Max-plus algebra in the history of discrete event systems

ContentslistsavailableatScienceDirect

Annual Reviews in Control

journalhomepage:www.elsevier.com/locate/arcontrol

Tutorial Article

Max-plus algebra in the history of discrete event systems ^R

J. Komenda

^a,∗

, S. Lahaye

^b

, J.-L. Boimond

^b

, T. van den Boom

^c

aInstitute of Mathematics, Brno Branch, Czech Academy of Sciences, Prague, Czech Republic, Argentina

bLARIS, Angers University, Angers, France

cDelft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands

a rt i c l e i n f o

Article history:

Received 12 January 2018 Revised 21 March 2018 Accepted 3 April 2018 Available online xxx Keywords:

Discrete-event system Max-plus algebra History

a b s t r a c t

Thispaperisasurveyofthehistoryofmax-plusalgebraanditsroleinthefieldofdiscreteeventsystems duringthelastthreedecades.Itisbasedontheperspectiveoftheauthorsbutitcoversalargevariety oftopics,wheremax-plusalgebraplaysakeyrole.

© 2018ElsevierLtd.Allrightsreserved.

Contents

1. Emergenceofmax-plusapproach.Asystemtheorytailoredforsynchronization ... 1

2. SomeextensionsfocusedonsynchronizationinDES... 2

3. TimedDESwithsharedresources... 3

4. Max-plus-algebraandtheoreticalcomputerscience... 4

4.1. Max-plusautomataandpropertiesoftheirseries... 4

4.2. Bisimulationproperties... 5

4.3. Supervisorycontrol... 5

5. Max-plusplanningandmodelpredictivecontrol... 6

6. Max-plusandmin-plusgeometry... 6

7. Applications ... 8

References ... 8

1. Emergenceofmax-plusapproach.Asystemtheorytailored forsynchronization

Thispapersummarizesthehistoryofmax-plusalgebra within the field ofdiscrete eventsystems. Itis based onbriefsurvey of the role of max-plus algebra in the field of discrete event sys- tems that appeared in Komenda, Lahaye, Boimond, and van den Boom(2017),butextendedinseveraldirections.Inparticular,there is a section, where computational aspects are discussed together withresultsaboutmax-plusalgebrafromthecomputersciencelit- erature.

Theemergenceofasystemtheoryforclassesofdiscreteevent systems (DES), in which max-plus algebra and similar algebraic

R The research was supported by GA ˇCR grant S15-2532 and by RVO 67985840.

∗ Corresponding author.

E-mail address: komenda@ipm.cz (J. Komenda).

toolsplaya centralrole, datesfromthe early1980’s.We empha- sizethat the idempotentsemiring(also calleddioid)ofextended realnumbers(R∪

{

−∞

}

,max,+)^{iscommonlycalledmax-plusal-} gebra,whileitisnotformallyanalgebrainthestrictlymathematic sense.

Its inspirationstems certainly fromthe following observation:

synchronization, whichis a verynon smooth andnonlinear phe- nomenonwithregardto“usual” systemtheory,canbemodeledby linearequationsinparticularalgebraicstructuressuchasmax-plus algebraandotheridempotentsemiringstructures(Cohen,Dubois, Quadrat,&Viot,1983;Cuninghame-Green,1979).

Twoimportantfeaturescharacterizethisapproachoftencalled max-pluslinearsystemtheory:

• mostofthecontributions haveusedasaguidelinethe“classi- cal” linearsystemtheory;

https://doi.org/10.1016/j.arcontrol.2018.04.004 1367-5788/© 2018 Elsevier Ltd. All rights reserved.

Pleasecitethisarticleas:J.Komendaetal.,Max-plusalgebrainthehistoryofdiscreteeventsystems,AnnualReviewsinControl(2018),

Fig. 1. A timed event graph.

• it is turned towards DES performance related issues (as op- posed to logical aspects considered in other approaches such asautomata and formal language theory)by including timing aspectsinDESdescription.

A considerationhassignificantly contributedto thepromotion and the scope definition of the approach: a class of ordinary¹ Petrinets,namely thetimedeventgraphs (TEGs) hasbeeniden- tifiedto capturethe class ofstationary² max-plus linearsystems (Cohen, Dubois, Quadrat, & Viot, 1985) and subsequent publica- tionsbyMaxPlusteam.³TEGsaretimedPetrinetsinwhicheach placehasa singleinput transitionandasingleoutput transition.

Asingleoutputtransitionmeansthatnoconflictisconsideredfor the tokens consumption in the place,in other words, the atten- tionisrestrictedtoDESinwhichall potentialconflictshavebeen solved by some predefined policy. Symmetrically, a single input transitionimpliesthatthereisnocompetitioninsupplyingtokens inthe place.Intheend,mostlysynchronizationphenomena(cor- respondingto theconfiguration inwhich atransitionhas several inputplacesand/or severaloutputplaces)canbe considered,and thisisthepricetopayforlinearity.

Example1. Fig.1depictsa TEG,thatisaPetrinetinwhicheach place (represented by a circle) has exactly one input transition (representedbya rectangle)andone outputtransition. Thenum- bernextto aplaceindicatesthe sojourntimeforatoken,that is thenumberofunitsoftimethatmustelapsebeforethetokenbe- comesavailableforthefiringoftheoutputtransition.Letu(k)de- notethedateofthek^thfiringoftransitionu(samenotationforx₁, x₂andy).Consideringtheearliestfiringrule(atransitionisfiredas soonasthereis an availabletoken ineach input place),we have thefollowingevolutionequations

x1

(

^k

)

=max

(

²+u

(

^k

)

,1+x2

(

^k−1

))

x2

(

^k

)

=3+x1

(

^k−1

)

y

(

^k

)

=1+x2

(

^k

)

.

Denoting(resp.)theadditioncorrespondingtomax operation (resp.the multiplicationcorresponding tousual addition),we ob- tainlinearequationsinmax-plusalgebra,thatis:

x1

(

^k

)

=2u

(

^k

)

1x2

(

^k−1

)

x₂

(

^k

)

⁼³x₁

(

^k−1

)

y

(

^k

)

=1x₂

(

^k

)

Rewritingtheresultingequationsinmax-plus-algebraicmatrixno- tationleadstoastate-spacerepresentation:

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

x1

(

^k

)

x₂

(

^k

)

=

ε

¹

3

ε

x1

(

^k−1

)

x₂

(

^k−1

)

2

ε

u

(

^k

)

y

(

^k

)

=

ε

¹

x1

(

^k

)

x2

(

^k

)

where

ε

^isequalto−∞.

1Petri nets in which all arc weights are 1.

2Stationarity is defined conventionally but over operators of max-plus algebra.

3Max Plus is a collective name for a working group on max-plus algebra, at INRIA Rocquencourt, comprising: Marianne Akian, Guy Cohen, Stéphane Gaubert, Jean-Pierre Quadrat and Michel Viot.

This new area of linear system theory has benefited from existing mathematical tools related to idempotent algebras such as lattice theory (Birkhoff, 1940), residuation theory (Blyth &

Janowitz,1972),graphtheory(Gondran&Minoux,1979),optimiza- tion(Zimmermann,1981)andidempotentanalysis(Kolokoltsov &

Maslov, 1997), howeverit is worth mentioning that the progress has probably been impeded by the fact that some fundamental mathematicalissuesinthisareaarestillopen.

The overview ofthe contributions reveals that main concepts from linear systemtheory havebeen step by step specified into max-pluslinearsystemtheory.Withoutaimingtobeexhaustive:

• severalpossiblerepresentationshavebeenstudied,namelystate- spaceequations,transferfunctionineventdomain(Cohenetal., 1983; 1985), time domain (Caspi & Halbwachs, 1986), and two-dimensional domain using series in two formal variables (Cohen, Moller, Quadrat, & Viot, 1986) (with more details in Cohen,Moller,Quadrat,&Viot,1989);

• performanceanalysisandstabilityaremostlybasedontheinter- pretationoftheeigenvalueofthestate-matrixintermsofcycle- time,withitsassociatedeigenspaceandrelatedcyclicityprop- erty(Baccelli,Cohen,Olsder,&Quadrat,1992;Gaubert,1997);

• awiderangeofcontrollawshavebeenadaptedsuchas:

– open-loop structures overcoming system output tracking (Baccellietal., 1992,chap.5.6),(Cofer& Garg,1996;Men- guy,Boimond,Hardouin,&Ferrier,2000)ormodelreference tracking(Libeaut&Loiseau,1996),

– closed-loopstructures takingintoaccountdisturbancesand model-systemmismatches(Cottenceau,Hardouin,Boimond, Ferrier etal., 1999; Lüders& Santos-Mendes, 2002), possi- bly includingastate-observer (Hardouin,Maia, Cottenceau,

&Lhommeau,2010),

– model predictive control scheme (De Schutter & van den Boom,2001;vandenBoom&DeSchutter,2002)withem- phasis onstabilityin Necoara, DeSchutter, vandenBoom, andHellendoorn(2007).

Fora large survey on max-plus linearsystems theory, we re- fer to books (Baccelli et al., 1992; Butkoviˇc, 2010; Gunawardena, 1998; Heidergott,Olsder,& vanderWoude, 2006), tomanuscript (Gaubert, 1992) andsurveys(Akian, Bapat, & Gaubert, 2003; Co- hen,Gaubert,&Quadrat,1999;Cohenetal.,1989;Gaubert,1997).

2. SomeextensionsfocusedonsynchronizationinDES

There is an important connection between min-max-plus systems, in which time evolution depend on both max and min, but also addition operation and the game theory. It goes back to Olsder (1991), where spectral properties of such sys- tems are studied. More recent references on this topic are Gunawardena (2003) and Akian, Gaubert, and Guterman (2012). The latter work establishes an equivalence with mean payoff games, an important open complexity problem in computer sci- ence, andit seems manyverificationproblems formax-plus sys- tems reduce to meanpayoff games. We mention that manythe- oreticalworkson max-plusalgebraandmax-plussystems donot make useof thewords “max-plus” butrather “tropical”. The ad- jectivetropical wasinvented byFrench mathematicians, inhonor oftheBrazilianmathematicianandcomputerscientistImreSimon (1943–2009).

A natural generalization of deterministic max-plus-linear sys- tems are stochastic max-plus-linear systems, which have been studied for more than two decades. Ergodic theory of stochastic timed eventgraphs is developed in Baccelli et al. (1992), where most of the theory is covered. In particular, asymptotic proper- ties of stochastic max-plus-linear systems are studied therein in termsoftheso-calledLyapunovexponentsthatcorrespondtothe

asymptotic meanvalue ofthe normof thestate variables. Inthe case the underlying event graph is strongly connected the Lya- punovexponentisthe uniquevalue towhichthe meanvalue al- mostsurelyconverges.Forgeneraleventgraphsthereisamaximal Lyapunovexponent.

Uncertainty can also be considered through intervals defin- ing the possible values for parameters of the system. In Lhommeau, Hardouin, Cottenceau, and Jaulin (2004) TEGs, in which thenumber ofinitial tokens andthe time delaysare only known to belong tointervals, are represented overa semiringof intervalsandrobustcontrollersaredesigned.

Another way ofextending techniques for linear systems is to consider that parameters of the models may vary, that is study non-stationarylinearsystems.This possibilityhasbeenexamined withinthemax-pluslinearsetting(Brat&Garg,1998;Lahaye,Boi- mond,&Hardouin,1999;2004)withcontributionsmainlyfocused onrepresentation,controlandperformanceanalysis.

Continuous TEGs inwhich fluidsholdrather thandiscrete to- kens and the fluid flow through transitions can be limited to a maximum value. Moreover, an initial volume offluid can be de- fined inplacesandtimescanbe associatedwithplacestomodel fluid transportation times(MaxPlus,1991). Such graphs are rele- vantforexampletoapproximatethebehaviorofhighthroughput manufacturingsystemsinwhichthenumberofprocessedpartsis very large.In parallel,a similar approachcallednetwork calculus hasbeendeveloped byconsidering computernetwork trafficasa flow(basedontheuseof‘leakybuckets’)toapproximatethehigh numberofconveyedpackets(Cruz,1991;LeBoudec,2001).Exten- siononfluidtimedeventgraphswithmultipliersinanewalgebra, analogous to the min-plus algebra, has beenproposed in Cohen, Gaubert,andQuadrat(1995,1998).

Switching Max-Plus-Linear (SMPL) systems are discrete-event systems that can switch between different modes of operation (van den Boom & De Schutter, 2006). The switching allows to changethestructureofthesystem,tobreaksynchronization,orto changetheorderofevents.Ineachmodethesystemisdescribed by amax-plus-linearstateequation andamax-plus-linearoutput equation.Notethatregularmax-plus-linearsystemsareasubclass ofSMPLsystem,namelywithonlyonemode.InvandenBoomand De Schutter (2011), authors describe the commutation between differentmax-plus-linear modesandshowsthat an SMPLsystem can be writtenasa piecewiseaffine systemwhichallows forus- ing similar techniques insuch seemingly differentclasses ofsys- tems.IncyclicDEStheoperationsappearinacyclicway.Afterall operations in a systemhave been completed, the cycleis closed andanewcyclebegins.Inthe caseofchanges inoperationsand resources per cycle the system is called semi-cyclic. SMPL mod- els can be used to describe the dynamics of various semi-cyclic DES.

Another extension of the class of systems that can be mod- eledinmax-plusalgebrasconsistsinconsideringhybridPetrinets, andmore particularly,hybridTEGs that consist ofa discretepart (a TEG) and a continuous part (a continuous TEG). It has been showninKomenda,ElMoudni,andZerhouni(2001) thatalinear modelcanbeobtainedbasedoncounterfunctionifonlyonetype of theinterface between continuousanddiscrete partis present.

However,forapplicationtojustintimecontrolthisconstraintcan be relaxed as it has been shown in Hamaci, Boimond, and La- haye(2006).

Theweighted⁴ TEGsmakeitpossibletodescribe batchingand duplication(unbatching)phenomena.InCottenceau,Hardouin,and Trunk (2017),authors show thatsuch graphs canbe also linearly modeledbytransferseriesinaparticularmax-plusalgebra.Inad-

4Arcs weights can be any positive integers.

ditiontoeventandtime shifts,two additionaloperatorsare used todescribethebatching/unbatchingoperations.

P-time Petri nets form an important extension of Petri nets, wherethetiming ofplaces/transitionsisnondeterministic. P-time EventGraphshavebeenstudiedinmax-plus algebrasinDeclerck andAlaoui(2004,2005).Theyfindtheirapplicationse.g.inmodel- ingofelectroplatinglinesorchemicalprocesses,wherebothupper andlowerboundconstraintsprocessingtimearerequired,seee.g.

Spacek,Manier,andMoudni(1999).

3. TimedDESwithsharedresources

A major issue withapplication of max-plus linear systems to modelingoftimedDES(importantamongothersinmanufacturing systems orin computer andcommunication networks) is that it appearsdifficult tomodel resourcesharing within TEGsthat cor- respondtostationarymax-pluslinearsystems.Inrealmanufactur- ingsystems,however,therearetypically severalprocesses(tasks) that share (and compete for) given resources such as robots in manufacturing systems or memory in computer systems. In the max-plus systems literature various resource allocations policies havebeenproposedtointegrateconflictresolutionwithothertyp- icalphenomenaoftimedDES,namelysynchronizationandparal- lelism. For instance, within timedPetri nets resource allocations policieshasbeenstudied basedonthe dualcounter functionde- scriptionintheidempotentsemiringmin-plus(Cohen,Gaubert,&

Quadrat,1997).GeneralPetri netmodelshavebeenaddressedby preselectionrulesinCohenetal.(1995),whichenablestodescribe theirevolutionusingmax-plusdynamics.Moregeneralmonotone homogeneous dynamics, relevant to free choice Petri nets, and theiroptimalroutingisstudiedinGaujalandGiua(2004).

More recently, conflicts among several TEGs have been stud- ied in Addad, Amari, and Lesage (2012), where conflicting TEGs (CTEG) have been proposed with some fairly restrictive assump- tions.It shouldbe statedthatresource allocationpolicies studied inAddadetal.(2012)are eitherFIFOorcyclic(periodic)policies.

Onone handtheperformance analysis(computation ofan upper boundonthecycletimeofCTEG)hasbeenproposedanditisde- pendentonthecycletimeofindividualTEGsandontimingofthe conflictplaces.Ontheotherhand,theapproachhasnotyetbeen appliedtocontrolproblems.

Unlikethe approach based on TEGs, where different resource allocation policies are handled one by one, there exists a max- plusautomatabasedapproachthatallowssimultaneousmodeling ofdifferent resource allocation policies within a singlemodel as long asthese policies can be representedby a regular language.

Anautomaton-basedmodel itcan handleseveralsuch policiesat thesametimewithinasinglemodelwithouthavingtorebuildthe modeleachtimethepolicyischanged.

The framework ofmax-plus automata has enableda deep in- vestigationofperformanceevaluation(Gaubert,1995)ofDESwith shared resources. Max-plus automata can be viewed asa rather specialclass ofautomatamodelsenriched withtime,becauseun- like timed automata they do not time non determinism, where both lower and upper bounds on timing of events can be de- fined. However, they have strong expressive power in terms of timed Petri nets as shown in Gaubert and Mairesse (1999) and Lahaye,Komenda, and Boimond(2015a). In particular, every safe timedPetri net can be representedby specialmax-plus automa- ton,calledheapmodel.

Example2. AsafetimedPetrinetisdepictedonFig.2.Thenum- ber next to the label of a transition specifies its firing duration, thatistheminimal timethatmustelapse,startingfromthetime atwhichitisenabled,until thetransitioncanfire. Alltheplaces areassumedtohaveasojourndurationequalto0unitoftime.Its

Fig. 2. A timed Petri net.

Fig. 3. Heap model and associated (max,+) automaton to represent the timed Petri net of Fig. 2 .

behaviorcanbedescribedbytheheapmodelontheleft-handside ofFig.3.Infewwords,aheapmodeliscomposedof

• slots which correspond to resources (tokens in the Petri net) andoneslot isassociatedtoeach place(possiblycontaininga token),

• pieces which represent activities (firings of transitions in the Petrinet)andonepieceisassociatedtoeachtransition.

Theactivitiesrequireresources(transitionsconsumetoken(s)to befiredinthePetrinetandpiecesoccupyslotsintheheapmodel) duringpredefined durations(transitionsfiringdurationsrendered byspecificheightsofpiecesinslots). Thedynamicsisthen mod- eledby thesequence ofpieces(transitions firingsequenceinthe Petrinet)pillingupaccordingtotheTetrisgamemechanism.Itcan beshownthattheheightofheapsofpiecesisrecognizedbyapar- ticular(max,+)automaton (especiallyusefulforalgebraic compu- tations).The(max,+)automatonderivedfromtheexampleofheap modelisdepicted⁵ontheright-handsideofFig.3.

S. Gaubert has presented several important analysis results in Gaubert (1995), where the worst case, the optimal case, and themean case performance of max-plus automata are examined in detail. Better results are naturally obtained for deterministic max-plusautomata, butmost ofthe paperis focused on general nondeterministicmax-plus automata. We emphasize that not all max-plus automata can be determinized and their determiniza- tion,i.e.existence of a deterministic max-plus automatonhaving thesamebehavior(recognizingthesameformal powerseries), is stillanopenproblemanditisnotevenknownifdeterminization ofa givennondeterministic max-plus automaton is decidable.In Gaubert(1995)asufficientconditionintermsofprojectivelyfinite semigroupsfordeterminizationofmax-plusautomataisprovided, whichcanbe usedasasemi-algorithm fordeterminization(with

5The graphical representation of a (max,+) automaton is such that: nodes cor- respond to states, an arrow from a state to another with label a / n denotes a state transition requiring n units of time before event a can occur, an input arrow sym- bolizes an initial state.

no guarantee of success). Several other works on determiniza- tion ofmax-plus automatahave appeared later. Mohrideveloped a semi-algorithm for determinization of max-plus automata in Mohri(1997) with avery successful applicationinspeech recog- nition.It isknown that thetwins property,generalizedto clones propertyforpolynomiallyambiguousinKirsten (2008),isasuffi- cientconditionfortheterminationofthisalgorithm.InKirstenand Lombardy (2009) the authors propose an algorithm for deciding unambiguity and sequentiality of polynomially ambiguous min- plus automata, which leaves the unambiguity and sequentiality problemopenonly fornon polynomiallyambiguousclass ofboth max-plusandmin-plusautomata.

4. Max-plus-algebraandtheoreticalcomputerscience

Inthissectionwe willaddressalgorithms inmax-plus-algebra fromacomputerscienceperspective,whereaspecialemphasisis put on complexity issues. Time and space complexities of algo- rithms are important in the whole theory of discrete-event sys- tems,whichincludesamongotherssupervisorycontrol,stochastic discrete systems (Markov Chains), and timeddiscrete event sys- tems.Max-plusalgebrafindsits applicationsmainly intimeddis- creteeventsystems.Althoughtheunderlyingsystemmodelsvary fromdeterministic-timemodelssuchastimedeventgraphstonon deterministic-timemodelssuchastimeeventgraphs,themainop- erationsused in equationsdescribing the evolution of thesesys- tems are the rational matrix operations: sum, product, and the Kleene star. Matrixmultiplication, includingthe one inthe max- plus-algebra is well known to have the worst case complexity O(n³) for square matriceswith n lines andcolumns. This rather naiveboundcanbeimproved,whichisamajortopicofresearchin algebraiccomplexitytheory,wherealgorithmsformatrixmultipli- cationandinversioninn^ω with

ω

≤2.373areknownformatrices overfield.ThecomplexityofthematrixproductandKleenestarin thetropicalsettingisamajoropenproblem, seeWilliams(2014). The important aspect is, however,that all rationaloperations on matrices (including the Kleene star, which reduces to the finite sumof thefirst n+1max-plus-powers)are of polynomial worst case complexity. The same complexity resultobviously holds for matrix residuationusedin solutions tovarious control problems, because matrix residuation can be viewed as a dual multiplica- tion with the so called conjugate matrix (Cuninghame-Green &

Butkovic,2008). This isa very good news,because control prob- lemsformax-plus-linearsystemslistedinSection 1arebasedon matrixmultiplicationandresiduation,i.e.canbesolvedinpolyno- mialtime.

Similarly,themax-plusandmin-plusoperationsonvectorsare ofpolynomial time complexity.We recall that min-plus convolu- tion oftwo vectorsplays acentral role inthe dynamicprogram- ming.Since theearly 1960s,itisknownthat itcan becomputed inO(n²) time. Thisbound can be furtherimproved insome spe- cial cases, e.g. for two convex sequences it can be computed in O(n) time (Brenier, 1989; Lucet, 1996) by a simple merge (the Minkowskisum) oftwo convexpolygons (Rockafellar,1970). This specialcaseisalreadyusedinimageprocessingandcomputervi- sion.

However, the above discussion mainly applies to approaches coveredinSections1and2.Oneshould bearinmindthatchoice phenomena (i.e. resource sharings) are then excluded and the modelscorrespondtorecurrentequationsonnaturalnumbersthat countevents,withoutmakingdistinction betweenthem.Asmen- tioned in Section 3, some DES including choice phenomena can be seen as max-plus linear systems, e.g. max-plus automata, if oneuses recurrentequation onwordsreflectingsequences ofdif- ferentiated events.The complexity picture is then very different,

because we will recallbelow that manyfundamental verification problemsarealreadyundecidable.

4.1. Max-plusautomataandpropertiesoftheirseries

Max-plus automata have been introduced by S. Gaubert in Gaubert(1995)asageneralizationofbothmax-plus-linearsystems and standard Boolean automata. Max-plus automata as weighted automata with weights (sometimes called multiplicities) in the max-plus semiringhave also been studied by the computer sci- ence community. The basic reference on the theory of automata of multiplicities developed by Eilenberg and Schutzenberger is Eilenberg(1974).Ithasbeenunderstoodin1990sthatseveralfun- damentalproblemsundecidableforgeneraltimedsystems,suchas timed automata, are alreadyundecidable formax-plus automata.

Inparticular,ithasbeenshowninKrob(1992)thatequalitiesand inequalitiesofrationalmax-plusformalpowerseriesareundecid- able. Sinceit iswell known thatrational max-plusformal power seriesarebehaviors(weightedlanguages)offinite(state)max-plus automata,theresultofD.Krobmeansthat,ingeneral,itisnotal- gorithmicallypossibletocomparethebehaviorsoffinitemax-plus automata.Someotherverificationproblemsaredecidableformax- plusautomata.Forinstance,itcanbedecided inpolynomial time (namely O(n³) withn thesize of the state set ofthe recognizer, seeLombardy &Mairesse,2006) ifa rationalmax-pluspowerse- ries has all coefficients non positive, i.e.

^S,w

≤0 forall w∈A^∗. This means that existence of w∈A^∗ with

^{S, w}

>0 can be de- cided in polynomial time aswell. On the other hand, the prob- lem

^S,w

≥0 forall w∈A^∗ is undecidable, cf.(Krob, 1992). It is also knownthat equality toa constant is decidable,i.e.for ara- tionalmax-plus powerseriesit canbe decidedifforall w∈A^∗it holds that

^S,w

=c forsome realconstant c,see Lombardy and Mairesse(2006).Intheliteratureoneencountersalsomin-plusau- tomata, which are weighted automata with weight in the (dual) min-plus semiring.Theyare alsonondeterministic:minimum,in- steadofmaximum,ofweightsofpathsthatsharedalabelistaken forcomputationof thecorresponding min-plus formalseries.For min-plus automata dual decidabilityresults holdmeaning that it is decidableinO(n³) tocheck if

^S,w

≥0forall w∈A^∗,while it is undecidable to check if

^S,w

≤0 forall w∈A^∗. Veryinterest- ingarecomplexityresultsconcerningthecomparisonsofmax-plus andmin-plusseries.ItisknownthatinequalityS(w)≤S(w)forall w∈A^∗canbedecidedifSisamax-plusrationalformalpowerse- riesandSisamin-plusrationalseries,buttheoppositeinequality isthen undecidable!InLombardy andMairesse(2006)theseries whichare recognizedboth bya finitemax-plusandafinitemin- plusautomaton,i.e.seriesatthesametimemax-plusandmin-plus rational, have been characterized. It has been shown that these series are preciselythe unambiguous max-plus (equivalently, un- ambiguousmin-plus)series.Werecallthatunambiguousmax-plus series are those recognized by unambiguous max-plus automata:

foreverywordw,thereisatmostonesuccessfulpathlabeledby w.Notethatinvertingthecoefficientofarationalmax-plusseries, i.e.multiplying allits coefficientsby −1doesnot yieldarational max-plus series,but rathera rational min-plus series.This helps understandingtheabovediscussedasymmetriesinthefundamen- taldecisionproblemsdiscussedabove.

4.2. Bisimulationproperties

Itisquitedisappointingthatseveralfundamentalproblemsare undecidable for max-plus automata. We point out that recently there arealsomoreoptimistic complexityresultsaboutmax-plus automata. The well known concept of bisimulation, which cap- tures behavioral equivalence of nondeterministic transition sys- tems,hasbeenintroducedformax-plusautomatainBuchholzand

Kemper (2003). It is a stronger property that equality of formal powerseries,butmayserve asapartialremedyto undecidability ofinequalitiesandequalitiesbetweenformal powerseries.Bisim- ulation between two max-plus automata means that the related (equivalent) states match each other’s transitions (not only from thelogicalviewpoint:existenceoftransitions,butalsofromquan- titativeviewpoint:theweightsoftwomatchingtransitionsshould be identical). Algebraic approach to the investigation of bisimu- lation relations encoded as Boolean matrices has been adopted inDamljanovi´c, ´Ciri´c,andIgnjatovi´c(2014),wherebisimulationis characterizedbymax-plus-linearmatrixinequalities(tobedistin- guishedfromMLI’sinclassical controltheory)anda fix-pointal- gorithm witha polynomial complexityfor algebraiccomputation oflargestbisimulationshasbeenproposed.

Inconcurrency theorythereisa conceptofweakbisimulation, whichweakens thebisimulation by notrequiring internal (exter- nallyinvisible)transitionstobe preserved.InBuchholzandKem- per(2003) theconcept ofprojectedmax-plus automatonhasap- pearedfirst (although itis not explicitly named so). The authors ofthat paperdefine weakbisimulations as(strong)bisimulations betweenprojected automata,whichenablestousetheiralgebraic characterization. Since bisimulations are stronger than language (formal power series) equalities, it immediately follows that ex- istence of a weak bisimulation between two max-plus automata implies the equality of their projected behaviors. This particular definitionofbisimulationisverymuch influencedby theconcur- rencytheorycommunity,whereinsteadofunobservableevents(as asubsetoftheeventset)ascustomaryinDEScommunitytheno- tion ofan internal action denoted by

τ

^{is used.We believe that}

conceptslikeweakbisimulationwillbeprovenveryusefulforpar- tiallyobservedmax-plusautomatainanearfuture,becausefirstly they admit nice algebraiccharacterization andsecondly they can becomputedinapolynomialtime.

4.3.Supervisorycontrol

Supervisorycontrol theory can be viewed as a generalization of verification. The idea is that if a property to be verified fails tobesatisfieditcanstill beimposedbyasupervisor.Supervisory controlisaformal approachintroduced firstforcontrol oflogical automata withpartial transitionfunctions that aims to solve the safetyissue(avoidanceofforbiddenstates givenby controlspec- ifications)andnonblockingness (avoidanceof deadlocksandlive- locks).Ifacontrolspecification(property)isgiveninformally,soft- ware engineersmusttranslate them into controlsoftware manu- ally.Theultimategoalofsupervisorycontrol istodevelopformal theorythatenablesan automatedsynthesisofcontrollersthatare correctby constructionso that furtherverificationis notneeded.

Givena control specification describing required behavior of the system, one hasto construct a supervisorthat observes a subset ofevents(yieldingapossiblypartialinformationaboutthestateof theplant)andselectsactuators,thatcan controltheexecutionof some controllableeventsin ordertomeet theprescribed specifi- cationlanguage,whichspecifies apropertyofthesystemsuchas certain statesmustbe forbidden.An interesting control approach tomax-plusautomata ispresentedin Klimann(2003), wherethe problemof(A,B)-invarianceforformalpowerseriesissolved.

Supervisory control theory of max-plus automata with com- plete observations has been proposed in Komenda, Lahaye, and Boimond (2009), where the basic elements of supervisory con- trol,suchassupervisor,closed-loopsystem,andcontrollabilityare extended fromlogical tomax-plus automata. However, it follows fromresultspresentedtherein that rational(i.e.finite state)con- trollerscanonlybe obtainedforsystems(plants)which havebe- haviors at the same time max-plus and min-plus rational. The problemisthatthecontrollerseriesisbasedonresiduationofthe

Hadamardproduct of series, which can be seen as a Hadamard productwitha serieshavingall itscoefficientsinverted.This op- eration has alreadybeen discussed in Section 4.1and it outputs a min-plus rational series for a given max-plus rational series.

We then need to work with the class of series that are at the same time max-plus and min-plus rational in order to have ra- tional (finite-state) controllers. Unfortunately, it has been shown in Lombardy and Mairesse (2006) that this class of series coin- cideswith theclass of unambiguous series(series recognized by anunambiguousautomaton).Althoughunambiguousseriesisless restrictivepropertythandeterministicseries(seriesrecognizedby a deterministic max-plus automaton), a typical approach for im- posingunambiguityistodeterminizeamax-plusautomaton.

More complete picture about rationality issues extended to more general setting is presented in Lahaye, Komenda, and Boi- mond (2015b). More specifically, minimally permissive and just- after-timesupervisorsarestudiedinordertoguaranteeaminimal requiredbehaviorandtodelaythesystemaslittleaspossibleso thatsequences ofeventsoccurlaterthan prescribeddates,which isimportant forapplicationsin transportation networks(e.g. im- provingtrain connections inrailwaysystems), but alsoin manu- facturingsystemsandcommunicationnetworks.Ithasbeenshown thatfinitestatecontrollersexistifthesystem-seriesandthespec- ification (reference-series) are both unambiguous. This assump- tionismetforseveralclassesofpractically relevantmax-plusau- tomata,e.g.thosemodelingatypeofmanufacturingsystemssuch assafeFlow-shops andJob-shops.Anotherclass of timedsystem calledtimedweightedsystemshasbeenstudiedinSu,vanSchup- pen,andRooda(2012).Timedweightedsystemsaresimply mod- ularautomata (collectionoflocalautomata)endowedwiththeso calledmutualexclusionfunctionaswellasatime-weightedfunc- tion.Timedweightedsystemscanbeunderstoodasasynchronous productofmax-plusautomata,whichisnotmadeexplicitandthe durationsofeventsaredescribedbytime-weightedfunction.

In our opinion max-plus automata form a gateway to the general timed automata, because systems modeled by max-plus automata exhibit most of decidability and determinization is- sues that are present for general timed automata, while they are conceptually simpler, which allows for better grasping the coreofthesefundamentalproblems.Fortunately, thereexist sev- eral ways how to deal with these issues. For instance, further progress in determinization of max-plus automata is possible as it is shown inLahaye, Lai, and Komenda (2017). There exist ap- proaches to approximate determinization of weighted automata (Filiot,Jecker,Lhote,A.Pérez,&Raskin,2017).Finally,onemayre- placethecontrolspecification(requirement)intermsofinequality offormalpowerseriesbyasimulation-basedspecificationandin- troducethesupervisorycontroltheoryforimpositionofsimulation properties.

5. Max-plusplanningandmodelpredictivecontrol

The ModelPredictiveControl(MPC)designmethodcanbeap- pliedto(switching)max-pluslinearsystems(vandenBoom&De Schutter, 2006). MPC for conventional (non-DES) systemsis very popularintheprocessindustry(Maciejowski,2002)andakeyad- vantageofMPCisthatitcanaccommodateconstraintsonthein- puts and outputs. For every cycle the future control actions are optimized by minimizing a cost function over a prediction win- dowsubjecttoconstraints.Ifthecostfunctionandtheconstraints arepiecewiseaffinefunctionsintheinput,output,andstate vari- ables,theresulting optimizationproblemwill bea mixed-integer linearprogramming(MILP)problem,forwhichfastandreliableal- gorithmsexist. Analternative approach is to useoptimistic opti- mization(Xu,De Schutter, & van denBoom, 2014). In De Schut- terandvandenBoom(2001)MPCforregularmax-plus-linearsys-

tems was studied using the just-in-timecost function withcon- straints that were monotonically nondecreasing in the output. In thatcasetheproblemturnsouttobealinearprogrammingprob- lem.

A natural generalization of deterministic max-plus-linear sys- tems are max-plus-linear systems with uncertainty. This uncer- taintycaneitherhaveaboundednatureorastochasticnature.The uncertainty will appearin a max-plus-multiplicative wayasper- turbationsofthesystemparameters (Olsder, Resing,Vries,Keane,

&Hooghiemstra,1990).

In the bounded uncertainty approach the parameters of the modelsmayvary, whichleadstothe studyof non-stationarylin- ear systems approach.This possibilityhas been examinedwithin themax-pluslinearsetting(Brat&Garg,1998;Lahayeetal.,1999;

2004)withcontributionsmainlyfocusedonrepresentation,control andperformanceanalysis.Boundeduncertaintycanalsobeconsid- eredthrough intervalsdefiningthepossiblevaluesforparameters ofthesystem.InLhommeauetal.(2004)TEGs,inwhichthenum- berofinitialtokensandthetimedelaysareonlyknowntobelong to intervals, are represented over a semiringof intervalsand ro- bustcontrollersaredesigned.Adynamicprogrammingapproachto robust state-feedbackcontrol ofmax-plus-linearsystemswithin- tervalboundedmatricesisgiveninNecoara,DeSchutter,vanden Boom,andHellendoorn(2009)inwhichitisshownthatthemin- maxcontrolproblemcanberecastasadeterministicoptimalcon- trolproblembyemployingresultsfromdynamicprogramming.

Stochastic Max Plus Linear systems, defined as MPL systems where the matrices entries are characterized by stochastic vari- ables (Heidergottetal., 2006; Heidergott,2007), havebeenstud- ied formore than two decades.As noticed in Section 2,mostof theergodictheory ofstochastic timedeventgraphsis coveredin Baccelli etal.(1992).Results for MPCof stochastic max-plus lin- ear systems are given invan denBoom andDe Schutter (2004), where the authorsshow that under quite generalconditions the resultingoptimization problemsturnout to beconvex. The main problemwiththismethodisthatthecomputationoftheexpected value canbe highlycomplexandexpensive, whichalsoresultsin a high computation time to solve the optimization problem. To thisend Farahani, vandenBoom, vander Weide, and DeSchut- ter(2016)useanapproximationmethodbasedonthemomentsof arandomvariabletoobtainamuchlowercomputationtimewhile stillguaranteeingacomparableperformance.

Switchingmax-pluslinearsystemswithbothstochasticandde- terministicswitchingarediscussedinvandenBoomandDeSchut- ter (2012). In general, the optimization in the model predictive control approach boils downto a nonlinear nonconvexoptimiza- tionproblem,wherethecostcriterionispiecewisepolynomialon polyhedralsetsandtheinequality constraintsarelinear.However, in thecase ofstochastic switching that dependson the previous modeonly,theresultingoptimizationproblemcanbesolvedusing linearprogrammingalgorithms.

InvandenBoom,Lopes,andSchutter (2013)a generalframe- work has been set up for model predictive scheduling of semi- cyclic discrete event systems. In a systematic way the main scheduling steps, i.e. routing, ordering, and synchronization, can bemodeled.Aswitchingmax-pluslinearmodelhasbeenderived withschedulingparameters foreach schedulingstep. Thesystem matrix is max-plus affine inthe max-plus binary scheduling pa- rametersandamodelpredictiveschedulingproblemhasbeenfor- mulated.This modelpredictivescheduling problemcanbe recast intoamixedintegerlinearprogrammingproblem.Thisscheduling techniquehasbeenappliedinKersbergen,Rudan,vandenBoom, and De Schutter (2016), where a railway traffic management al- gorithm has been derived that can determine new conflict-free schedulesandroutesforarailwaytrafficnetworkwhendelaysoc-

Fig. 4. Dutch railway network.

cur.SeeFig.4fortheschemeoftherailwaynetworkintheNether- lands.

Scheduling using switching max-plus linear models has also beendescribedin Lopes,Kersbergen,vandenBoom,De Schutter, and Babuška (2014), where the transition between different gait transitionschemes inlegged robotshasbeendiscussed andopti- maltransitions arederived such that thestance time variationis minimized, allowingforconstantaccelerationanddeceleration.In Alirezaei,vandenBoom,andBabuška(2012)anoptimalscheduler forpaper-handlinginaduplexprinterispresented.Thescheduling isbasedonthemax-plusmodelingframework.Itisshownthatthe proposed methodsuccessfullyfindsthe globallyoptimalschedule fordifferenttypesofthesheets.

6. Max-plusandmin-plusgeometry

Itiswellunderstoodthatlinearalgebraiscloselyconnectedto geometry andthat geometric concepts play an important role in controloflinearsystems.

As we have argued in previous sections, control theory for max-plus-linear systems has beeninspired mainly by the theory of linear systems. A fundamental concept in both linear algebra andgeometryisthat ofvectorspacesormoregenerallymodules.

Theirmax-plus counterparts are knownasidempotent semimod- ules, which are module-like structures, but over an idempotent semiring(suchasmax-plussemiring)ratherthanoveraring.The basic properties of idempotent semimodules including the con- ceptsofindependenceanddimensionhavebeenstudiedsincelate 1980’s by Wagneur (1991) orRussian school (Litvinov, Maslov, &

Shpiz,2001).Afundamentalcontroltheoreticconceptis(A,B) in- variant space, which isa controlled invariance of a semimodule.

Namely,itrequiresthatanytrajectorystartinginthissemimodule can be controlled such that it remains forever within this semi- module.IthasbeeninvestigatedinKatz(2007),whereaclassical algorithmforthecomputationofthemaximal(A,B)-invariantsub- space contained in a given spaceis generalized to the max-plus

linear systems. Although the algorithm needs not converge in a finitenumberofsteps, thesufficient conditions(of demonstrated practicalinterest fora classofsemimodules)havebeenproposed fortheconvergenceinafinitenumberofsteps.

Thestudyofinvariancepropertiesformax-pluslinearsystems areinspiredby theWonham’sgeometrictheory oflinearsystems (Wonham, 1974).We emphasize that thistheory hasbeen atthe very origin of the supervisory control theory developed in early 1980’sinparallelwiththetheoryofmax-plus-linearsystems.The geometric framework has enabled among others to solve distur- bance decoupling problem for linear systems. Following similar ideas,modifieddisturbancedecouplingproblemformax-pluslin- earsystemshasbeenstudiedinShang,Hardouin,Lhommeau,and Maia(2016).

Another interesting work is Cohen, Gaubert, and Quadrat (1996), where projection onto images of operators and parallel to the kernels of operators have been studied. It shouldbenotedthattheseoperatorsareusefulnotonlyincontrol ofmax-pluslinear systems,butadmit alsospecific applicationto aggregationandotherproblemsforMarkovchains.

Veryimportantconcept,convexity,apowerfultoolinoptimiza- tion and operational research, has been extended to the max- plus framework. Well known Minkowskitheorem fromlinear al- gebra states that a non-empty compact convex subset of a fi- nite dimensional space is the convex hull of its set of extreme points.Themax-pluscounterpartofMinkowskitheorempresented inGaubert and Katz (2007)extends thisresultto max-plus con- vex sets. Thisresult is very important,because max-plus convex setsariseinmanydifferentdomains,rangingfrommax-plus-linear systems,abstractionsoftimedautomatatosolutionsofHamilton–

Jacobi equations associated with a deterministic optimal control problem,seee.g.Litvinovetal.(2001).

More recently, an interesting relation has been discovered between geometric approach to max-plus-linear systems pro- posed in Gaubert and Katz (2007) and reachability analysis of timed automata, cf. Allamigeon, Fahrenberg, Gaubert, Katz, and Legay (2014). Timed automata are very general models of timed DESthatinvolveseveralparallelclocksvariablesthatmeasuretime elapsedsincetheirlastresetanddefinetimeconstraints(knownas guards)forenablinglogicaltransitionsintimedautomata.

Interestingly,max-plusgeometrycanbeappliedinreachability analysisoftimedautomata.Timedautomatawithinfinite(butfi- nitedimensional) clockspacesare abstractedintofiniteautomata calledregion orzone automata,where theinfinite clockspace is abstractedby afinitenumberofregions orgeometriczones. This abstractionisshowntobe atimedbisimulationsandthisenables tosolveseveralfundamentalproblemsfortimedautomatasuchas nonemptiness. Thezonesarerepresented byefficientdatastruc- turescalleddifference bound matrices(DBM)that represents the boundsondifferencesbetweenstate variables.Thereachabilityof differentzonescanbestudiedusingmax-plus-conesfromgeomet- rictheoryofmax-plus-linearsystems.

IthasbeenshowninLuetal.(2012)thateverymax-pluscone (also called max-plus polyhedron) can actually be described as a union of finitely many DBM’s as shown in Adzkiya, De Schut- ter,andAbate(2013).These geometricobjects haveproven tobe extremelyusefulforbothforwardandbackwardreachabilityanal- ysis,seee.g.Allamigeonetal.(2014).Forwardreachabilityanalysis aimsat computingtheset ofpossiblestates that canbe reached underthemodeldynamics,overasetofconsecutiveeventsfrom asetofinitialconditionsandpossiblybychoosingcontrolactions (Adzkiya,DeSchutter, &Abate,2015).Backwardreachabilityanal- ysisconsistsincomputingthe setofstatesthatentera givenset offinalstates,possiblybychoosingcontrolactions.Thisisofprac- ticalimportance in safetycontrol problemsconsisting inthe de- terminationofthesetofinitialconditionsleadingtounsafestates.

However,forbackwardreachabilityanalysisthesystemmatrixhas to be max-plus invertible, i.e. in each row and in each column thereshould be a singlefinite element (notequal to -∞), which isrestrictive.The mainadvantage ofusingmax-plus polyhedrais in saving computational complexity, because time complexity of theseapproaches is polynomial asall standard DBM based algo- rithms.

From a practical perspective, there are two basic ways of describing max-plus polyhedra. The first one, internal, gives the extreme points and rays, the second one, exter- nal, gives linear inequalities over max-plus semiring. It has been shown in Allamigeon, Gaubert, and Goubault (2010), see Allamigeon, Gaubert, andGoubault (2013) for more recentwork, howtopassfromtheexternaldescriptionofa polyhedronto the internaldescription.Namely, theextremalpointsarecomputedin a recursive way, where the problem of checking the extremality ofapointreducestocheckingwhetherthereisonlyoneminimal strongly connected component in an hyper-graph. For the latter problemthereexists a fast(almostlinear time)algorithm,which allowsquickeliminationofredundantgenerators,butthenumber ofgeneratorscanbeexponentialingeneral.

7.Applications

Itcanappearsomewhatsurprisingthatmethodsbasedonvery particularstructure of max-plusalgebra can find a large number ofapplications.Butitturns out that max-plussystemtheory has beenindeedappliedtoalargevarietyofdomains,suchas:

• capacityassessment,evaluationandcontrol ofdelaysintrans- portationsystems(Braker,1991;Heidergottetal.,2006;Houssin, Lahaye,&Boimond,2007;Kersbergenetal.,2016)andcartraf- fic(Farhi,Goursat,&Quadrat,2011),

• sizing, optimization and production management in manufac- turing systems (Cohen et al., 1985; Cottenceau, Hardouin, &

Ouerghi, 2008; Imaev & Judd, 2008; Martinez & Castagna, 2003),

• performanceguaranteesincommunicationnetworksthroughso- callednetworkcalculus(Cruz,1991;LeBoudec,2001),

• high throughput screening in biology and chemistry (Brunsch,Raisch,&Hardouin,2012),

• modeling, analysis and control of legged locomotion (Lopes etal., 2014; Lopes, Kersbergen,DeSchutter, van denBoom,&

Babus˘ka,2016),

• speechrecognition (Mohri, Pereira,&Riley, 2002)orimagepro- cessing(Culik&Kari,1997)throughweightedautomatasuchas max-plusautomata,

• optimizationofcroprotationinagriculture(Bacaër,2003),

• schedulingofenergyflowsforparallelbatchprocesses(Mutsaers, Özkan,&Backx,2012),

• max-plus model of ribosome dynamics during mRNA translation (Brackley,Broomhead,Romano,&Thiel,2012),

• paperhandlinginprinters(Alirezaeietal.,2012),

• performanceevaluation oftheemergencycallcenter17-18-112in theParisarea(Allamigeon,Bœuf,&Gaubert,2015),

• control of cluster tools in semiconductor manufacturing (Kim &

Lee,2015),

• biologicalsequencecomparisons(Comet,2003).

Thisdiversityistobeemphasizedall themoresincetheseap- plicationshavesometimessuggestednewtheoreticalquestions.

Wecannotfinishthisbriefoverviewoftheroleofmax-plusal- gebrainthehistoryofDESwithoutmentioningtheimportantcon- nectionswithother fields ofresearch: dynamic programmingand optimalcontrolwithsolutionstoHamilton-Jacobi-Bellman(partial) differentialequations(Maslov&Kolokoltsov,1994;Quadrat&Max-

Plus,1994),statisticalmechanics(Quadrat&Max-Plus,1997),oper- ationsresearch(Zimmermann,2003).

References

Addad, B. , Amari, S. , & Lesage, J.-J. (2012). Networked conflicting timed event graphs representation in (max,+) algebra. Discrete Event Dynamic Systems, 22 (4), 429–449 .

Adzkiya, D. , De Schutter, B. , & Abate, A. (2013). Finite abstractions of max-plus linear systems. IEEE Transactions on Automatic Control, 58 (12), 3039–3053 .

Adzkiya, D. , De Schutter, B. , & Abate, A. (2015). Computational techniques for reach- ability analysis of max-plus-linear systems. Automatica, 53 (3), 293–302 . Akian, M. , Bapat, R. , & Gaubert, S. (2003). Handbook of linear algebra . Chapman and

Hall .

Akian, M. , Gaubert, S. , & Guterman, A. (2012). Tropical polyhedra are equivalent to mean payoff games. International Journal of Algebra and Computation, 22 (01), 125–168 .

Alirezaei, M. , van den Boom, T. , & Babuška, R. (2012). Max-plus algebra for optimal scheduling of multiple sheets in a printer. In Proceedings of american control conference (pp. 1973–1978). Montreal, Canada .

Allamigeon, X. , Bœuf, V. , & Gaubert, S. (2015). Performance evaluation of an emer- gency call center: Tropical polynomial systems applied to timed petri nets. In Formal modeling and analysis of timed systems (p. 1026). Springer International Publishing .

Allamigeon, X. , Fahrenberg, U. , Gaubert, S. , Katz, R. D. , & Legay, A. (2014). Tropical Fourier–Motzkin elimination, with an application to real-time verification. IJAC, 24 (5), 569–608 .

Allamigeon, X. , Gaubert, S. , & Goubault, E. (2010). The tropical double description method. In Symposium on theoretical aspects of computer science 2010 (stacs 2010) (pp. pp.47–58) .

Allamigeon, X. , Gaubert, S. , & Goubault, E. (2013). Computing the vertices of tropi- cal polyhedra using directed hypergraphs. Discrete and Computational Geometry, 49 (2), 247–279 .

Bacaër, N. (2003). Modèles mathématiques pour l’optimisation des rotations. Compte rendus de l’Académie d’Agriculture de France, 89 (3) .

Baccelli, F. , Cohen, G. , Olsder, G. , & Quadrat, J. P. (1992). Synchronization and Linear- ity . Wiley .

Birkhoff, G. (1940). Lattice theory. American mathematical society colloquium publica- tions : XXV. Providence, Rhode Island .

Blyth, T. , & Janowitz, M. (1972). Residuation Theory . Pergamon Press .

van den Boom, T. , Lopes, G. , & Schutter, B. D. (2013). A modeling framework for model predictive scheduling using switching max-plus linear models. In Pro- ceedings of the 52st IEEE conference on decision and control (cdc 2013) . Florence, Italy .

Brackley, C. A. , Broomhead, D. S. , Romano, M. C. , & Thiel, M. (2012). A max-plus model of ribosome dynamics during mrna translation. Journal of Theoretical Bi- ology, 303 , 128–140 .

Braker, J. (1991). Max-algebra modelling and analysis of time-dependent trans- portation networks. In Proceedings of first european control conference (pp. 1831–1836). Grenoble, France .

Brat, G. P. , & Garg, V. K. (1998). A (max,+) algebra for non-stationary periodic timed discrete event systems. In Proceedings of the 4th international workshop on dis- crete event systems (pp. 237–242) .

Brenier, Y. (1989). Un algorithme rapide pour le calcul de transform ¸E es de leg- endre-fenchel discr ȿ tes. Comptes Rendus de l Académie des Sciences - Series I - Mathematics, 308 (20), 587589 .

Brunsch, T. , Raisch, J. , & Hardouin, L. (2012). Modeling and control of high-through- put screening systems. Control Engineering Practice, 20 (1), 14–23 .

Buchholz, P. , & Kemper, P. (2003). Weak bisimulation for (max/+) automata and re- lated models.. Journal of Automata, Languages and Combinatorics, 8 , 187–218 . Butkovi ˇc, P. (2010). Max-linear systems: Theory and algorithms. Springer Mono-

graphs in Mathematics . Springer-Verlag London .

Caspi, P. , & Halbwachs, N. (1986). A functional model for describing and reasoning about time behaviour of computing systems. Acta Informatica, 23 , 595–627 . Cofer, D. D. , & Garg, V. K. (1996). Supervisory control of real-time discrete event sys-

tems using lattice theory. IEEE Transactions on Automatic Control, 41 (2), 199–209 . Cohen, G. , Dubois, D. , Quadrat, J. P. , & Viot, M. (1983). Analyse du comportement périodique des systèmes de production par la théorie des dioïdes. Rapport de recherche . Le Chesnay, France: INRIA .

Cohen, G. , Dubois, D. , Quadrat, J. P. , & Viot, M. (1985). A linear system theoretic view of discrete event processes and its use for performance evaluation in manufac- turing. IEEE Transactions on Automatic Control, 30 , 210–220 .

Cohen, G. , Gaubert, S. , & Quadrat, J. (1997). Algebraic system analysis of timed Petri nets. In Idempotency (pp. 145–170). Cambridge University Press .

Cohen, G. , Gaubert, S. , & Quadrat, J. P. (1995). Asymptotic throughput of continuous timed Petri nets. In Proceedings of the 34th conference on decision and control (pp. 2029–2034). New Orleans .

Cohen, G. , Gaubert, S. , & Quadrat, J.-P. (1996). Kernels, images and projections in dioids. In Proceedings of 3rd international workshop on discrete event systems (pp. 151–158) .

Cohen, G. , Gaubert, S. , & Quadrat, J. P. (1998). Timed event graphs with multipli- ers and homogeneous min-plus systems. IEEE Transactions on Automatic Control, 43 (9), 1296–1302 .

Cohen, G. , Gaubert, S. , & Quadrat, J. P. (1999). Max-plus algebra and system theory:

Where we are and where to go now. Annual Reviews in Control, 23 , 207–219 .