-

vodistributor and the atuator are respetively

about 200 Hz and 1.5 Hz. Using the singular

perturbationtheory,thedynamisoftheservodis-

tributors are negleted and their models an be

reduedtoastationedesribedbytworelation-

shipsq

mP (u

P

;p

P )andq

mN ( u

N

;p

N

)betweenthe

massowratesq

mP andq

mN

,theinputvoltages

u

P andu

N

,andtheoutputpressuresp

P andp

N .

The pressure and temperature evolution laws in

ahamberwithvariable volume areobtainedas-

sumingthefollowingassumptions(Shearer,1956):

airis aperfet gas andits kinetienergy is neg-

ligible, thepressureand thetemperatureareho-

mogeneousineahhamber.Moreovertheproess

issupposedtobepolytropiandharaterizedby

the oeÆient k

(P orN)

. The following eletrop-

neumatisystemmodelisobtained:

8

>

>

>

>

<

>

>

>

>

: _ p

P

= k

P rT

P

V

P (x)

q

mP ( u

P

;p

P )

S

P

rT

P p

P _ x

_ p

N

= k

N rT

N

V

N (x)

q

m N

(u

N

;p

N )+

S

N

rT

N p

N _ x

Mx=S

P p

P S

N p

N F

ext F

f (x )_

(1)

Where:

F

ext

=(S

P S

N )p

ext

(2)

And:

V

P (x)=V

P ( 0)+S

P x

V

N (x)=V

N (0) S

N x

With: 8

<

: V

P (0)=V

DP +S

P l

2

V

N (0)=V

DN +S

N l

2 (3)

beingvolumes ofthe hambersfor thezeroposi-

tionand V

D(P orN)

thedeadvolumespresenton

eahextremities ofthe ylinder.ThetermF

f (x )_

in (1)represents allthe frition fores whih at

onthemovingpart.Itwasobservedduringexper-

imentaltestingthatCoulombfritiondependson

thediretion ofmotion.Thus thefuntionF

f (x )_

F

f (x )_ =

<

: f

v _ x+F

+

if x_ >0

f

v _ x F

if x_ <0

0 if x_ =0

(4)

2.2 Identiation model

Thedynami model(1) an bewritten in arela-

tionlinearto thedynamiparametersasfollows:

y=D

s X

s

(5)

With:

y= 2

4

q

mP (u

P

;p

P )

q

mN (u

N

;p

N )

S

P p

P S

N p

N (S

P S

N )p

ext 3

5

(6)

And:

Ds=

"

_ p

P xp_

P _ xp

P

0 0 0 0 0 0 0

0 0 0 p_

N xp_

N _ xp

N

0 0 0 0

0 0 0 0 0 0 x x_ f(x)_ g(x)_

#

(7)

Thefuntionsf andg in(7)aredened by:

8

>

<

>

: f(x )_ =

1+sign(x)_

2

g(x)_ =

1 sign(x)_

2

(8)

ThevetorofunknownparametersX

s is:

X T

s

= X T

sP X

T

sN M f

v F

+

F

(9)

WhereX

sP andX

sN

arerespetivelytheparam-

etersdesribingthepressureevolutionlawsinthe

hambersPandN:

X T

sP

=

V

P (0)

k

P rT

P S

P

k

P rT

P S

P

rT

P

=X

s (1) X

s (2) X

s (3)

(10)

X T

sN

=

V

N (0)

k

N rT

N S

N

k

N rT

N S

N

rT

N

=X

s (4) X

s (5) X

s (6)

(11)

3. APPROXIMATIONOFTHE

SERVODISTRIBUTORFLOWSTAGE

CHARACTERISTICS

The main diÆulty formodel (5)is to knowthe

mass ow rates q

mP (u

P

;p

P

) and q

mN (u

N

;p

N ).

In order to establish a mathematial model of

the power modulator ow stage, many works

present approximations based on physial laws

(Araki,1981)(Mo, 1989)by themodellingof the

geometrial variations of the restrition areas of

the servodistributor as well as by the experi-

mental loal haraterization(Rihard andSav-

arda,1996).Thesemethodsarebasedonapprox-

imationsofuidowthroughaonvergentnozzle

in turbulentregime,orretedbyaoeÆientC

q

(M Cloyand Martin, 1980)oron thethe norm

In this paper, we propose to use the results of

theglobal experimentalmethod givingthestati

harateristisoftheowstage(SesmatandSav-

arda,1996).Theglobalharaterization(gure2)

orrespondstothestatimeasurementoftheout-

put mass ow rate q

m

depending on the input

ontroluandtheoutputpressurepforaonstant

sourepressure.Figure2learlyshowsthenonlin-

earbehaviouroftheowrateevolutionaording

to thepressureandtheinputontrol.Theglobal

haraterization has the advantage of obtaining

simply, by projetionofthe harateristisseries

q

m

(u;p)ondierentplanes:

the mass ow rate harateristis series

(plane"p q

m

"),

themassowrategainharateristisseries

(plane"u q

m

"),

thepressuregainharateristisseries(plane

"u p").

−10

−5 0 5

10 0

2 4 6 8

−40

−30

−20

−10 0 10 20 30 40

pressure p (bar) input control u (V)

Mass flow rate q m (u,p) (g/s)

Fig.2.Global statiharateristis

The authors in (Belgharbiet al., 1999) havede-

velopedanalytialmodelsforbothsimulationand

ontrol purposes. Two ases have been studied

to approximate the ow stage harateristis by

polynomialfuntions:

amultivariablepolynomialfuntion,

apolynomialapproximationaÆneinontrol

suhas:

q

m

(u;p)='(p)+ (p;sign(u))u (12)

In this paper we will used the seond approxi-

mationbeauseitallowstogiveaphysialsigni-

anetothepolynomialfuntions.'(p)in(12)isa

polynomialfuntion whoseevolutionorresponds

to the mass ow rate leakage, it is idential for

allinput ontrol valueu. (p;sign(u))is apoly-

nomial funtionwhose evolutionissimilar tothe

onedesribedby themethods basedon approxi-

mations of mass ow rate through a onvergent

nozzle in turbulent regime (M Cloy and Mar-

tin,1980).Itisafuntionoftheinputontrolsign

and the exhaust (u < 0). For a disussion and

moredetails onthe hoie offuntions and their

degrees please refer to (Belgharbi et al., 1999).

Figure 3shows themassowrate errorbetween

theanalytialmodelandthemeasurementswhen

the polynomial funtions '(p), (p;u > 0) and

(p;u < 0) have respetively degrees equal to

seven,sevenandfour.

1 2

3 4 5 6 7 8

−10

−5 0 5 10

−1.5

−1

−0.5 0 0.5 1 1.5

pressure p (bar) input control u (V)

Mass flow rate error (g/s)

Fig.3.Massowrateerror

The error tted in gure 3 desribes a polyno-

mial approximation whih ts the atual data

extremely losely. This approximationis used to

estimate the mass ow rates q

mP (u

P

;p

P ) and

q

mN (u

N

;p

N

)in (6).

4. IDENTIFICATIONMETHOD

4.1 Weighted LeastSquares

The vetor X

s

is estimated as the solution of

the Weighted Least Squares (WLS) of an over

determinedsystemobtainedfromthesamplingat

thevarious moments t

i

, i =1,..., r =neof the

system(5)(CanudasdeWitetal.,1996):

Y =WX

s

+ (13)

where:W isa(rNp)observationmatrix,whih

isasamplingoftheregressor(7), Y isa(r 1)

vetorwhih is a sampling of (6), is a (r1)

vetoroferrorsduetomodelerrorandnoisemea-

surements,r>Npisthenumberofequations.

TheW.L.S.solutionminimizesthe2normjjjjof

thevetorof errors.The uniityof theW.L.S.

solution depends on the rank of the observation

matrix W. The rank deieny of W an ome

fromtwoorigins:

- struturalrankdeienywhih standsfor any

samples of (x ;_ x; p

P

;p

N

;p_

P

;p_

N

) in (7). This is

ters(Gautier,1991).

- datarankdeienydue toabad hoie ofthe

trajetory(x ;_ x; p

P

;p

N

;p_

P

;p_

N

)whihis sampled

inW.Thisistheproblemofoptimalmeasurement

strategieswhihissolvedusinglosedloopidenti-

ationtotrakexitingtrajetories(Gautierand

Khalil,1992).

CalulatingtheW.L.S.solutionof(13)fromnoisy

disrete measurements or estimations of deriva-

tives, may lead to bias beause W and Y may

be non independent random matries. Then it

is essential to lter data in Y and W, before

omputingtheW.L.S.solution.

4.2 Filteringaspets

Thederivativesin(13)areobtainedwithoutphase

shift usinga entral dierenealgorithm.A low-

pass lterwithout phaseshiftand without mag-

nitude distortion into the bandwidth is applied

on the measurements to redue the noise. This

lowpasslteriseasilyobtainedwithanonausal

zero-phase digital ltering by proessing the in-

put data through an IIR lowpass Butterworth

lter in both the forward and reverse diretion

using a 'ltlt' proedure from Matlab (Pham

et al., 2001). The ut-o frequeny !

H

of the

lowpassltershould be hosento avoid any dis-

tortion of magnitude on the ltered signals into

the bandwidth of the system. A seond lter is

implemented to eliminate the high frequenies

noises. ThevetorY and eah olumn of W are

ltered(parallel ltering)byalowpasslterand

are resampled at a lower rate. This step is not

sensitive to lter distortion beause error intro-

duedbythislteringproessisthesameineah

memberofthelinearsystem(13).Thekeypointof

thisidentiationmethod istohoosetheut-o

frequeny !

H

and the sampling frequeny !

s to

keepusefulsignalofthedynamibehaviorofthe

systeminthelterbandwidth.In(Gautier,1996),

the author proposes to hoose the sampling fre-

queny!

s

ofmeasurementsinpratie,ifpossible,

suhas:

!

s

100!

dyn

(14)

Where !

dyn

is the bandwidth of the position

losedloop.Astrategyoftuningforthefrequeny

!

H

and the sampling frequeny !

s

is presented

in (Pham et al., 2001). This method suggests to

bound thedistortion of amplitudeintroduedby

the derivative lter and the lowpass lter at a

frequeny xed with regard to the dynamis of

An experimental identiation is performed on

the testing bed. The sampling frequeny for the

aquisition of measurements isequalto 5kHz in

order to satisfy the relation (14). A losed loop

identiation, usingaproportionalfeedbakon-

trol, has been performed. A hirp sweeping be-

tween0Hzand2Hzinordertoexitethesystem

losetoitsapriori naturalfrequenywhihises-

timatedaround1.5Hz.Severalsquaretrajetories

forthedesiredpositionwithdierentamplitudes

are used to exite the frition parameters. The

results of the experimental identiation are re-

ported in thetable 1.The estimatedparameters

aregiven withtheirondeneintervalandtheir

relative standard deviation. Standard deviations

^

Xsi

areestimated using lassialand simplere-

sults from statistis, onsidering the matrix W

to be a deterministi one, and to be a zero

mean additive independent noise, with standard

deviation

suhlike:

C

= 2

I

rr

(15)

Where I

rr

is the matrix identity (rr). The

ovariane matrix of the estimation error and

standarddeviationsanbealulatedby:

C

^

Xs

^

Xs

= 2

W T

W

1

(16)

2

^

X

si

=C

^

X

s

^

X

s ii

, is the i th

diagonal oeÆient of

C

^

X

s

^

X

s

. Therelativestandarddeviation%

^

Xsr is

givenby:

%

^

Xsr i

=100

^

Xsi

^

X

si

(17)

Aparameterwith%

^

Xsr

10%anberemoved

from the model beause it is not identiable on

the given trajetory and it poorly inreases the

relativeerrornorm.

Table1.Identiationresults

Parameters

^

X

s

2

^

Xs

%

^

Xsr

(ISOunits)

X

s

(1) 3.49e-009 8.62e-012 0.1234

X

s

(2) 9.95e-009 6.41e-011 0.3223

X

s

(3) 1.04e-008 2.74e-011 0.1278

X

s

(4) 1.78e-009 2.74e-011 0.7695

X

s

(5) 7.00e-009 2.89e-010 2.0603

X

s

(6) 7.17e-009 8.30e-011 0.5784

M 1.69e+001 7.40e-002 0.2187

f

v

1.10e+001 8.75e-001 3.9679

F +

1.03e+001 4.80e-001 2.3245

F

2.38e+001 4.55e-001 0.9565

From the table 1, we notie that the dynami

parameters present a very small relative stan-

dard deviation, whih translates the good iden-

mass M is lose to the manufaturer data (17

kg). In general, onerning the pneumati part,

itisassumedthatthepolytropioeÆientklies

between 1 (isothermal evolution) and 1.4 (adia-

bati evolution).It is noteworthy that theratios

X

s (3)=X

s

(2)=1:04andX

s (6)=X

s

(5)=1:02give

a non lassial result. They orrespond to the

polytropi oeÆients of the gas k

P and k

N in

eahhamber.

0 0.1 0.2 0.3 0.4 0.5

−5

−4

−3

−2

−1 0 1 2 3 4

5 x 10 ⁻³ cross validation : model −. actual −

time (s) Mass flow rate q mP (kg/s)

Fig.4.Massowrateq

m

0 0.1 0.2 0.3 0.4 0.5 0.6

−150

−100

−50 0 50 100 150 200 250

cross validation : model −. actual −

time (s)

Effort (N)

Fig.5.Eort:S

P p

P S

N p

N F

ext

A ross-validation of the identiation is per-

formedtotestthemodel.Itonsistsinomparing

theestimationsofthemassowratesandtheef-

fortofthemodelwithexperimentalsignalswhih

had not been used in the identiation proess.

On gures4and 5,wepresentaomparison be-

tweenthesimulatedandtheatualmassowrate

andeort.Theseguresshowthatthesimulation

and themeasurementsareverylose,thismeans

a good identiation of the parameters for the

testingbed.

6. CONLUSION

This paperdealswith the estimation of physial

parametersofapneumatiservopositioningsys-

a polynomial approximation.Next, the dynami

parametersareestimatedusingtheweightedleast

squaressolutionofanoverdeterminedlinearsys-

tem obtained from the sampling of the dynami

modelalongalosedlooptrakingtrajetory.An

rstexperimental study exhibits goodidentia-

tionresults.Nevertheless,theseresultsshould be

viewed with some degree of reservation beause

more investigations should beadress in order to

hekthesensivityofidentied parametersinre-

lationtothestatimassowrateapproximation.

NOMENCLATURE

f

v

:visousfritionoeÆient(N/m/s)

k:polytropioeÆient

M :totalloadmass(kg)

p:pressureintheylinderhamber(Pa)

qm : mass ow rate provided from servodistributor to

ylinderhamber(kg/s)

r:perfetgasonstantrelatedtounitmass(J/kg/K)

S:areaofthepistonylinder(m 2

)

T :temperature(K)

V :volume(m 3

)

x,x,_ x:position(m),veloity(m/s),aeleration(m/s 2

)

u:spoolposition(v)

!:pulsation(rad/s)

'(:):leakagepolynomialfuntion(kg/s)

(:):polynomialfuntion(kg/s/V)

l:lengthofstroke(m)

Subsript

Coulombfrition,Ddeadvolume,extexternal,ffrition,

N hamberN,P hamberP

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