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Model Reference Control for Timed Event Graphs in Dioids
Bertrand Cottenceau, Laurent Hardouin, Jean-Louis Boimond, Jean-Louis Ferrier
To cite this version:
Bertrand Cottenceau, Laurent Hardouin, Jean-Louis Boimond, Jean-Louis Ferrier. Model Reference Control for Timed Event Graphs in Dioids. Automatica, Elsevier, 2001, 37, pp.1451-1458. �hal- 00845151�
B. Cotteneau
, L. Hardouin, J.L.Boimond, J.L.Ferrier
Laboratoired'Ingenierie des Systemes Automatises,
62av. Notre-Dame du La, 49000 ANGERS,FRANCE.
Tel: (33) 241 3657 33
Fax: (33) 2 4136 5735.
Abstrat
Thispaperdealswithfeedbakontrollersynthesisfor TimedEvent Graphsindioids. We
disuss here the existene and the omputation of a ontroller whih leads to a losed-loop
system whosebehavior isaslose aspossible totheoneofagivenreferene modelandwhih
delaysas muhas possible the input of tokens inside the (ontrolled) system. The synthesis
presentedhereismainlybasedonresiduationtheoryresults andsomeKleenestarproperties.
Keywords: Disrete Event Systems, Timed Event Graphs, Dioid, Residuation Theory,
FeedbakSynthesis.
bertrand.otteneauistia.univ-angers.fr
TimedEventGraphs(TEG)onstituteasublassoftimedPetrinetsofwhiheahplaehasexatly
oneupstream and one downstream transition. It is well known that the timed/event behaviorof
aTEG,under theearliestfuntioning rule 1
, anbe expressedby linearrelationsoversomedioids
(Baellietal.,1992)(DeShutter,1996). Stronganalogiesthenappearbetweenthelassiallinear
systemtheory and the (max,+)-linearsystem theory. Inpartiular, the onept ofontrol is well
dened intheontext ofTEGstudy. Itreferstothering-ontrol oftheTEGinputtransitions in
order to reah desiredperformane (see for instane (Cofer &Garg, 1996) (Takai, 1989)). Inthe
(max,+)literature,anoptimalontrolforTEGexistsandisproposedin(Cohenetal.,1989)(Menguy
et al., 2000). Fora given referene input, this open-loop struture ontrol yields thelatest input
ringdatein ordertoobtaintheoutputbeforethedesireddate.
Thispaperaimstotransposesomelosed-loopontrolstruturestoTEG.Morepreisely,wefous
onontrollersynthesis suhasoutputfeedbakontroller,statefeedbakontroller oroutputfeedbak
onstateontroller. Theontrollersynthesisis donein orderthat theontrolledsystem willbehave
asloseaspossibletoagivenreferenemodel. Furthermoretheproposedontrollersallowdelaying
asmuhaspossiblethetokeninputinside theTEG.
Appliations of these ontrollers are possible within the framework of produtionmanagement.
Indeed, TEG are well adapted to represent a lass of manufaturing systems whih present only
delaysandsynhronizationphenomena(Ayhan&Wortman,1999). Therefore,inthemanufaturing
ontextourontrollersallowmodifyingthedynamisofasystem(produtionlineormanufaturing
workshop) aording to agiven referene model and delaying as muh as possible the raw parts
input into the system. Thelatter propertyontributes to dereasing thework-in-proess amount
whihisapermanentonernforthejust-in-timeprodution.
Inthenextsetion,wereallsometheoretialresultsfrom the(max,+) literatureandintrodue
thealgebraifoundations. Setion3isdevoted toreallsomeelementsof TEGrepresentationover
partiular dioids. The problem ofontrollersynthesisis stated andsolvedin setion 4. Setion5
aimstopresentanillustrativeexample.
2 Elementsof Dioidand Residuation Theories
2.1 Dioid Theory
Werst reallin thissetion somenotionsfromthe dioid theory. The readerisinvitedtoonsult
(Baelliet al.,1992)foraompletepresentation.
Denition1 (Dioid) A dioid is a set D endowed with two inner operations denoted and .
Thesum isassoiative,ommutative, idempotent(8a2D;aa=a)andadmits aneutral element
denoted". Theprodutisassoiative,distributesoverthesumandadmitsaneutralelementdenoted
e. The element" isabsorbingfor the produt.
1
i.e.atransitionisredassoonasitisenabled.
equivalene: 8a;b2D;ab () a=ab.
Denition3 (CompleteDioid) A dioidDisomplete ifitislosedfor innitesumsandif the
produtdistributesoverinnite sumstoo.
Example1(Z
max
dioid) SetZ=Z[f 1;+1gendowedwiththe maxoperator assumandthe
lassial sum +asprodutisaompletedioid, usuallydenotedZ
max
,of whih "= 1 ande=0.
Thefollowingtheoremallowssolvingertainimpliitequationsdenedoverompletedioids.
Theorem1 Over a omplete dioid D, the impliit 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 aomplete dioid D, will be sometimesrepresented by
the following mapping
K: D ! D
x 7!
L
i2N x
i
:
Thefollowingtheoremreallssomelassialformul involvingKleenestarmapping.
Property1 LetD aompletedioidanda;b2D.
(a
)
= a
(1)
a
a
= a
(2)
a(ba)
= (ab)
a: (3)
2.2 ResiduationTheory
Theresiduationtheoryprovides,undersomeassumptions,optimalsolutionstoinequalitiessuhas
f(x)b,wheref isanorder-preservingmappingdenedoverorderedsets. Sometheoretialresults
are realledbelow. Complete presentations aregiven in (Blyth &Janowitz,1972)(Baelli 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
℄
isalledthe residual off. Whenf isresiduated, f
℄
isthe uniqueisotonemapping suhthat
fÆf
℄
Idandf
℄
Æf Id; (4)
whereIdisthe identity mappingrespetivelyon F andE.
Theorem2((Baelli etal.,1992)) Let f : E ! F where E and F are omplete dioids of
whihbottomelementsarerespetivelydenoted"
E and"
F
. Then,f isresiduatedif("
E )="
F and
8AE f( L
x2A x)=
L
x2A f(x).
Theirresidualsare usuallydenotedrespetivelyx7!a Æ
nx andx7!
xÆ
= ain(max,+) literature.
proof: aordingto def.3, ifDis aompletedioid then theprodutdistributes overinnitesums
and"isabsorbingwhihsatisestherequirementofth.2.
Somelassialresultsonerningprodutresidualaregiveninthefollowingtheorem.
Theorem3((Baelli 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
(9)
Theorem4 LetDaomplete dioidandA2D pn
. Then,A Æ
nA2D nn
and
A Æ
nA=(A Æ
nA)
: (10)
proof: see(MaxPlus,1991)foranotherproof. First,aordingto(5), A Æ
nAe,wheree2D nn
is the neutralelement formatrix produt. Moreover, aordingto (6), A = A(A Æ
nA). Therefore,
we have A Æ
nA = A Æ
n[A(A Æ
nA)℄. Furthermore, thanks to (8), we an 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)
whihnally leadstoequality sine,aordingto thedioid orderdenition (def.2) andtheKleene
stardenition (th.1),wealsohave(A Æ
nA)
=eA Æ
nAA Æ
nA.
2.3 Mapping restrition
In this subsetion, we address the problem of mapping restrition and its onnetion with the
residuationtheory. Inpartiular,weshowthattheKleenestarmapping,whihanbeshowntobe
notresiduated,beomesresiduatedassoonasitsodomainisrestritedto itsimage.
Denition6 (Restritedmapping) 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 anonial
injetion. Identially, letB F withImf B. Mapping
Bj
f :E!B isdenedby f =Id
jB Æ
Bj f,
whereId
jB
:B !F;x7!x isthe anonial injetion.
alosuremapping iff IdandfÆf =f.
Remark1 Aording to(1), mappingK isalosuremappingsinea
a and(a
)
=a
.
Proposition 1 Let a losure mapping f : E ! E . Then,
Imfj
f is a residuated mapping whose
residual isthe anonial injetion Id
jImf
:Imf !E;x7!x.
proof: aording to (4),
Imfj
f is residuated if there exists a mapping g suh that
Imfj
f Æg Id
andgÆ
Imfj
f Id,whereidentitymappings arerespetivelyidentityonImf and onE. Bysetting
g = Id
jImf
, we both verify
Imfj f ÆId
jImf
=
Imfj f
jImf
= Id (identity on Imf) sine 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: theproofisdiretsineKisalosuremapping.
Remark2 We an state from or.2 that x = a
is the greatest solution to inequality x
a
.
Atually,this greatestsolutionahieves equality.
3 TEGdesription on dioids
3.1 Transferfuntion
We reall that TEG an be seen as linear disrete event dynamial systemsby using some dioid
algebras(Cohenetal.,1989)(Baellietal.,1992). Forinstane,byassoiatingwitheahtransition
x a\dater"funtionfx(k)g
k 2Z
,inwhihx(k)isequaltothedatewhenwhihtheringnumbered
k ours,it ispossibleto obtainalinear staterepresentationin Z
max
. Asin onventionalsystem
theory,outputfy(k)g
k 2Z
of aSISOTEGisthenexpressed asaonvolutionofitsinputfu(k)g
k 2Z
byitsimpulseresponse 2
fh(k)g
k 2Z .
Analogous transforms to z-transform (used to represent disrete-time trajetories in lassial
theory)anbeintroduedforTEG.Indeed,oneanrepresentadaterfx(k)g
k 2Z
byits-transform
whihisdenedasthefollowingformalpowerseries: X()= L
k 2Z x(k)
k
. Variablemayalsobe
regardedasthebakwardshiftoperatorineventdomain(formally,x(k)=x(k 1)). Consequently,
oneanexpressTEGbehavioroverthedioidofformalpowerseriesinonevariableandoeÆients
in Z
max
. Thisdioidisusually denotedZ
max
JKinliterature.
Forinstane, onsidering theTEGdrawn insolid blaklines ing.2(without takingaountof
thegreyars),datersx
1 ,x
2 andx
3
arerelatedasfollowsoverZ
max :
x
3
(k)=3x
1
(k 1)8x
2
(k)2x
3 (k 1):
2
whihistheouputduetoaninnityofinputringsatdatezero(MaxPlus,1991).
max
x
3
()=3x
1
()8x
2
()2x
3 ():
Consequently,forthis TEG,weanobtainthefollowingrepresentationoverZ
max JK:
8
<
:
X = AX BU
Y = CX
(11)
with
A= 0
B
B
B
2 " "
" 3 "
3 8 2
1
C
C
C
A
; B= 0
B
B
B
e "
" e
" "
1
C
C
C
A
; X= 0
B
B
B
x1
x2
x
3 1
C
C
C
A
;
C=
" " 2
; U = 0
u
1
u
2 1
A
and Y =y:
Bysolvingthestateequationof(11)aordingtoth.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 suh a transfer
relationstartingfromthe staterepresentation (Gaubert,1992) (Cotteneau,1999).
3.2 Periodiity,ausality and realizability
ThetransferrelationofaTEGisharaterizedbysomeperiodiandausalpropertiesthatwereall
hereafter. Let usonsider 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 orresponds to the lower bound of Supp(s),
i.e. val(s)=minfk2 Zjs(k)6="g. A series s2 Z
max
JK suh that Supp(s) is nite issaid to be
polynomial.
Denition8 (Causality) A series s 2 Z
max
JK is ausal if s = " or if fval(s) 0 and s
val(s)
g. Theset ofausalelementsof Z
max
JK hasaomplete dioidstruturedenotedZ +
max JK.
Denition9 (Periodiity) A series s 2 Z
max
JK is said to be periodi if it an be written as
s=pq(
)
withpandqtwopolynomialsand ; 2N. Amatrixissaidtobeperiodi ifallits
entries areperiodi.
Denition10 (Realizability) A seriess2Z
max
JK issaid tobe realizableif it existsthree ma-
tries A,B andC withentries in N[f 1;+1g suhthat s=C(A)
B. A matrixissaid tobe
realizableif allitsentries arerealizable.
Inotherwords,aseriessisrealizableifitorrespondsto atransferrelationofaTEG.
Theorem5((Cohen et al.,1989)) The following statements areequivalent:
A seriessisrealizable.
Theset of periodiseries ofZ
max
JK hasadioidstruture whihis notomplete. Nevertheless,
wehavethefollowingproperty.
Theorem6 Lets
1 ands
2
twoperiodi seriesof Z
max
JK. Then, s
1 Æ
ns
2
isalso aperiodi series.
proof: see(MaxPlus,1991).
Theorem7 The anonial injetion Id
j+
: Z +
max
JK ! Z
max
JK is residuated. We denote Pr
+ :
Z
max
JK!Z +
max
JKitsresidual,i.e. Pr
+
(s)isthe greatestausal serieslessthanor equaltos.
proof: see(Cotteneauetal.,1999).
Fromapratialpointofview,foralls2Z
max
JK,theomputationofPr
+
(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 Letsaperiodi(notneessaryausal)seriesofZ
max
JK. Then,Pr
+
(s)isthegreatest
realizableseries lessthanorequal tos.
proof: (sketh of proof) thePr
+
mappingsimply amountsto zeroing termsof aseries whih are
notwith positiveoeÆientorexponent. Then,if sis periodi, Pr
+
(s) remainsperiodi. Finally,
Pr
+
(s)isbothperiodiandausal,i.e.realizable(f. th.5).
4 Feedbak ontroller synthesis
4.1 Problem statement
As presented previously, in dioid Z
max
JK, the behavior of an m-inputs p-outputs TEG an be
desribedbyastaterepresentationsuhas(11)whereU 2Z
max JK
m
andY 2Z
max JK
p
. Aording
toth.1,bysolvingthestateequationinX,theinput-outputtransferrelationisthenexpressedby
Y =HU; (13)
whereH =CA
B belongsto Z
max JK
pm
.
Wefoushereonontrollersynthesissuhas:
outputfeedbakontroller: aontroller,denotedF,isaddedbetweenoutputY andinputU of
thenominal system(seeg.1-a-). Therefore, theproessinputveriesU =V FY,andthe
output is desribedby Y =H(V FY). Aording to th.1, thelosed-looptransferis then
equalto
Y =(HF)
HV: (14)
state feedbak ontroller: a ontroller, denoted L, is added between internal state 3
X and
3
suhaontrolstrutureimpliesthattheinternalstateusedfortheontrolismeasurable.
solvingthestateequationof(11)aordingtoth.1,wehaveX =A
BU =A
B(V LX)=
A
BLXA
BV. Therefore,bysolvingthisnewimpliitequationaordingtoth.1,weobtain
X = (A
BL)
A
BV. Finally, by replaing X in the output equation of (11) and by using
(3) we haveY = C(A
BL)
A
BV =CA
B(LA
B)
V, whih orresponds to the following
transferrelation:
Y =H(LA
B)
V: (15)
output feedbak on state ontroller: a ontroller, denoted S, is added between output and
internalstate(seeg.1--). ThestateevolutionisthendesribedbyX =AXBUSY. It
isthereader'sonerntohekthat theinput-outputtransferisgivenby
Y =(CA
S)
HU: (16)
TheontrollerS,loatedbetweenoutputandinternalstate,behaveslikeinhibiting 4
ars. For
instane, these ars aredepited in greylines in g.2. Therefore,suhastruture of ontrol
preservesasuitablemeaningprovidedthatoneaneetivelyontroltheinternaltransitions,
i.e.oneandelaytheirringswhenneessary.
The objetive of the model referene ontrol is to impose adesired behavior(G
ref
) to a given
system(H)whilendingthebestontrollerarryingoutthisobjetive. Morepreisely,bydenoting
G
C
thetransferoftheontrolledsystemwithontrollerC, wetrytodetermineC suhthat
G
C G
ref
: (17)
Constraint (17) may be literally expressed as: the losed-loop system is at least as fast as the
referenemodel.
In addition, by assumingthat it may exist several ontrollers C
i
;i 2f1;:::;ng, leadingto the
sameontrolledtransfer,i.e.G
C
0
==G
C
n
,wefousonthegreatestone(whensuhanoptimal
exists): thegreatestistheonewhihdelaysasmuhaspossibletheinputinthesystem. Therefore,
intheTEGontext,thissupremalontrollerminimizestheamountoftokensintheontrolledTEG.
In short, for a given referene model, the problem takled here onsists in nding the greatest
ontroller C (when it exists) heking G
C
G
ref
. Therefore, within the framework of feedbak
synthesis and aording 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 withrespettotheresiduationtheory.
4
thesupplementaryarsduetotheontrollerauthorizeorprohibittheringoftheontrolledtransitions.
