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Algebra
Sébastien Lahaye, Jean-Louis Boimond, Laurent Hardouin
To cite this version:
Sébastien Lahaye, Jean-Louis Boimond, Laurent Hardouin. Analysis of Periodic Discrete Event
Sys-tems in (Max,+) Algebra. WODES’2000, Workshop on Discrete Event SysSys-tems, Aug 2000, Ghent,
Belgium. pp.93-102. �hal-00845026�
EVENT SYSTEMS IN (MAX,+) ALGEBRA
S.Lahaye, J.L.Boimond, L.Hardouin
Laboratoire d'Ingenierie des Systemes Automatises, 62 Avenue Notre-Dame du La , 49000Angers, Fran e
[lahaye,boimond,hardouin℄istia.univ-angers.fr
Keywords: Dis reteEventDynami Systems,(max,+)Algebra,Periodi Systems
Abstra t Dis reteEventDynami Systemsmodeledby(max;+)linearequations with periodi ally varying oeÆ ients are studied. It turns out that spe tralpropertiesoftheso- alled monodromymatrix anbeusedfor theperforman eevaluationofthesesystems.
1. INTRODUCTION
Dis rete EventDynami Systems(DEDS) subje tto syn hronization phenomena an bemodeledbylinearequationsinaparti ularalgebrai stru ture alled (max ;+) algebra. A linear system theory analogous to the onventional theory has been developed forthis lassof systems whi h an be, for example, manufa turing systems or ommuni ation networks [2℄. In parti ular, linear time invariant systems, whose be-haviors are usually represented by Timed Event Graphs with onstant timings,have beenstudied extensively [2℄, [5℄, [7℄. In a manufa turing system,timeinvarian e orrespondsforexampletoassumethat pro ess-ingtimes are onstant.
Lots of systems arising in pra ti e are time-varying, that is, the v al-uesof the output response dependon when theinputis applied. Time variationisa resultofsystemparameters hanging: inamanufa turing system, pro essing times of parts may depend on their type. Systems des ribedbystate modelswith varying oeÆ ientsin (max;+) algebra have been onsidered in [10℄. The output tra king under just-in-time riterionhasinparti ular beenextendedto su h systems.
In this paper, the fo us is on linear systems whose state models have periodi ally varying oeÆ ients. Expli itly, ea h entry of the matri es
inthe state model satisfy a(k+K) =a(k), k 2Z. We aim at extend-ing to the (max ;+) ontext some established on epts and results of the onventional periodi linear system theory [3℄, [6℄, [4℄. In parti u-lar,thespe tralproperties oftheso- alled monodromymatrix, i.e.,the transitionmatrixoverone periodK,areusedtoshowthatautonomous periodi systems ouplein nitetime to a periodi regime. This result an for example be applied to the performan e evaluation of manufa -turingsystemsin whi htasks ares heduledperiodi ally.
The outline of the paperis as follows. In x2, we re all the elements of (max;+) algebra we shall use throughout the paper. In x3, (max ;+) linear time-varying systems are presented. Se tion 4 is devoted to the analysisofperiodi systems. Anappli ationto theperforman e evalua-tionof a parti ular lassof DEDS isproposedinx5.
2. PRELIMINARIES
We onsider the semi- eld (R [f 1g;;) in whi h the law is max,and is theusualaddition. We denoterespe tively"= 1 and e=0 the neutralelementsof and . The element "is absorbing for . The lawis idempotent,i.e., aa=a.
(R [f 1g;;) is an idempotent semi-ring or dioid [2℄, [5℄, and is usuallyreferredto as(max,+) algebra. We shalldenote itbyR
max . In thefollowing, we shall onsider ve tors and matri es with entries in R
max
. Theprodu tofave tor u2R n max byas alara2R max isde ned as (au) i =au i =a+u i :
The sum and produ t of matri es are de ned onventionally, repla ing +and byand ,respe tively. LetA;B 2R
nn max , (AB) ij = A ij B ij (AB) ij = n L l =1 A il B l j = max 1l n (A il +B l j ):
Thematrix-ve torprodu tisde nedinasimilarway. Mostofthetime, thesymbol''is omittedas isthe ase in onventionalalgebra.
Let us re all basi de nitions and results about the (max ;+) spe tral problem(see [2℄, [7℄ forexhaustive presentations), that is the existen e of(nonzero) eigenvalues2R max andeigenve torsv2R n max foragiven amatrixM 2R nn max ,su h thatM v=v. De nition1 A matrix M 2R nn max is irredu ible if
8i;j 9l0 su h that (M l
) ij
>":
Theorem 1 An irredu ible matrix M 2R nn max
has a unique eigenvalue denoted .
There might be several eigenve tors of an irredu ible matrix with the unique orresponding eigenvalue . A linear ombination (in R
max ) of
eigenve tors is an eigenve tor. An eigenve tor has all its oordinates di erent from ". Finally, let us re all that in R
max
every irredu ible matrixis y li inthe senseofthefollowingtheorem.
Theorem 2 Let M 2R nn max
be an irredu ible matrix whose eigenvalue is. Thereexists integers N and su h that
8mN; M m+ = M m :
Theleast value of is alled the y li ity of M.
3. TIME-VARYING (MAX,+) LINEAR
SYSTEMS
We study time-varying (max ;+) linear systems represented by equa-tions:
x(k) = A(k 1)x(k 1)B(k)u(k) (1a)
y(k) = C(k)x(k) (1b ) inwhi h fork2Z: A(k)2R nn max ,B(k)2R np max , andC(k)2R qn max ; u(k) 2 R p max (resp. x(k) 2 R n max , y(k) 2 R q max
) is alled the input (resp. state, output) ve tor.
There ursive equation(1a) an also bewritten
x(k)=(k;k 0 )x(k 0 ) k M j=k0+1 (k;j)B(j)u(j) (2) inwhi h(k;k 0
)is alledtransitionmatrix byanalogywith onventional time-varyinglinear systemstheory [8℄,and isgiven by
(k;k 0 )= 8 < : notde ned ;k 0 >k Id (identityelement ofR nn max ) ;k 0 =k A(k 1)A(k 2)A(k
0 ) ;k
0 <k
(3)
Remark1: Thetransition matrixsatis esthe ompositionproperty
k lk 0 ; (k;k 0 )=(k;l)(l;k 0 ): (4) In parti ular, for k > k 0 , we have (k;k 0 ) = A(k 1)(k 1;k 0 ), whi h shows that thetransition matrix issolution of the homogeneous state equation (Eq. (1a)withu(k)=";8k).
The input-output relationship is dedu ed from Eq. (2) with x(k 0 ) = u(k 0 )="fork 0 <0,and isgiven by y(k)= L j2Z h(k;j)u(j); with h(k;j)= C(k)(k;j)B(j) ;k j; " ;k <j;
4. ANALYSIS OF PERIODIC SYSTEMS
Inthisse tion,we de neand study(max;+) linear periodi systems by analogy with linear periodi systems over onventional algebra [3℄, [6℄, [4℄. Using basi properties and (max;+) spe tral theory, we show that autonomous periodi systems ouple in nite time to a periodi regime.
De nition2 A system represented by Eqs. (1) is said to be periodi of period K (or shortly K-periodi ) if K is the least integer su h that 8k2Z; A(k+K)=A(k); B(k+K)=B(k); C(k+K)=C(k):
Remark2: The period K of the system is equal to the least ommon multiplieroftheperiodsofentriesA(k)
ij ,B(k) ik andC(k) l j ,i=1:::n, j=1:::n, k=1:::p,l=1:::q,k 2Z.
Proposition 1 Thetransition matrix isK-periodi , i.e.,
k 0 <k ; (k+K ;k 0 +K)=(k;k 0 ) (5)
if, andonly if, A(k) isK-periodi .
Proof: Let us suppose that A(k) is K-periodi . We have 8k;k 0
< k, (k+K ;k0+K)=A(k+K):::A(k0+1+K)=A(k):::A(k0+1)=(k;k0): Conversely,theK-periodi ityofthetransitionmatrixgivesfork
0
=k 1 (k+K ;k 1+K)=(k;k 1);
whi h, a ording to the de nition of the transition matrix (Eq. (3)),
leadsto 8k; A(k+K)=A(k):
TheK-periodi ityof alsowrites k 0 <k ; 8m2Z; (k+mK ;k 0 +mK)=(k;k 0 ): Settingk =i+mK withk 0
i<K,m2N,andusingthe omposition property(4)aswellastheperiodi ity(5) of, we have:
(i+mK ;k 0 ) =(i+mK ;k 0 +mK)(k 0 +mK ;k 0 +(m 1)K):::(k 0 +K ;k 0 ) =(i;k 0 )(k 0 +K ;k 0 ):::(k 0 +K ;k 0 ) | {z } mtimes =(i;k 0 )[(k 0 +K ;k 0 )℄ m :
De nition3 Thematrix M k0
=(k 0
+K ;k 0
)is alled themonodromy matrix at k
0
(as in onventional theory [3℄).
For autonomous systems, that is systems for whi h the input is null (u(k)=",8k2Zineq. (1a), thestate ve tor obeys:
x(i+mK) = (i+mK ;k 0 )x(k 0 ) = (i;k 0 )[(k 0 +K ;k 0 )℄ m x(k 0 ) = (i;k 0 )M m k 0 x(k 0 ): (6)
In other words, the monodromy matrix des ribes the evolution of the stateoveroneperiod. Thisrelationallowsshowingthatanautonomous periodi system ouplein nitetime to aperiodi regime.
Proposition 2 If themonodromymatrix M k
0
isirredu iblewith eigen-value ,then thereexists two integers N and su h that for mN
x(k+(m+ )K)=
x(k+mK):
Proof: From equation (6), a dire t appli ation of theorem 2 leads to: x(i+(m+ )K)=(i;k 0 )M m+ k 0 x(k 0 ) = (i;k 0 ) M m k0 x(k 0 ) = (i;k 0 )M m k 0 x(k 0 ) = x(i+mK) inwhi h (resp. ) isthe eigenvalue (resp. the y li ity) ofM
k0
.
Inthisperiodi regime,thelengthofthepatternisequalto K. Inthe appli ation of se tion 5, entries of x(k) shall point out dater variables asso iatedwithaDEDS:x
i
(k)willdenotethedateofthek-tho urren e of event labeled x
i
. The ratio ( )=( K) = =K (the numeri al evaluation of
in the formula equals in onventional analysis) shall then be interpreted as the y le time (inverse of the throughput) ofthesystem;every K o urren esofevents arespa edoutof unitsof times. The following proposition, laiming that the y le time is independent of k
0
, ompletes the des riptionof this periodi regime. Remark3: Ifx(k 0 ) is aneigenve tor of M k 0 , wehave x(k 0 +K)=M k0 x(k 0 )=x(k 0 ): Fromk 0
,thestate isthenperiodi . Thepatternisshorter(equalto K), butthe y letime isstillequalto =K.
Proposition 3 Thespe trum of M k 0 =(k 0 +K ;k 0 ) isindependentof k 0 . Furthermore, if x(k 0 ) is an eigenve tor of M k0 with orresponding eigenvalue,thenx(k)=(k;k 0 )x(k 0 )isaneigenve torofM k =(k+ K ;k) with orresponding eigenvalue .
Proof: Foranypair(k 0 ; 0 )with 0 +Kk 0 0
,themonodromymatri es at k 0 and at 0 an respe tivelybe written (k 0 +K ;k 0 )=(k 0 +K ; 0 +K)( 0 +K ;k 0 )=(k 0 ; 0 )( 0 +K ;k 0 ) ( 0 +K ; 0 )=( 0 +K ;k 0 )(k 0 ; 0 )
Inother words,themonodromymatri es an beexpressed intheforms (k 0 +K ;k 0 )=FG and ( 0 +K ; 0 )=GF. If is a nonzero eigenvalue of (k 0 +K ;k 0 ), i.e., FGx = x, x 6= ", thenGFGx=Gx=Gx, orGFy=y,withy =Gx.
Sin e 6= " and x 6= ", y = Gx 6= "; so that is an eigenvalue of (
0
+K ; 0
Byreversingtheroleof(k 0 +K ;k 0 )and( 0 +K ; 0
)intheabove ar-gument, it onverselyfollows thatallthenonzero eigenvaluesof(
0 + K ; 0 )are eigenvaluesof (k 0 +K ;k 0 ). Letusassumethatx(k
0 ) isan eigenve torofM k 0 with orresponding eigenvalue . We have M k x(k)=(k+K ;k)x(k) = (k+K ;k)(k;k 0 )x(k 0 ) = (k+K ;k 0 )x(k 0 ) = (k+K ;k 0 +K)(k 0 +K ;k 0 )x(k 0 ) = (k;k 0 )x(k 0 ) = x(k)
whi hshowsthatx(k)isaneigenve torofM k
with orresponding
eigen-value .
5. APPLICATION TO THE PERFORMANCE
EVALUATION OF DEDS
In this se tion, we apply the pre eding results to DEDS. More pre- isely, we de ne a lass of Timed Petri Nets suitable to model time-varying (max;+) linear systems (introdu ed in [10℄). Essentially,those areTimedEventGraphs(TEGs,Petrinetforwhi hea hpla ehasonly one inputar and one output ar )whose holdingtimes asso iatedwith pla es are variable. We give their representation in (max;+) algebra. When sequen es of holding times are periodi , the representation is a state modelwith periodi allyvarying oeÆ ients.
5.1. FIFO TEGS
WedenotebyP (respe tively,Q)the nitesetofpla es(respe tively, transitions)ofa TEG,and M
p
2N thenumberoftokensbeinginitially in pla e p 2 P; p
(respe tively,
p) refers to the output transition (respe tively, inputtransition) of p. We de ne similarlythe sets q
,
q as the set of output pla es, and the set of input pla es, of transition q2Q.
We allholding time theminimumamount oftime tokenshave to stay in a pla e (without loss of modeling power, the ringof transitions is supposed to be instantaneous): the k-th token in pla e p in urs the holdingtimedenoted
p (k).
De nition4 We de ne the earliest First-In-First-Out (FIFO) fun -tioningrule of a TEG as follows.
1 A transition q res as soon as ea h pla e upstream q ontains at least one available token.
2 We denote q(n) the date of n-th ringof transition q. This ring onsumes one token in ea h upstream pla e and produ es one to-ken in ea h downstream pla e. A token added in pla e p 2 q
at time q(n)is indexed k withk =n+M
p
and be omes availablefor transition p
from instant max 1in fq(i)+ p (i+M p )g.
ATEGfun tioning underthisruleis alleda FIFO TEG.
The only originality in this de nition on erns the token's availability. Indeed,the token indexed k inpla e p isusually saidto be availableas soonasits holdingtime
p
(k) is over [1℄.
The reason why we have de ned a new fun tioning rule is that TEGs anbemodeledbylinearequationsin(max;+)algebraonlyiftokensdo notovertake oneanotherwhentraversingpla es[1℄,[2, h. 2℄. Previous studies have onsequently onsidered onditions on sequen es of hold-ingtimes( onstantornon-de reasingholdingtimesforexample)and/or stru tural onditions (preventing several tokens to be simultaneously present in a pla e with variable holding time). The above fun tioning ruleensures withoutstru tural onditions that pla es operate asFIFO hannels foranysequen es ofholding times,and willnotably enableto easily model mixed-model assembly lines (whi h are intrinsi ally over-take free) on whi h several parts an be simultaneously handled at a same"station" orma hine. Let us onsiderforexampletheautomobile produ tion linepartly shown in gure 1.(a) where ars are handledby a linear a umulative onveyor rossing su essive working areas sepa-rated by bu er zones. In ea h working area, a xed number of ars
x 2 bu er zone (a) working area (b) - - p 2 () - -p 2 x 1 x 3 - -p 1 R R i i
Figure1 (a)Aportionofanautomobileprodu tionline,(b)aFIFOTEG.
an be pro essed simultaneously(two in therepresented working area) andbu erzoneshave limited apa ities(onlyone ar an besto ked in therepresented bu erzone). Cars annot overtake one another on the onveyor; bu erzones and working areas work as FIFO hannels. The evolutionof arsinthe onsideredportionoftheline an bemodeledby theFIFOTEGof gure1.(b). Atokeninpla e p
1
(resp. p 2
a ar sto ked in the bu er zone (resp. being pro essed in the working area). Pro essing times inthe working areaare equal to
p 2
(), and for sakeofsimpli ity,travellingtimeinthebu erzoneaswellassetuptimes areassumed to be null. Finally,let usnote thatthissystem would not be modeled byaTEG ifthe lassi alfun tioningrulewasused.
5.2. REPRESENTATION OF FIFO TEGS
Withea htransitionq 2Qweasso iateadater variable alsodenoted q: q(k)denotesthedateofthek-th ringoftransition q. Thesequen es of holdingtimes
p
(k), p2P, k 2Zareassumed to be given nonnega-tive and niteintegers.
Assertion 1 The dater variables of a FIFO TEG satisfy the following equation: q(k)= M fp2 qjq 0 = pg M ik p (i)q 0 (i M p ) ;k2Z; or equivalently, q(k)=q(k 1) M fp2 qjq 0 = pg p (k)q 0 (k M p ) ; k2Z:
The methodology to obtain a state representation for FIFO TEGs in (max;+) algebra is the same one as for TEGs fun tioning with the lassi al rule [2, hap. 2℄. One an partition the set of transitions Q = U [X [Y where U is the set of transitions with no prede essors (inputtransitions),Y isthesetoftransitionswithnosu essors(output transitions),and X =Qn(U[Y) (state transitions). We denote byu (respe tivelyx,y)theve torofinput(respe tively,state,output)daters q,q2U (respe tively,X,Y). One anobtainafterseveral ombinatorial manipulationsthestandard state model given byEqs. (1)(see [10℄). Example1: We onsidertheFIFOTEGof gure1.(b)(non-dotted part ofthegraph) whi hmaypartlyrepresent aworkingarea ofan automo-bileprodu tion line, as des ribed inse tion 5.1. This graph has ex lu-sivelystate transitions,its dynami behavior an berepresentedbythe followingequation: x(k)= 0 x1(k) x2(k) x3(k) 1 A = 0 " " " e " " " " " 1 A 0 x1(k) x2(k) x3(k) 1 A 0 e e " " e e " p 2 (k) e 1 A 0 x1(k 1) x2(k 1) x3(k 1) 1 A :
Thisequ. an bewritteninan expli itform (see[2, th. 2.66, p. 79℄):
x(k)= 0 x 1 (k) x2(k) x 3 (k) 1 A = 0 e e " e e e " p 2 (k) e 1 A 0 x 1 (k 1) x2(k 1) x 3 (k 1) 1 A .
AFIFOTEGrepresentedbyEqs. (1)willbeseenasaperiodi system if, and only if, all the entries of matri es A(k), B(k) and C(k) ( orre-sponding to the holding times asso iated with pla es of thegraph) are periodi . Theperiodi ityofthesystem,denotedK,isthenequaltothe lowest ommon multiplierof theperiodi ities of the sequen es of hold-ing times. In the manufa turing system des ribed above, sequen es of holdingtimesrepresentthepro essingtimesof ars rossingtheworking area. Su h asystem isthenperiodi ifforexamplethesu essive types of ars releasedontheprodu tionlineareorderedinaperiodi manner. In su h a periodi setting, the reasoning of se tion 4 an be used to show that daters asso iated with transitions of the FIFO TEG rea h a periodi regime, and to assess the y le time of these daters whi h orrespond to theaverage timebetweentwo su essive rings.
Example2: Letus onsideragaintheportionoftheautomobile produ -tionlinemodeled inexample1.
Weassumethatthreetypesof ars,denotedR 1 ,R 2 andR 3 ,arehandled inthisline. Thepro essingtimeforR
1 (resp. R 2 ,R 3 )inthe onsidered working areais equalto 3(resp. 2,1) unitsof times. The s heduling is supposedtobea y li permutationof thedi erent typesof ars. More pre isely,the su essive typesof ars released inthe lineare : R
1 , R 2 , R 3 ,R 1 ,R 2 ,R 3 ;::: We then have8j 2Z; p2 (k 0 +3j)=3; p2 (k 0 +3j+1)=2; p2 (k 0 +3j+2)=1. The system is 3-periodi sin e A(k +3) = A(k), 8k 2 Z. The mon-odromymatrixat k 0 isequalto M k0 =(k 0 +3;k 0 )= 0 0 3 0 2 3 2 2 4 2 1 A : M k 0
is irredu ible (see de nition 1). Its unique eigenvalue is equal to 3 and its y li ity is equal to 1. The state of the systems rea hes a periodi regime in nite time. The length of the pattern is equal to K = 1 3 = 3 and the y le time of the system is equal to =K=3=3=1.
Remark4: Repetitivemanufa turingsystemshavepreviouslybeen stud-ied in [9℄. Nevertheless, the approa h presented in this referen e does notallow onsideringsystems whereseveralparts an behandled simul-taneouslyon asame ma hineasinthe onsideredexample.
6. CONCLUSION
Wehaveta kledthestudyof(max;+)linearsystemswithperiodi ally varying parameters. We have given basi properties, and in parti ular we have shown that su h autonomous systems rea h a periodi regime
in nitetime. Thisresult an beusedfortheperforman eevaluation of parti ularDEDS.
Wethinkthatfurtherresultson onventionalperiodi systems ouldbe adapted to the onsidered DEDS.In parti ular, forthe analysis of dis- retetimeperiodi systems,itisoftenusefultoresorttoatime-invariant reformulation[6℄,[4℄. Inasimilarway,(max ;+)linearperiodi systems admit a time-invariant reformulation whi h ould extend the study of these systems.
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