Model reference control for timed event graphs in dioids

(1)

B. Cottenceau

, L. Hardouin, J.L.Boimond, J.L.Ferrier

Laboratoired'Ingenieriedes Systemes Automatises,

62av. Notre-Damedu Lac, 49000 ANGERS,FRANCE.

Tel: (33) 241 3657 33

Fax: (33) 2 4136 5735.

Abstract

Thispaperdealswithfeedbackcontrollersynthesisfor TimedEvent Graphsindioids. We

discuss here the existence and the computation of a controller which leads to a closed-loop

system whosebehavior isasclose aspossible totheoneofagivenreference modeland which

delaysas muchas possible the input of tokens inside the (controlled) system. The synthesis

presentedhereismainlybasedonresiduationtheoryresults andsomeKleenestarproperties.

Keywords: Discrete Event Systems, Timed Event Graphs, Dioid, Residuation Theory,

FeedbackSynthesis.

bertrand.cottenceau@istia.univ-angers.fr

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TimedEventGraphs(TEG)constituteasubclassoftimedPetrinetsofwhicheachplacehasexactly

oneupstream and one downstream transition. It is well known that the timed/event behaviorof

aTEG,under theearliestfunctioning rule 1

, canbe expressedby linearrelationsoversomedioids

(Baccellietal.,1992)(DeSchutter,1996). Stronganalogiesthenappearbetweentheclassicallinear

systemtheory and the (max,+)-linearsystem theory. Inparticular, the concept ofcontrol is well

dened inthecontext ofTEGstudy. Itreferstothering-control oftheTEGinputtransitions in

order to reach desiredperformance (see for instance (Cofer &Garg, 1996) (Takai, 1989)). Inthe

(max,+)literature,anoptimalcontrolforTEGexistsandisproposedin(Cohenetal.,1989)(Menguy

et al., 2000). For a given reference input, this open-loop structure control yields thelatest input

ringdatein ordertoobtaintheoutputbeforethedesireddate.

Thispaperaimstotransposesomeclosed-loopcontrolstructurestoTEG.Moreprecisely,wefocus

oncontrollersynthesis suchasoutputfeedbackcontroller,statefeedbackcontroller oroutputfeedback

onstatecontroller. Thecontrollersynthesisis donein orderthat thecontrolledsystem willbehave

ascloseaspossibletoagivenreferencemodel. Furthermoretheproposedcontrollersallowdelaying

asmuchas possiblethetokeninputinside theTEG.

Applications of these controllers are possible within the framework of productionmanagement.

Indeed, TEG are well adapted to represent a class of manufacturing systems which present only

delaysandsynchronizationphenomena(Ayhan&Wortman,1999). Therefore,inthemanufacturing

contextourcontrollersallowmodifyingthedynamicsofasystem(productionlineormanufacturing

workshop) according to agiven reference model and delaying as much as possible the raw parts

input into the system. Thelatter propertycontributes to decreasing thework-in-process amount

whichisapermanentconcernforthejust-in-timeproduction.

Inthenextsection,werecallsometheoreticalresultsfrom the(max,+) literatureandintroduce

thealgebraicfoundations. Section3isdevoted torecallsomeelementsof TEGrepresentationover

particular dioids. The problem ofcontrollersynthesisis stated andsolvedin section 4. Section5

aimstopresentanillustrativeexample.

2 Elements of Dioidand Residuation Theories

2.1 Dioid Theory

Werst recallin thissection somenotionsfromthe dioid theory. The readerisinvitedtoconsult

(Baccelliet al.,1992)foracompletepresentation.

Denition1 (Dioid) A dioid is a set D endowed with two inner operations denoted and .

Thesum isassociative,commutative, idempotent(8a2D;aa=a)andadmits aneutral element

denoted". Theproductisassociative,distributesoverthesumandadmitsaneutralelementdenoted

e. The element" isabsorbingfor the product.

1

i.e.atransitionisredassoonasitisenabled.

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equivalence: 8a;b2D;ab () a=ab.

Denition3 (CompleteDioid) A dioidDiscomplete ifitisclosedfor innitesumsandif the

productdistributesoverinnite sumstoo.

Example1(Z

max

dioid) SetZ=Z[f 1;+1gendowedwiththe maxoperator assumandthe

classical sum +asproductisacompletedioid, usuallydenotedZ

max

,of which "= 1 ande=0.

Thefollowingtheoremallowssolvingcertainimplicitequationsdenedovercompletedioids.

Theorem1 Over a complete dioid D, the implicit equation x =axb admits x = a

b as least

solution,wherea

= L

i2N a

i

(Kleene staroperator)with a 0

=e.

Notation1 The Kleene star operator, over acomplete dioid D, will be sometimesrepresented by

the following mapping

K: D ! D

x 7!

L

i2N x

i

:

Thefollowingtheoremrecallssomeclassicalformul involvingKleenestarmapping.

Property1 LetD acompletedioidanda;b2D.

(a

)

= a

(1)

a

= a

(2)

a(ba)

= (ab)

a: (3)

2.2 ResiduationTheory

Theresiduationtheoryprovides,under someassumptions,optimalsolutionstoinequalitiessuchas

f(x)b,wheref isanorder-preservingmappingdenedoverorderedsets. Sometheoreticalresults

are recalledbelow. Complete presentations aregiven in (Blyth &Janowitz,1972)(Baccelli et al.,

1992).

Denition4 (Isotone mapping) A mapping f dened over ordered sets is isotone if a b )

f(a)f(b).

Denition5 (Residual and residuated mapping) Let f : E ! F an isotone mapping, where

(E;) and (F;) are ordered sets. Mapping f issaid residuated if for all y 2 F, the leastupper

bound of subsetfx2Ejf(x)ygexistsandlies inthis subset. Itis thendenotedf ]

(y). Mapping

f ]

iscalledthe residual off. Whenf isresiduated, f ]

isthe uniqueisotonemapping suchthat

fÆf ]

Idandf ]

Æf Id; (4)

whereIdisthe identity mappingrespectivelyon F andE.

Theorem2((Baccelli etal.,1992)) Let f : E ! F where E and F are complete dioids of

whichbottomelementsarerespectivelydenoted"

E and"

F

. Then,f isresiduatedif("

E )="

F and

8AE f( L

x2A x)=

L

x2A f(x).

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Theirresidualsare usuallydenotedrespectivelyx7!a Æ

nx andx7!x Æ

= ain(max,+) literature.

proof: accordingto def.3, ifDis acompletedioid then theproductdistributes overinnitesums

and"isabsorbingwhichsatisestherequirementofth.2.

Someclassicalresultsconcerningproductresidualaregiveninthefollowingtheorem.

Theorem3((Baccelli etal.,1992)) Mappingsx7!a Æ

nxandx7!

xÆ

=averifythe followingprop-

erties:

a Æ

n[ax] x [xa]

Æ

=a x (5)

a[a Æ

nax] = ax

[xaÆ

= a]a = xa (6)

[ab]

Æ

nx = b Æ

n[a Æ

nx]

xÆ

= [ba] =

[xÆ=a]Æ

=b (7)

[a Æ

nx]b a Æ

n[xb]

b[xÆ

=a]

[bx]Æ

=a (8)

a

x = a

Æ

n[a

x] xa

= [xa

]Æ

=a

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Theorem4 LetDacomplete dioidandA2D pn

. Then,A Æ

nA2D nn

and

A Æ

nA=(A Æ

nA)

: (10)

proof: see(MaxPlus,1991)foranotherproof. First,accordingto(5), A Æ

nAe,wheree2D nn

is the neutralelement formatrix product. Moreover, accordingto (6), A = A(A Æ

nA). Therefore,

we have A Æ

nA = A Æ

n[A(A Æ

nA)]. Furthermore, thanks to (8), we can show that A Æ

n[A(A Æ

nA)]

A Æ

nAA Æ

nA. We thus obtain the following inequality e (A Æ

nA) 2

A

Æ

nA; and more generally

8n2N, e(A Æ

nA) n

A Æ

nA. Therefore,weverifye L

n2N (A

Æ

nA) n

A Æ

nA(i.e.(A Æ

nA)

A Æ

nA)

whichnally leadstoequality since,accordingto thedioid orderdenition (def.2) andtheKleene

stardenition (th.1),wealsohave(A Æ

nA)

=eA Æ

nAA Æ

nA.

2.3 Mapping restriction

In this subsection, we address the problem of mapping restriction and its connection with the

residuationtheory. Inparticular,weshowthattheKleenestarmapping,whichcanbeshowntobe

notresiduated,becomesresiduatedassoonasitscodomainisrestrictedto itsimage.

Denition6 (Restrictedmapping) Let f : E ! F a mapping and A E. We will denote

f

jA

: A ! F the mapping dened by f

jA

= f ÆId

jA

where Id

jA

: A ! E;x 7! x is the canonical

injection. Identically, letB F withImf B. Mapping

Bj

f :E!B isdenedby f =Id

jB Æ

Bj f,

whereId

jB

:B !F;x7!x isthe canonical injection.

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aclosuremapping iff IdandfÆf =f.

Remark1 According to(1), mappingK isaclosuremappingsincea

a and(a

)

=a

.

Proposition 1 Let a closure mapping f : E ! E . Then,

Imfj

f is a residuated mapping whose

residual isthe canonical injection Id

jImf

:Imf !E;x7!x.

proof: according to (4),

Imfj

f is residuated if there exists a mapping g such that

Imfj

f Æg Id

andgÆ

Imfj

f Id,whereidentitymappings arerespectivelyidentityonImf and onE. Bysetting

g = Id

jImf

, we both verify

Imfj f ÆId

jImf

=

Imfj f

jImf

= Id (identity on Imf) since f Æf = f, and

Id

jImf Æ

Imfj

f =f Id(bydef.7).

Corollary 2 Mapping

ImKj

Kisaresiduatedmappingwhose residual is(

ImKj K)

]

=Id

jImK .

proof: theproofisdirectsinceKisaclosuremapping.

Remark2 We can state from cor.2 that x = a

is the greatest solution to inequality x

a

.

Actually,this greatestsolutionachieves equality.

3 TEGdescription on dioids

3.1 Transferfunction

We recall that TEG can be seen as linear discrete event dynamical systemsby using some dioid

algebras(Cohenetal.,1989)(Baccellietal.,1992). Forinstance,byassociatingwitheachtransition

x a\dater"functionfx(k)g

k 2Z

,inwhichx(k)isequaltothedatewhenwhichtheringnumbered

k occurs,it ispossibleto obtainalinear staterepresentationin Z

max

. Asin conventionalsystem

theory,outputfy(k)g

k 2Z

of aSISOTEGisthenexpressed asaconvolutionofitsinputfu(k)g

k 2Z

byitsimpulseresponse 2

fh(k)g

k 2Z .

Analogous transforms to z-transform (used to represent discrete-time trajectories in classical

theory)can beintroducedforTEG.Indeed,onecanrepresentadaterfx(k)g

k 2Z

byits-transform

whichisdenedasthefollowingformalpowerseries: X()= L

k 2Z x(k)

k

. Variablemayalsobe

regardedasthebackwardshiftoperatorineventdomain(formally,x(k)=x(k 1)). Consequently,

onecanexpressTEGbehavioroverthedioidofformalpowerseriesinonevariableandcoeÆcients

in Z

max

. Thisdioidisusually denotedZ

max

JKinliterature.

Forinstance, considering theTEGdrawn insolid blacklines ing.2(without takingaccountof

thegreyarcs),datersx

1 ,x

2 andx

3

arerelatedasfollowsoverZ

max :

x

3

(k)=3x

1

(k 1)8x

2

(k)2x

3 (k 1):

2

whichistheouputduetoaninnityofinputringsatdatezero(MaxPlus,1991).

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max

x

3

()=3x

1

()8x

2

()2x

3 ():

Consequently,forthis TEG,wecanobtainthefollowingrepresentationoverZ

max JK:

8

<

:

X = AX BU

Y = CX

(11)

with

A= 0

B

@

2 " "

" 3 "

3 8 2

1

C

A

; B= 0

B

@ e "

" e

" "

1

C

A

; X= 0

B

@ x1

x2

x

3 1

C

A

;

C=

" " 2

; U = 0

@ u

1

u

2 1

A

and Y =y:

Bysolvingthestateequationof(11)accordingtoth.1,i.e.Y =CA

BU,weobtainthefollowing

transferrelationin Z

max JK:

Y =

5(2)

10(3)

U: (12)

Remark3 Algorithms and software tools are now available in order to establish such a transfer

relationstartingfromthe staterepresentation (Gaubert,1992) (Cottenceau,1999).

3.2 Periodicity,causality and realizability

ThetransferrelationofaTEGischaracterizedbysomeperiodicandcausalpropertiesthatwerecall

hereafter. Let usconsider aseries s= L

k 2Z s(k)

k

in Z

max

JK. Thesupport of sis then dened

by Supp(s) = fk 2 Zjs(k) 6= "g, and its valuation corresponds to the lower bound of Supp(s),

i.e. val(s)=minfk 2Zjs(k)6="g. A series s2 Z

max

JK such that Supp(s) is niteis said to be

polynomial.

Denition8 (Causality) A series s 2 Z

max

JK is causal if s = " or if fval(s) 0 and s

val(s)

g. Theset ofcausalelementsof Z

max

JK hasacomplete dioidstructuredenotedZ +

max JK.

Denition9 (Periodicity) A series s 2 Z

max

JK is said to be periodic if it can be written as

s=pq(

)

withpandqtwopolynomialsand ; 2N. Amatrixissaidtobeperiodic ifallits

entries areperiodic.

Denition10 (Realizability) A seriess2Z

max

JK issaid tobe realizableif it existsthree ma-

trices A,B andC withentries in N[f 1;+1gsuchthat s=C(A)

B. A matrixissaid tobe

realizableif allitsentries arerealizable.

Inotherwords,aseriessisrealizableifitcorrespondsto atransferrelationofaTEG.

Theorem5((Cohen et al.,1989)) The followingstatements areequivalent:

A seriessisrealizable.

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Theset of periodicseries ofZ

max

JK hasadioidstructure whichis notcomplete. Nevertheless,

wehavethefollowingproperty.

Theorem6 Lets

1 ands

2

twoperiodic seriesof Z

max

JK. Then, s

1 Æ

ns

2

isalso aperiodic series.

proof: see(MaxPlus,1991).

Theorem7 The canonical injection Id

j+

: Z +

max

JK ! Z

max

JK is residuated. We denote Pr

+ :

Z

max

JK!Z +

max

JKitsresidual,i.e. Pr

+

(s)isthe greatestcausal serieslessthanor equaltos.

proof: see(Cottenceauetal.,1999).

Fromapracticalpointofview,foralls2Z

max

JK,thecomputationofPr

+

(s)isobtainedby:

Pr

+ (

L

k 2Z s(k)

k

)= L

k 2Z s

+ (k)

k

wheres

+ (k)=

8

<

:

s(k)if(k;s(k))(0;0)

"otherwise

:

Theorem8 Letsaperiodic(notnecessarycausal)seriesofZ

max

JK. Then,Pr

+

(s)isthegreatest

realizableseries lessthanorequal tos.

proof: (sketch of proof) thePr

+

mappingsimply amountsto zeroing termsof aseries which are

notwith positivecoeÆcientor exponent. Then,if sis periodic, Pr

+

(s) remainsperiodic. Finally,

Pr

+

(s)isbothperiodicandcausal,i.e.realizable(cf. th.5).

4 Feedback controllersynthesis

4.1 Problem statement

As presented previously, in dioid Z

max

JK, the behavior of an m-inputs p-outputs TEG can be

describedbyastaterepresentationsuchas(11)whereU 2Z

max JK

m

andY 2Z

max JK

p

. According

toth.1,bysolvingthestateequationinX,theinput-outputtransferrelationis thenexpressedby

Y =HU; (13)

whereH =CA

B belongsto Z

max JK

pm

.

Wefocushereoncontrollersynthesissuchas:

outputfeedbackcontroller: acontroller,denotedF,isaddedbetweenoutputY andinputU of

thenominal system(seeg.1-a-). Therefore, theprocessinputveriesU =V FY,andthe

output is describedby Y =H(V FY). According to th.1, theclosed-looptransferis then

equalto

Y =(HF)

HV: (14)

state feedback controller: a controller, denoted L, is added between internal state 3

X and

3

suchacontrolstructureimpliesthattheinternalstateusedforthecontrolismeasurable.

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solvingthestateequationof(11)accordingtoth.1,wehaveX =A

BU =A

B(V LX)=

A

BLXA

BV. Therefore,bysolvingthisnewimplicitequationaccordingtoth.1,weobtain

X = (A

BL)

A

BV. Finally, by replacing X in the output equation of (11) and by using

(3) we haveY = C(A

BL)

A

BV =CA

B(LA

B)

V, which corresponds to the following

transferrelation:

Y =H(LA

B)

V: (15)

output feedback on state controller: a controller, denoted S, is added between output and

internalstate(seeg.1-c-). ThestateevolutionisthendescribedbyX =AXBUSY. It

isthereader'sconcerntocheckthat theinput-outputtransferisgivenby

Y =(CA

S)

HU: (16)

ThecontrollerS,locatedbetweenoutputandinternalstate,behaveslikeinhibiting 4

arcs. For

instance, these arcs aredepicted in greylines in g.2. Therefore,suchastructure of control

preservesasuitablemeaningprovidedthatonecaneectivelycontroltheinternaltransitions,

i.e.onecandelaytheirringswhennecessary.

The objective of the model reference control is to impose adesired behavior(G

ref

) to a given

system(H)whilendingthebestcontrollercarryingoutthisobjective. Moreprecisely,bydenoting

G

C

thetransferofthecontrolledsystemwithcontrollerC, wetrytodetermineC suchthat

G

C G

ref

: (17)

Constraint (17) may be literally expressed as: the closed-loop system is at least as fast as the

referencemodel.

In addition, by assumingthat it may exist several controllers C

i

;i 2f1;:::;ng, leadingto the

samecontrolledtransfer,i.e.G

C

0

==G

C

n

,wefocusonthegreatestone(whensuchanoptimal

exists): thegreatestistheonewhichdelaysasmuchaspossibletheinputinthesystem. Therefore,

intheTEGcontext,thissupremalcontrollerminimizestheamountoftokensinthecontrolledTEG.

In short, for a given reference model, the problem tackled here consists in nding the greatest

controller C (when it exists) checking G

C

G

ref

. Therefore, within the framework of feedback

synthesis and according to (14)-(16), we have to nd, for agiven G

ref

, a greatest solutionin F

(resp. L andS)forinequality(18)(resp. (19)and(20))

(HF)

H G

ref

(18)

H(LA

B)

G

ref

(19)

(CA

S)

H G

ref

: (20)

In other words, this amounts to being interested in the properties of mappings x 7! (Hx)

H,

x7!H(xA

B)

andx7!(CA

x)

H withrespecttotheresiduationtheory.

4

thesupplementaryarcsduetothecontrollerauthorizeorprohibittheringofthecontrolledtransitions.

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Letusdene

M

H : Z

max JK

mp

! Z

max JK

pm

X 7! (HX)

H:

This mappingclearlyrepresentshowan outputfeedback X inuencesthe closed-looptransferdy-

namics. Clearly, inequality (18) admits a greatest solution for all reference models G

ref

only if

M

H

is residuated. However,accordingto th.2, one easily checks that M

H

is notresiduated since

M

H

(")=H 6=". Nevertheless,thefollowingresultshowsthattherearerestrictionsofmappingM

H

whichareresiduated. Thatamountstosayingthattheinequality(18)admitsanoptimal solution

onlyforspecic right-handsides.

Proposition 2 LetG2Z

max JK

pm

andD2Z

max JK

pp

. Letus consider the following sets:

G

1

=fGj9D periodic andcausal s.t. G=D

Hg

G

2

=fGj9D periodic andcausal s.t. G=HD

g:

Mappings

G1j M

H and

G2j M

H

are both residuated. Their residuals are such that (

G1j M

H )

]

(x) =

(

G

2 j

M

H )

]

(x)=H Æ

nx Æ

= H.

proof: accordingtodef.5, weremarkthatthetwofollowingassertionsareequivalent:

G

1 j

M

H

isresiduated

8Dperiodicandcausal;(HX)H

D

H admitsagreatestsolution.

So, wecanconcentrateonthe secondpoint. Since mappingx 7!Hx isresiduated (cf. cor.1)and

accordingto (3),wehave:

(HX)

H =H(XH)

D

H () (XH)

H Æ

n(D

H):

Accordingto(9)and(7), wecanrewrite

H Æ

n(D

H)=H Æ

n[D

Æ

n(D

H)]=(D

H) Æ

n(D

H):

Accordingto(10),thislastexpressionshowsthatH Æ

n(D

H)belongstotheimageofK:Z

max JK

mm

!

Z

max JK

mm

. Since

ImKj

Kisresiduated(cfcor.2),thereisalsothefollowingequivalence:

(XH)

H Æ

n(D

H) () XH H Æ

n(D

H):

Finally, sincemappingx 7!xH is residuated too(cf. cor.1),weverifythat X =H Æ

n(D

H )Æ

=H is

thegreatestsolutionofH(XH)

D

H,8D2Z

max JK

pp

. Thatamountstosayingthat

G1j M

H

isresiduated. Wewould showthat

G

2 j

M

H

isresiduatedwithanalogsteps.

As recalledinsection 3,workingonTEGcomesdownto consideringonlythesubsetofperiodic

andcausal seriesofZ

max

JK (cf. th.5). Then,the resultsobtainedin prop.2mustberestrictedto

thatcaseinorder tobeapplied toTEGcontrol.

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ref 1 2 r

(HF

r )

H G

ref

. Thisgreatestcontroller isgiven by

F

r

=Pr

+ (H

Æ

nG

ref Æ

=H):

proof: accordingto prop.2,H Æ

nG

ref Æ

=H isthegreatestsolutionto(HX)

H G

ref

. SinceG

ref 2

G

1 [G

2 ,G

ref

isperiodicandcausal. Therefore,accordingtoth.6,H Æ

nG

ref Æ

=H isperiodic. Eventu-

ally,accordingtoth.8,Pr

+ (H

Æ

nG

ref Æ

=H)isthegreatestrealizablesolution.

4.3 State feedback,feedback between outputand state.

For these two feedback synthesis problems, it is still a question of checking whether mappings

x 7! H(xA

B)

and x 7! (CA

x)

H are residuated or not. According to th.2, it is clear that

theyarenotresiduated. Nevertheless,theproblemofreferencemodelcontrolmayhaveanoptimal

realizablesolution,ineachcase,ifG

ref

isconstrainedtobelongtoparticularsubsetsofZ

max JK

pm

.

Proposition 4 LetH =CA

BbeaTEGtransfermatrix. ForallreferencemodelG

ref 2G

1 ,there

existsagreatestrealizable statefeedback L

r

suchthat H(L

r A

B)

G

ref

. This optimal solutionis

thencomputedby

L

r

=Pr

+ (H

Æ

nG

ref Æ

=(A

B)):

proof: as in the prop.2 proof, we rst have to show that for all D 2 Z

max JK

pp

, equation

H(LA

B)

D

H admitsagreatestsolution. Sincemappingx7!Hx isresiduated,wehave:

H(LA

B)

D

H () (LA

B)

H Æ

n(D

H):

Moreover, we have shown in the prop.2 proof that element H Æ

n(D

H) belongs to the image of

K:Z

max JK

mm

!Z

max JK

mm

. Then,since

ImKj

Kisresiduated,

(LA

B)

H Æ

n(D

H) () LA

B H Æ

n(D

H):

Sincex 7!xH is residuatedtoo,wethen obtainthat H Æ

n(D

H) Æ

=(A

B)isthe greatestsolutionto

H(LA

B)

D

H. Finally,ifG

ref

belongstoG

1 ,H

Æ

nG

ref Æ

=(A

B)isaperiodicmatrix(byapplying

th.6),andPr

+ (H

Æ

nG

ref Æ

=(A

B))isthenthegreatestrealizablesolution(byapplyingth.8).

Proposition 5 LetH =CA

B beaTEGtransfermatrix. ForallreferencemodelG

ref 2G

2 ,there

existsa greatest realizable output feedback on stateS

r

suchthat (CA

S

r )

H G

ref

. This optimal

solution isthencomputedby

S

r

=Pr

+ ((CA

) Æ

nG

ref Æ

=H):

proof: similartothepreviousproof.

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Remark4 (Particular caseG

ref

=H.) Since the identity matrixe issuch that e =e,we can

easily check that H 2G

1

and H 2 G

2

. Therefore, for any TEG, itis possible to preserve its own

transfer with either a greatest realizable output feedback, a greatest realizable state feedback or a

greatestrealizable output feedback on state. In (Cottenceau et al., 1999), that particular case has

already been studiedfor output feedback control.

5 Example

In order to illustrate results presented previously, we describe a complete synthesis of an output

feedback on state forthe TEGdepictedwith solidblacklines in g.2. Weassumethat this model

representsaworkshopwith3machines(M

1 , M

2 ,M

3

)ofwhich inputsaredescribedbytransitions

x

1 , x

2 and x

3

. Therefore, this example corresponds to a short application of our results in the

domainofmanufacturingmanagement.

We propose to compute a greatest output feedback on state so that the system has a transfer

relationclosetoagivenreferencetransferG

ref

. ForthisTEG,accordingtosection3,wehave

H =

5(2)

10(3)

:

Thistransfershowsthedierencethat exists betweentheproductionrateofpathu

1

!y,namely

1/2token/time unit,andthoseof pathu

2

!y,namely1/3token/timeunit.

According tothestructure oftheworkshop,anunstability 5

problem arisesassoonastoomany

partsareadmittedatthesametimeatinputsu

1 andu

2

becauseofthedierenceofproductionrates

ofmachinesM

1 andM

2

. Indeed,insuchacase,themarkingoftheplacelocatedbetweenx

1 andx

3

willgrowwithoutbound. So,arealisticobjectivewouldbeheretoimpose,thankstothecontroller,

theproductionrateoftheslowestmachine (M

2

)to thewhole system(i.e.1partper3timeunits).

Accordingtoprop.5,themodelreferencecontrolhasanoptimalsolutionifG

ref

bothbelongstoG

2

andreectsthedesiredproductionrate. Forinstance,ifwechoosehereG

ref

=H(3)

,weobtain

G

ref

=

5(3)

10(3)

;

whichsatisesbothconstraints.

Then,accordingtoprop.5,thegreatestrealizablefeedbackS

r

isgivenbycomputingPr

+ ((CA

) Æ

nG

ref Æ

=H).

Accordingtostaterepresentation(11),wehave

CA

=

5(2)

10(3)

2(2)

:

Therefore,wecancomputethecontroller. First,weobtain

(CA

) Æ

nG

ref Æ

=H = 0

B

@ 5

1

(3)

10(3)

2(3)

1

C

A

5

aTEGissaidstableifthemarkingofallitsinternalplacesremainsboundedforallinputsequence(MaxPlus,

1991). TheproblemofTEGstabilizationhasrecentlybeenreconsideredin(Commault,1998).

(12)

S

r

=Pr

+ ((CA

) Æ

nG

ref Æ

=H)= 0

B

@ 1(3)

2 4

(3)

1(3)

1

C

A

Arealizationofthat optimalcontrollerisdrawningreylinesing.2.

Remark5 Some otherexamplesaredevelopedin(Cottenceauetal., 1999)(outputfeedback),(Co-

henetal., 1998) (outputfeedback) and(Cottenceau,1999) (allthese structuresareillustrated). Let

usnotethatsuchasynthesisisindierentlyobtainedineitherdioidZ

max

JKordioidsZ

min JÆKand

M ax

in J;ÆK.

(13)

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Takai, S. (1989). A characterization of realizable behavior in supervisory control of timed event

Model reference control for timed event graphs in dioids

B U Y

Å C

A X S

H U Y

Å F

V B U Y Å L

V Å A C X

- a - - b -

- c -

y x !

!

u ! x M

u ! & x M M !