A Black-box Identification Method for Automated Discrete Event Systems

Transcription

1 A lack-box Idenificaion Mehod for Auomaed Discree Even Sysems Ana Paula Esrada-Vargas, E López-Mellado, Jean-Jacques Lesage To cie his version: Ana Paula Esrada-Vargas, E López-Mellado, Jean-Jacques Lesage A lack-box Idenificaion Mehod for Auomaed Discree Even Sysems IEEE Transacions on Auomaion Science and Engineering, Insiue of Elecrical and Elecronics Engineers, 25, 4 (3), pp <9/TASE > <hal-26998> HAL Id: hal hps://halarchives-ouveresfr/hal Submied on 5 Feb 26 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés

2 A lack-box Idenificaion Mehod for Auomaed Discree Even Sysems Ana Paula Esrada-Vargas, Erneso López-Mellado, and Jean-Jacques Lesage, Members, IEEE Absrac This paper deals wih he idenificaion of discree even manufacuring sysems ha are auomaed using a programmable logic conroller (PLC) The behavior of he closed loop sysem (PLC and Plan) is observed during is operaion and is represened by a single long sequence of observed inpu/oupu (I/O) signals vecors The proposed mehod follows a black-box and passive idenificaion approach ha allows addressing large and complex indusrial DES and yields compac and expressive inerpreed Peri ne (IPN) models I consiss of wo complemenary sages; he firs one obains, from he I/O sequence, he reacive par of he model composed by observable places and ransiions The I/O sequence is also mapped ino a sequence of he creaed ransiions, from which he second sage builds he non observable par of he model including places ha ensure he reproducion of he observed inpu oupu sequence This mehod, based on polynomial-ime algorihms on he size of he inpu daa, has been implemened as a sofware ool ha generaes and draws he IPN model; i has been esed wih inpu/oupu sequences obained from real sysems in operaion The ool is described and is applicaion is illusraed hrough a case sudy Noe o praciioners Auomaed modeling of conrolled discree manufacuring sysems can be achieved by efficien idenificaion algorihms ha cope wih large and complex plans performing concurren and repeiive asks a priori unknown The black-box idenificaion procedure processes an inpu/oupu sequence recorded during he sysem funcioning for a long period of ime, and hen yields a comprehensive model of he closed-loop conrolled sysem; his model approximaes closely he acual behavior of he compound sysem conrollerplan A ool based on idenificaion algorihms consiues an excellen resource for compuer-aided reverse engineering of conrolled manufacuring sysems The mehod proposed herein allows processing sequences composed by housands of I/O vecors in few seconds Index Terms Discree Even Sysems, lack-box Idenificaion, Inerpreed Peri Nes I I INTRODUCTION DENTIFICATION of discree even sysems (DES) allows building sysemaically a mahemaical model (Peri nes, auomaa) ha describes he behavior of an unknown or illknown sysem based on he observaion of is evoluion Observaions consis of daa revealing he sysem aciviy: sequences of operaions, evens, messages, signals ec, and he models allow reproducing he observed behavior A Relaed works DES idenificaion has been firs addressed as a problem of grammaical inference [], [2] for obaining finie auomaa (FA) ha represens a given language Aferwards, Peri ne (PN) models have been proposed for coping wih more complex sysems exhibiing concurren behavior; in [3] an algorihm for consrucing PN models is presened Several approaches for idenificaion of DES have been proposed in lieraure in various formulaions and from diverse approaches elow here is an overview of he main approaches; oher works can be found in deailed surveys on idenificaion mehods [4] and [5] In [6], [7] mehods based on Ineger Linear Programming (ILP) are proposed; hey allow obaining accurae Peri nes from a se of ransiion sequences ha can be fired from he iniial marking These mehods require he a priori knowledge of he se of ransiions and of he number of places, wha makes difficul heir applicaion o idenify real DES as black boxes, since he only available informaion afer observaion is he evoluion of inpu and oupus signals exchanged beween he conrol sysem and he plan In [8], [9] i is described an efficien mehod o incremenally consruc an IPN model from a single oupu vecors sequence The considered DESs o idenify mus be even-deecable by he oupus Applying his mehod o an I/O sequence would lead o models in which same oupu changes caused by differen inpu evoluions would no be disinguished, and hen incorrec behavior could be inroduced The mehod presened in [] is dedicaed o faul deecion and isolaion (FDI) I allows obaining a finie auomaon represening precisely a se of cyclic I/O sequences An exension o disribued idenificaion and disribued FDI has been presened in [] However, due o he usage of finie auomaa, srucural informaion such as parallelism canno be explicily expressed ino he models, wha makes his approach inefficien for applicaions such reverse engineering Oher proposals relaed wih FDI in he PN framework are presened in [2] and [3] E López-Mellado is wih CINVESTAV Unidad Guadalajara Av del osque 45, Col El ajío 459 Zapopan, Mexico elopez@gdlcinvesavmx J-J Lesage is wih LURPA, ENS Cachan, Univ Paris-Sud, F Cachan, France Jean-Jacqueslesage@lurpaenscachanfr A P Esrada-Vargas was wih boh insiuions; she has been sponsored by CONACYT (Mexico) under Gran No 532, and by Région Île-de- France aesrada@gdlcinvesavmx

3 In [4], [5] an even sequence is observed, as well as he corresponding oupu symbols of a DES o produce an IPN model, in which he sequence and he observed oupu vecors are reproducible This mehod requires he definiion of an even lis, which is no available a priori in he conex of black-box idenificaion problem addressed in his work An alernaive o his lack of evens lis could be he consideraion of all he observed inpu changes In his case, models wih several pahs describing inpu changes would be consruced, in which some inpu/oupu relaions would no be explicily observed This work has been exended in [6] owards he deerminaion of sochasic ransiions for FDI purposes In [7] a echnique for consrucing a Peri ne-like model ha describes he relaionship beween asks from a sequence of workflow evens is presened This echnique allows he discovering of evens belonging o cerain hreads and synchronizaion poins (forks and joins of asks) hrough a probabilisic analysis of merics such as he enropy, number and regulariy of ask occurrences I is assumed ha all he workflow operaions are observable In [8] he modeling of a workflow is also considered The inpu of he algorihm is a workflow log of several workflow insances composed by several asks Workflow insances are recorded sequenially, even if asks may be execued in parallel ased on he informaion in he workflow log and by making some assumpions abou compleeness of he log, a process model in he form of a workflow ne is deduced lack-box approach eyond he heoreical ineres of defining model synhesis mehods from even sequences, he challenges of applying idenificaion mehods o acual indusrial auomaed sysems are relaed o he scalabiliy of he algorihms and echnological issues: he echniques mus be efficien o cope wih large and complex sysems ha handle acual signals In our approach we deal wih Programmable Logic Conroller (PLC) based auomaed sysems The aim is o discover, from observaions of he sysem behavior expressed as a single sequence of PLC inpu and oupu signals how componens of he sysem are inerrelaed, and o consruc a concise model which can explicily show he discovered behavior, in paricular, concurrency, synchronizaion, resource sharing, ec Idenificaion of sysems in operaion involves wo imporan aspecs o consider: he sysem operaion and he observaion process Technological issues of boh aspecs mus be considered in he proposed algorihms o consruc suiable absracions In previous works [9], [2] an I/O sequence is considered o compue an IPN including cyclic behavior Alhough he proposed mehodology is scalable due o he algorihms efficiency, he obained models are close o finie auomaa and can be huge, due o he explici represenaion of observed inpu changes ha could no be relevan o define he oupu evoluion C Conribuion In his paper we address hese problems by analyzing he observed sequence o esablish a clearer relaion beween inpus and oupus of he conroller The proposed mehod allows building a reduced represenaion of he observable par of he model which yields consequenly, a reduced complee IPN I consiss of wo complemenary sages; he firs one obains, from he I/O sequence, he reacive par of he model composed by observable places and ransiions A firs version of his sage has been presened in [2] The I/O sequence is mapped ino a sequence of he creaed ransiions, from which he second sage builds he non-observable par of he model including places ha ensure he reproducion of he observed inpu oupu sequence This mehod, based on polynomial-ime algorihms on he size of he inpu daa, has been implemened as a sofware ool ha generaes and draws he IPN model [22] None of he black-box idenificaion approaches in relaed works allows obaining such well srucured models The presen aricle gahers boh sages of he mehod; i includes a deailed presenaion of he revised resuls, addiional illusraive examples, all he proofs omied in he conference papers, and a case sudy regarding he idenificaion of a real process D Conens The paper is organized as follows In Secion II IPN basic noions are overviewed Secion III saes he problem of indusrial auomaed sysems idenificaion and overviews he wo seps mehod Such seps are explained in Secion IV and Secion V Finally, he implemenaion deails and a case sudy are presened in Secion VI II INTERPRETED PETRI NETS This secion conains he basic conceps and noaion of PN and IPN used in his paper Definiion : An ordinary Peri Ne srucure G is a biparie digraph represened by he 4-uple G = (P, T, Pre, Pos) where: P = {p, p 2,, p P } and T = {, 2,, T } are finie ses of verices named places and ransiions respecively; Pre(Pos) : P T {,} is a funcion represening he arcs going from places o ransiions (from ransiions o places) The incidence marix of G is W = W + W, where W = [w ij ]; w ij = Pre(p i, j); and W + + = [w ij+ ]; w ij = Pos(p i, j) are he pre-incidence and pos-incidence marices respecively A marking funcion M : P Z + represens he number of okens residing inside each place; i is usually expressed as a P -enry vecor Z + is he se of nonnegaive inegers In paricular, in his paper M : P {,}; he PN is referred as - bounded or safe Definiion 2: A Peri Ne sysem or Peri Ne (PN) is he pair N = (G,M ), where G is a PN srucure and M is an iniial marking In a PN sysem, a ransiion j is enabled a marking M k if p i P, M k(p i) Pre(p i, j); an enabled ransiion j can be fired reaching a new marking M k+ This behavior is

4 represened as M k j M k+ The new marking can be compued as M k+ = M k + Wu k, where u k(i) =, i j, u k(j) = ; his equaion is called he PN sae equaion The reachabiliy se of a PN is he se of all possible reachable markings from M firing only enabled ransiions; his se is denoed by R(G,M ) Definiion 3 A Peri ne circui is a pah of verices linked by arcs saring and ending in he same node A circui is said o be simple if i does no use he same ransiion more han once, and elemenary if i does no use he same place more han once Now i is defined IPN, an exension o PN ha allows associaing inpu and oupu signals o PN models This definiion is adaped from [23] Definiion 4 : An inerpreed Peri ne (IPN) (Q, M ) is a labeled ne srucure Q = (G, Σ, Φ, λ, ϕ) wih an iniial marking M where: - G is a PN srucure, - Σ= {I, I 2,, I r} is he inpu alphabe, - Φ = {O, O 2,, O q} is he oupu alphabe, - λ : T C E is a labeling funcion of ransiions, where - C={C, C 2, } is he se of inpu condiions in which every C i is a oolean funcion on Σ; when a Ci is always rue i is denoed as =, and - E={E, E 2, } is he se of inpu evens condiions; every E i is a oolean funcion of inpu evens, build on Σ; evens are denoed as Ii_ and Ii_ for represening ha he inpu value changes from o, or from o respecively A condiion E i may no exis; his is denoed as ε In an IPN, a ransiion j will be fired if a) j is enabled, and b) condiion C( j) is rue, and c) he even in E( j) occurs - ϕ : R(Q,M ) (Z + ) q is an oupu funcion, ha associaes wih each marking in R(G,M ) a q-enry oupu vecor, where q= Φ is he number of oupus ϕ is represened by a q P marix, such ha if he oupu symbol O i is presen (urned on) every ime ha M(p j), henϕ (i, j) =, oherwise ϕ(i, j) = The sae equaion of PN is compleed wih he marking projecion Y k = ϕm k, where Y k (Z + ) q is he k-h oupu vecor of he IPN Definiion 5: A place p i P is said o be observable if he i- h column vecor of ϕ (denoed as ϕ(,i)) is no null Oherwise i is non-observable P = P obs P nobs, and P obs P nobs = ; where P obs is he se of observable places and P nobs he se of non-observable places III IDENTIFICATION OF INDUSTRIAL AUTOMATED SYSTEMS A The process PLC+Plan In his work we consider sysems composed by a Conroller (a PLC) and a Plan denoed as {PLC + Plan} working on a closed loop The inpu signals of he PLC (oupus of he Plan) are generaed by he sensors of he Plan The oupu signals of he PLC (inpus of he Plan) conrol he acuaors of he Plan The idenificaion is made wih respec o he inpus-oupus of he PLC (Fig ) A PLC cyclically performs hree main seps: i) Inpu reading, where signals are read from he sensors; ii) Program execuion, o deermine he new oupus values for he acuaors; and iii) Oupu wriing, where he conrol signals o he acuaors are se A each end of he Program execuion phase, he curren value of all r inpus and q oupus, called I/O vecor, is capured and recorded in a daa base Regarding he implemenaion of he daa link beween PLC and idenificaion daa base, we use he UDP (User Daagram Proocol) connecion presened in [24] Tess performed using a Siemens PLC (CPU 35-2 DP) equipped wih a program leading o a PLC-cycle ime of 25 o 3ms have shown ha his connecion is reliable and efficien: no daa packes go los during he ransmission and he execuion of he PLC program is no delayed by he capure of daa The only available daa for he idenificaion procedure is herefore a single sequence of I/O vecors whose lengh depends on he observaion duraion: I() I(2) I(3) I( w =,,,,, () () (2) (3) ( ) O O O O k I( and O( are vecors whose enries are respecively he values of he r inpus I, I 2, I r and q oupus O, O 2, O q a he k-h PLC cycle Furhermore we denoe I i( and O i( he values of inpu I i and oupu O i respecively a he k-h cycle l l2 s L L2 R R2 r r2 Inpu Reading s R ( ) R r Program execuion End of I/O calculaion Oupu Wriing I( O( Daa link (UDP) I/O sequence Idenificaion algorihm R r_ L s R2 r2_ Daa collecion and Plan Conroller IPN idenificaion Fig {PLC + Plan} compound and idenificaion procedure Even ypes In order o analyze signals evoluion, we compue even vecors, ie, he difference beween wo consecuive I/O vecors Each even vecor can be decomposed ino inpu and oupu even vecors: IE( I( k + ) I( E ( = =, OE( O( k + ) O( IE( OE( where = IE2 ( and IE( OE ( (2) OE( = IEr ( OEq ( Regarding inpu and oupu even vecors and he PLC cycle described in he previous subsecion, here only exis four L2 l_ l2_

5 siuaions (behavior ypes) beween consecuive I/O vecors ha could be observed, which are explained by differen occurring phenomena: Type IE( and OE( An inpu change has provoked direcly an oupu change, and consequenly, a sae evoluion This I/O reacive causaliy is observed a he same PLC cycle Type 2 IE( = and OE( The conroller has arrived a sep k- o a sae in which, given he inpu values, an oupu (and sae) evoluion is allowed a sep k Type 3 IE( and OE( = Le X( be he inernal curren sae of he conroller, a) X(k ) X( An inpu evoluion has provoked a nonobservable sae evoluion of he conroller b) X(k ) = X( I has occurred an inpu evoluion o which he conroller is no sensiive Type 4 IE( = and OE( = a) X(k ) X( I has occurred a non-observable sae evoluion of he conroller which is no exhibied by any inpu nor oupu change b) X(k ) = X( The conroller remains in a sable sae, ie, no sae evoluion condiion is saisfied All of hese siuaions should be aken ino accoun o represen he sysem dynamics Our aim in his work is o express he sysem s behavior exracible from he I/O vecor sequence as an IPN C Inpu-Oupu idenificaion approach C Overview of he mehod The purpose in his research is no only o compue an IPN model in which he observed sequence is reproducible, bu also o achieve expressiviy and compacness in he idenified model allowing represening causal relaionship and concurrency of he involved operaions The mehod processes off-line he I/O-sequence w capured during he process operaion and delivers an IPN model ha reproduces he observed behavior (w) The mehod is oulined here wih he help of a simple example I regards a conroller handling 3 inpus (s, x, y) and 3 oupus (A,, C), from which he following I/O sequence is obained: = C A y x s w The mehod consiss of wo main seps which are oulined below Sep Discovering he reacive inpu/oupu behavior In his sep is deermined he observable par of he IPN consising of subnes, named fragmens, composed by observable places labeled wih oupu symbols, and ransiions labeled wih algebraic expressions of inpu symbols (Fig 2) From he sequence w, a corresponding sequence of ransiions S= is obained A s_ 2 p x_ x= 3 4 p 2 C y_ y= 5 6 p 3 s= Fig 2 Firs sep of he idenificaion mehod: IPN fragmens Sep2 Deermining he non-observable par of he IPN and he iniial marking M The sequence S is processed for obaining causal and concurrency relaionships useful for deermining he non-observable places ha relae he fragmens such ha S (hus w) can be execued from M (Fig 3) A s_ s= x_ x= C y_ y= p p 5 p p 3 p 2 2 Fig 3 Second sep: assembled IPN fragmens C2 Dealing wih even ypes Since siuaions Type and Type 2 (cf secion III) are direcly observable by an oupu change, hey can be sraighforwardly modeled in an IPN Such a modeling is performed by he firs sep of our mehod The Type siuaion represens a direc inpu/oupu reacive behavior, and hus he modeling is quie easy: he inpu change is associaed wih he label of a ransiion and he oupu change is represened as arcs relaing such a ransiion wih he observable places represening oupus involved In he Type 2 siuaion he inpu values which lead o he oupu evoluion are no observed a he same PLC cycle (ie a he same even vecor) In order o represen such a behavior, he conex (he values of he inpus) in which he oupu changes occur is analyzed; in his case, he oupu change is modeled such as in he Type siuaion, bu he label of he corresponding ransiion conains only a condiion on inpus levels (he inpu change is ε) The Type 3 siuaion is divided in wo, depending on wheher or no here is an inernal sae evoluion of he conroller Siuaion Type 3a is he case of he inpu evens which provoke inernal sae evoluions and evenually lead o an oupu even of Type 2 Such inernal evoluions canno be direcly compued, bu can be inferred y looking in he sequence buil in Sep, he order in which ransiions appear can be deermined Such inernal sae inference will be performed by he second sep of our mehod and will be

6 modeled by he addiion of non observable places assuring he order of he ransiion firings, such as in Fig 3 In he siuaion Type 3b here is no inernal sae evoluion, and hus here is nohing o be inferred, as well as he siuaion Type 4, where here are neiher inpu nor oupu evens occurring in a PLC cycle Consequenly, he sequence sored in he daabase will be buil by adding a new I/O vecor only when i is differen o he las one Noice ha in his work we can only infer inernal sae evoluions by means of ransiion firing order Oher ype of inernal evoluions, such as imers or couners, is ou of he scope of his work We can now make he descripion of he wo idenificaion seps IV IDENTIFICATION OF THE OSERVALE EHAVIOR In his secion he firs sep of he mehod is presened The inroduced conceps and algorihms are illusraed hrough a simple case sudy inspired from a manufacuring example A Overview and case sudy descripion Algorihm summarizes he seps of he procedure o idenify he {PLC + Plan} observable behavior; he seps will be described in deail in he nex sub-secions Algorihm Compuing he observable IPN componens Inpu: I/O sequence w Oupu: Observable incidence marix φw and labeling ransiion funcion λ ) Analyze sequence w in order o Compue evens vecor sequence Compue elemenary evens Compue Direc and Indirec Causaliy Marices Consruc Oupu Even Firing Funcions Find Inpu evens wih differed influence 2) Use compued daa in he previous sep o Compue ransiions of he IPN and heir labeling λ Compue observable incidence marix φw Example The purpose of his sysem (Fig 4) is o sor parcels according o heir size I has 9 signal sensors from he sysem: a, a, a 2, b, b, c, c, k, k 2, and 4 signals o he acuaors: A+, A-,, C This example has been used in oher publicaions [9], [2] and we describe i o confron his work agains previous resuls Fig 4 Layou of he sysem case sudy Evens vecor sequence In Fig 5 he beginning of an I/O vecor sequence is shown for illusraive purposes; however, recall ha reaed sequences are usually very much longer (housands of vecors) We have included in he sequence he resul of firs sub-sep of he algorihm, ie, he compued even vecors (below he arrows) beween each wo consecuive I/O vecors E() E(2) E(3) E(4) E(5) E(6) E(7) k k2 a a a2 b b w = c c + A A C Fig 5 eginning of I/O vecor sequence C Elemenary evens In order o analyze he sysem behavior in a deeper way, even vecors can be decomposed ino a se of elemenary evens (simply called evens): i= r IE( = { IE, IE,, IE } = IE s I ( k + ) I ( (3) OE( = { OE k k 2 kr ki i i i= i= q k, OEk 2,, OEkq} = OEki s Oi ( k + ) Oi ( i= (4) If no elemenary inpu (oupu) even occurs in E(, we denoe i as IE(j)={ε} (OE(j)={ε}) The rising edge even of inpu I i (oupu O i) is denoed as I i_ (O i_) The falling edge even of inpu I i (oupu O i) is denoed as I i_ (O i_) Table shows he elemenary evens compued for he example sequence Even vecor TALE I ELEMENTARY EVENTS LIST FOR EXAMPLE Elemenary inpu evens Elemenary oupu evens E() IE() = {k_} OE() = {A+_} E(2) IE(2) = {a_} OE(2) = {ε } E(3) IE(3) = {k_} OE(3) = {ε} E(4) IE(4) = {a_} OE(4) = {A+_, A _, _} E(5) E(6) E(7) IE(5) = {b_} IE(6) = {a_} IE(7) = {b_} OE(5) = {ε } OE(6) = {ε } OE(7) = {_ } D Direc IE(6) and Indirec = a_ Causaliy Marices As saed in Secion III, he influence of some inpu signals over he oupus seing is observed a he same PLC cycle In order o discover such an inpu/oupu direc relaionship, we analyze he relaive frequency of he occurrence of boh inpu evens IE i and oupu evens OE k, wih respec o he occurrence of OE k along he whole sequence of evens This relaionship can be naurally expressed as he condiional probabiliy of he occurrence of an oupu even OE k, given

7 ha a cerain inpu even IE i has occurred a he same PLC cycle: N Obs ( OEk, IEi ) Prob ( OEk IEi ) = (5) N ( OE ) Obs where N Obs() denoes he number of observed occurrences Using all values Prob(OE k IE i), a marix can be filled We call such a marix he Direc Causaliy Marix (DM), in which every DM ik = Prob(OE k IE i) Fig 6 presens he compued DM marix for he Example, considering a sequence much longer han he presened one k Fig 7 Indirec Conex Marix of he Example Fig 6 Direc Causaliy Marix for he Example Similarly, condiional probabiliy has been used in [7] for deermining he relaionship beween workflow operaions Wih he DM marix, we can find Evoluion Type simply by looking a each column he values ha add up o, since his represens he oal number of occurrences of even OE k For example, from Fig 6 we can discover ha he oupu even A+_ is always provoked by even a_ (in 444% of he observed cases) or by even a2_ (in 556% of he observed cases) The general case where several inpu evens can provoke an oupu even is formalized on he nex secion Similarly, o discover inpu/oupu non direc relaionship, we look a he inpu values when a cerain oupu even occurs We compue he occurrence probabiliy of an oupu even OE k, given ha cerain inpu has a given value IL i a he same PLC cycle: N Obs ( OEk, ILi ) Prob ( OEk ILi ) = (6) N Obs ( OEk ) We consruc he Indirec Conex Marix (IM) in which every IM ik = Prob(OE k IL i) The IM marix for Example is shown in Fig 7 Using he IM marix we can discover evoluion Type 2 by inspecing in every column he values ha add up o which are no zero in he DM marix In he Example, k= and k2= are inpu values which can provoke A+_ oupu even, even if hey were no always observed a he same PLC cycle Now we will presen how hese relaions can be auomaically discovered from he DM and IM marices E Compuing Firing Funcions of Oupu Evens I can be noiced ha he occurrence of every oupu even OE k is caused by one or several inpu evens occurring a he same PLC cycle and by a condiion on he inpu values In order o represen such condiions, a firing funcion χ(oe has o be defined for every OE k I is called he Oupu Even Firing Funcion (OEFF): χ ( OE k ) = G( OEk ) F( OEk ) where G(OE is a funcion of inpu evens and F(OE is a funcion of inpus levels which allow he riggering of he oupu even OE k We compue G(OE as a conjuncion of disjuncions of inpu evens E j: G( OE k ) = ΠDisjE j (8) where each disjuncion DisjE j = (IE x IE z) involves hose variables corresponding o non-zero column values of he DM marix, which add up o, ie hose saisfying condiions: DM xj, DM yj, DM zj (9) DM DM + + DM = () xj + yj zj Similarly, F(OE is compued as a conjuncion of disjuncions of inpu levels L j: F( OE k ) = ΠDisjL j () wih DisjL j = (IL x IL z) such ha IM xj, IM yj, IM yj (2) IM xj + IM yj + + IM zj = (3) DM xj, DM yj, DM yj (4) A he end of he compuing, for every oupu signal O i, we will have he inpu evens and inpu condiions o produce is rising and falling edges O i_ and O i_ respecively This can be easily ranslaed ino IPN fragmens, as shown in Fig 8

8 F(OE j ) G(OE j ) O i OE j = O i _ O i F(OE j ) G(OE j ) OE j = O i _ such condiions ino a unique ransiion, which is labeled by a firing funcion compued from individual firing funcions of each oupu even This is capured in he model as a fusion of IPN fragmens as shown in Fig F Inpu evens wih differed influence on he oupus Noice ha condiion DM xj, DM yj,, DM zj requires ha he inpus relaed o he oupu change were observed a leas once changing is value a he same PLC cycle ha he considered oupu This condiion may be resricive if he inpu-oupu reacion is no observed in he same even vecor For example, in order o avoid componen damages, in he absence of an inpu sensor o indicae ha a pusher has been reraced, here may be some securiy emporizaions which do no allow anoher acuaor reacing a he momen an inpu condiion has been saisfied In such cases, he inpu-oupu reacion would no be found and hus here may be an oupu even wih empy condiions on is firing funcion In order o find he correc OEFF, we can relax he condiion o consider inpu evens which have been observed in previous even vecors insead of he same even vecor Formally, we can compue: NObs ( OEk, IEi ) (previous vecor) Prob ( OEk IEi ) = (5) NObs ( OEk ) u his ime, he compuaion is done by considering IE i occurred a he previous even vecor han OE k A new OEFF can be compued using new values insead of hose of he DM marix If he compued OEFF has sill empy condiions, we can ake he previous o he previous even vecor and successively while empy condiions are compued In he Example, such relaxing condiion is no necessary, since, as i can be noiced, no empy condiions have been compued However, in he experimenal case sudy of Secion VI, such a echnique is applied For he Example, D = ; he compued PN fragmens are shown in Fig 9 (( k k2) a b c) ε A + = ( a_ a2 _) = ( a_ a2 _) A = a _ Fig 8 Rising and falling edges of oupu Oi Fig 9 IPN fragmens for Example = a_ = a2 _ G Fusion of IPN fragmens As saed below, a each PLC cycle, several inpu and sae condiions could lead o he simulaneous occurrence of several oupu evens This behavior is reproduced by merging = b_ C = c_ I( j) E( j) I( j + ) O( j) O( j + ) OE ( j) = OE jp OE jq OE jr λ( T j ) = F( OE( j)) G( OE( j)) Fig IPN represenaion of several oupu evens a he same cycle The consrucion of he observable IPN can be sysemaically done wih he nex procedure: Algorihm 2 Inpu: I/O sequence w, I/O evens sequence E, Marices DM and IM, Differed inpu se D Oupu: Observable incidence marix φc, labeling ransiion funcion λ, and sequence of ransiions S P {p, p 2,, p q} //Creae q observable places, one for every oupu of he sysem 2 S ε //Iniialize he sequence S 3 E( j), j=,, E do //Consider all he compued I/O evens in E 3 If OE(j) = and IE s,,ie u IE(j) D //There is no an oupu change in E(j), bu IE(j) conains elemenary inpu evens IEs,,IEu belonging o D hen 3 T T { j}; λ( j) IE s IE u; W( j,p i), p i P //If i has no been creaed before, creae a new zero ransiion j (a zero column in he incidence marix) represening inpu changes IEs,,IEu 32 S S j //Concaenae j o S 32 else if OE(j) //There is an oupu change in E(j) hen 32 OE jk OE(j) //Consider all he elemenary oupu evens in OE( j) in order o compue G(OE(j)) and F(OE(j)) 32 DisjE i G(OE j, do DisjE i' DisjE i IE jk // Look ino IE(j) he inpu even IEjk which has saisfied DisjEi and assign i o DisjEi' 322 G (OE j ΠDisjE i' //Combine ino G (OEj all he condiions DisjEi' which have saisfied G(OEj 323 G(OE(j)) ΠG'(OE j //Combine ino G(OE(j)) all he inpu even condiions G'(OEj which have saisfied all he evens OEjk 324 DisjL i F(OE j, do DisjL i' DisjL i I(j+) // Looking he I(j+) vecor as a se of oolean variables, save ino DisjLi' he inpu value ILik which has saisfied DisjLi 325 F (OE j ΠDisjL i' //Combine ino F (OEj all he condiions which have made rue F(OEj 326 F(OE(j)) ΠF'(OE j //Combine ino F(OE(j)) all he condiions which have produced all he OEjk OE jp T j OE jq OE jr

9 322 T T { j}, λ( j) = F(OE(j)) G(OE(j)) //If i has no been creaed before, creae a new ransiion j and label i wih he compued F(OE(j)) and G(OE(j)) 323 p i P, do If O q_ OE(j) hen W( j,p q), else If O q_ OE(j) hen W( j,p q) -, else W( j,p jq) //for all elemenary oupu evens in OE(j) = OEjp OEjq OEjr, pu a ino he line corresponding o OEjk if i is a rising even, and a - if i is a falling even; for he res of he lines, assign a 324 S S j //Concaenae j o S Complexiy of Algorihm 2 Le r <r and q <q be respecively he maximum number of inpu and oupu elemenary evens appearing simulaneously in an even vecor, and E be he lengh of he evens sequence The Algorihm 2 processes each one of he evens in E When a ransiion should be added o represen one of such evens, an appropriae firing funcion should be compued If only inpus changed, i is only necessary o include in he firing funcion he elemenary inpu evens wih differed influence This is achieved in O(r ) If here is a leas an oupu change, for each one of he oupu elemenary evens which have occurred, we need search for each individual firing funcion he inpu evens and inpu condiions ha produced he evoluion This is performed in O(q (r log r )) Thus, he complexiy of he procedure for building he ransiion sequence and fragmens is O((q r log r ) E ) Consequenly, he Algorihm 2 can be execued in polynomial ime on he size of he inpu daa Propery The ransiions sequence S is a ranslaion of he I/O sequence w ino ransiion firings of he PN-fragmens buil by Algorihm 2 Proof I is easy o see ha a he seps 32 and 324 of Algorihm 2, S is formed by concaenaing he compued ransiions from he even sequence produced by w This allows ha he reacive behavior can be reproduced in he creaed IPN model Fig shows how evens E() and E(4) are reaed by he algorihm For E() he elemenary oupu even A+_ in OE() is analyzed and funcion λ( ) = (k a b c) (ε) is exraced considering ha k=, a=, b=, and c= are he inpu values which have saisfied χ(a+_) For E(4) all elemenary oupu evens A+_, A-_ and _ in OE(4) are considered and heir Firing Funcions χ(a+_) = (=) (a_ a2_), χ(a-_)=(=) (a_ a2_) and χ(_)=(=) (a_) are combined ino λ( 2) = (=) (a_) Noice ha ineresing labeling funcions have been compued For example, he oupu even A+_ is provoked by he presence of a piece (k= or k2=) and i occurs only when he hree componens corresponding o oupus A+,, and C are on is iniial posiion (a=, b= and c=) A he end of he procedure, he following observable incidence marix ϕw and labeling funcions are obained, as well as he ransiion sequence S, which is he projecion of w over T: S = A + A C IE() = {k_} OE() = {A+_} (( k k2) a b c) ε A + I(2) = k a b c 7 IE(4) = {a_} OE(4) = {A+_, A _, _} A + = ( a_ a2 _) = ( a_ a2 _) A I(5) = a b c λ( ) = ( k a b c) ( ε ) λ( 2 ) = ( = ) ( a_) λ( 3 ) = ( = ) ( b_) λ( 4 ) = ( = ) ( a _) λ( 5 ) = ( k2 a b c) ( ε ) λ( 6 ) = ( = ) ( a2 _) λ( ) = ( = ) ( c_) Fig Treamen of E() and E(4) by he Algorihm 2 The corresponding parial model is shown in Fig 2 The inferring procedure which allows discovering he nonobservable behavior is described in nex secion = a_ 4 A + A Fig 2 Observable IPN model V IDENTIFICATION OF THE NON-OSERVALE EHAVIOR A Problem (re)saemen The previously described procedures allow obaining an observable srucure which represens he reacive behavior of he sysem Given ha evens and ransiions of he ne are compleely defined, we need o add non-observable places o ranslae an aggregaion of he non-observable dynamics of he process in such a way ha he global PN will reproduce he whole behavior of he sysem y adding non-observable places (depiced as grey circles), we make he inference of siuaion Type 3a described in Secion III, which is he case of inpu evens provoking inernal sae evoluions 7 (k a b c) (ε) A + λ( ) = (k a b c) (ε) 2 C 7 A + (=) (a_) A λ( 2 ) = (=) (a_)

10 The problem of deermining he non-observable par of he IPN model complemenary o ha describing he observable (reacive) behavior can be saed as follows Given an observable IPN model whose srucure is G obs =(P obs, T, Pre obs, Pos obs ) and a ransiions sequence S = 2 j T* reproducing he I/O sequence w, an ordinary PN srucure G nobs =(P nobs, T, Pre nobs, Pos nobs ) ha reproduces S and an iniial marking M enabling S mus be found; (G obs, M ) mus be safe Thus, he PN srucure of he complee idenified model is G=(P, T, Pre, Pos) wih P= P obs P nobs, Pre= Pre obs Pre nobs, Pos= Pos obs Pos nobs Observe ha in S here are no consecuive appariions of he same ransiion, due o he naure of he considered evens (rising and falling edges of binary signals) In he lieraure here are many approaches which ackle he idenificaion problem as saed above However, our problem exceeds he hypohesis held in such works or hey are no enough efficien o cope wih long sequences In paricular, a) he sysem cycles are no know a priori, b) he whole language of he sysem is no known, c) he size of S is very large; hus finding efficien algorihms is required, d) he aim is building IPNs ha shows srucural parallelism New places and arcs mus be deermined such ha hey join he IPN fragmens found in he firs par of he mehod Since he asks in differen processes can occur simulaneously or a some predefined order, each wo fragmens can be relaed in wo manners only: sequenially or concurrenly Thus, several connecing forms are possible Some of hem are illusraed in Fig 3, where clouds represen he fragmens Fig 3 Some differen possibiliies for fragmens assembling In his secion, we presen a procedure o build a nonobservable PN srucure ha is able o reproduce he sequence S of ransiions firings This consrucion principle is based on he precedence and concurrency relaions among ransiions, which will deermine he final srucure of he idenified model Algorihm 3 given below provides an overview of he procedure Algorihm 3 Non-observable behavior consrucion Inpu: Transiions sequence S Oupu: Non-observable model represening S Compue basic srucures and relaions (Seq, SP, and TC) from S 2 From he informaion in Seq, SP and TC compue he causal relaion beween ransiions CausalR 3 From Seq and CausalR, compue concurrency relaion (ConcR) beween ransiions 4 uild a PN model represening CausalR and ConcR 5 Verify he okens flow and correc par of he srucure if needed The seps of Algorihm 3 are deailed in he following subsecions Firs some properies derived from he sequence S are inroduced Aferwards, based on such properies, an analysis echnique allowing deermining causal and concurrency relaionships among he ransiions in S is proposed Then, he seps for building a PN srucure observing he causal and concurrency relaionships are presened Dynamical properies Since he consrucion mehod is based on he analysis of causal and concurrency relaionships, some noions mus be defined before inroducing he non-observable behavior consrucion procedure Definiion 6 The relaionship beween ransiions in S ha are observed consecuively is expressed in a relaion Seq T T which is defined as Seq ={( j, j+) j < S } If ( a, b) Seq, his is denoed by a< b Example 2 Le us reuse he sequence S of example, which is he projecion of he observed I/O sequence w over he se of observable ransiions T: S = We can compue Seq = {(, 2), ( 2, 3), ( 3, 4), ( 4, ), ( 2, 4), ( 4, 3), ( 3, 5), ( 5, 6), ( 6, 7), ( 7, 4), ( 4, 5), ( 3, )}, which can be expressed also as { < 2, 2< 3, 3< 4, 4<, 2< 4, 4< 3, 3< 5, 5< 6, 6< 7, 7< 4, 4< 5, 3< } In a PN model every pair in Seq may in fac be represened differenly If a, b were observed consecuively in S, his behavior could be issued from one of wo siuaions in G nobs described in he following definiion Definiion 7 Every couple of consecuive ransiions a, b in Seq can be classified in one of he following siuaions: Causal relaionship If he occurrence of a enables b In a PN srucure, his implies ha here mus be a leas one place from a o b (Fig 4a) Concurren relaionship If boh a and b are simulaneously enabled, bu a occurs firs and is firing does no disable b In an ordinary PN srucure, his implies ha i is impossible he exisence of a place from a o b In his case, a and b are said o be concurren, denoed as a b (Fig 4b)

11 In order o find which is he siuaion occurring beween every pair of ransiions in Seq, some oher definiions are now inroduced The following noion is he sysemaical precedence of a ransiion j wih respec o anoher ransiion k; i esablishes a necessary condiion for j o occur repeaedly Definiion 8 A ransiion j is preceded sysemaically by k, denoed as k j iff k is always observed beween wo appariions of j in S y convenion, we say ha j j if j was observed a leas wice in S Then he Sysemaical Precedence Se of a ransiion j is given by he funcion SP: T 2 T, ha indicaes which ransiions mus be fired o re-enable he firing of j, ie SP( j)={ k k j} If j was observed only once in S, hen SP( j) = In he sequence S from Example 2, one may compue ha, 2, 3, and 4, hus SP( )={, 2, 3, 4} Noice ha SP( j) is he se of ransiions ha mus invarianly occur o fire j repeaedly The res of he SP ses are : SP( 2)={, 2, 3, 4}, SP( 3)={, 2, 3}, SP( 4)={ 4}, SP( 5)={ 4, 5, 6, 7}, SP( 6)={ 4, 5, 6, 7}, SP( 7)={ 4, 5, 6, 7} a b a a) b) Fig 4 Srucures ha represen a< b a) shows a causal relaionship from a o b, whereas b) shows a concurren relaionship beween a and b Definiion 9 Two ransiions a, b are called ransiions in a wo lengh cycle relaionship (named wo-cycle ransiions) if S conains he subsequence a b a or he subsequence b a b The wo-cycle ransiions se TC of S is given by TC={( a, b) a, b are in a wo-cycle} From he sequence S in Example 2, we observe ha he se of ransiions in a wo-cycle is TC= Remark Compuing Seq, SP and TC can be execued in polynomial ime on he size of S We will now exrac some srucural properies regarding N from S The previously defined erms will be used o deermine which siuaion beween causaliy and concurrence is he mos appropriaed for every pair of consecuively observed ransiions in S C Causal and concurrency relaionships ) Causal relaionship In order o deermine ha wo ransiions are causally relaed as shown in Figure 4a, several condiions saed below mus be fulfilled Proposiion If a b ( a SP( b)) hen, here mus exis in N a simple elemenary circui (SE circui) o which boh a and b belong Proof Suppose ha here is no a SE circui conaining a and b Thus, righ afer he firing of b, all he okens in b (he oupu places of b) could be displaced by ransiion firings b hrough some pah o b (he inpu places of b), enabling b wihou needing o fire a, which implies ha a SP( b) Proposiion 2 If a < b and a b, hen here mus exis in N a place from a o b Proof Suppose ha here is no a place from a o b In order o allow he observaion a < b, boh a and b should be enabled simulaneously y Proposiion, here is a leas one SE circui conaining a and b and hus, a leas one pah from a o b Thus, if a and b are enabled simulaneously and a is fired, all pahs from a o b conain wo okens If all ransiions in a pah from a o b are fired, hen here will be wo okens in one of he inpu places of b, resuling in a non-safe ne Then, a leas one of he ransiions i in each pah from a o b mus be condiioned o he previous firing of b u if b is fired, all he ransiions in pahs from a o b can be fired and all he ransiions in pahs from b o b which do no include a can be fired; hus b will be enabled before a fires and as a consequence a SP( b) Proposiion 3 If a < b and b a, hen here mus exis in N a place from a o b Proof Suppose ha here is no a place from a o b Then, before he observaion of a < b, boh a and b mus be enabled, and hus he occurrence of b< a is possible Furhermore, ogeher wih b a and by Proposiion 2 implies ha here should be a place from b o a However, a he firing of b here are wo okens in such a place, and hus he ne is no safe Proposiion 4 If ( a, b) TC, hen here mus exis in N a place from a o b and a place from b o a Proof The sequence a b a mus be reproducible in N Righ afer he firing of a here is a oken on is oupu places, and hus b mus be a he oupu of such places; oherwise, here would be wo okens in such places afer he second firing of a Similarly, righ afer he firs firing of a, here are no okens on is inpu places, and hus b mus be a he inpu of such places; oherwise, a could no be fired again The same reasoning can be applied o reproduce he sequence b a b Noice ha when wo ransiions are observed consecuively and one is sysemaically preceded by he oher, a causal relaionship is found Also, when wo ransiions are involved in a wo-cycle relaion, hey are in a causal relaionship each oher Observe ha all of hese relaionships are srucural, and hus hey do no depend of he iniial marking of he ne Definiion The causal relaionship se CausalR keeps rack of all he causal relaionships in S CausalR = {( a, b) ( a< b) and ( a b or b a or ( a, b) TC)} From he Seq se in Example 2 (see Definiion 6), he SP ses (see Definiion 8) and he TC se (see Definiion 9) we compue CausalR={(, 2), ( 2, 3), ( 4, ), ( 2, 4), ( 5, 6), ( 6, 7), ( 7, 4), ( 4, 5), ( 3, )} If a couple of ransiions ( a, b) in he Seq se, belongs also o CausalR, hen here mus be a place from a o b in order o

12 consrain he observed firing order For he res of he ransiion couples in Seq, we mus decide if a place should exis o relae hem Nex, we will discuss some cases where he exisence of a place can be discarded 2) Concurrency relaionship If wo ransiions a and b are concurren, here mus no exis a place neiher from a o b nor from b o a; oherwise, he firing of one would consrain he firing of he oher one Definiion The se of all pairs of concurren ransiions is called ConcR={( a, b) a b} If he sequence w is complee, (consequenly, S) ie, if i exhibis all of he possible behavior of he observed sysem, we can find concurrence beween ransiions ha are no in a causal relaion, as shown in he nex proposiion Proposiion 5 Le a, b be wo ransiions which have been observed consecuively in a complee sequence S in boh orders, ie ( a, b) Seq, ( b, a) Seq Then ( a, b) CausalR and ( b, a) CausalR if and only if a b Proof Suppose ha ( a, b) ConcR Wihou loss of generaliy, we suppose here is a place p ab from a o b Since ( b, a) Seq, here mus also be a place p ba from b o a; oherwise, a could be enabled simulaneously wih b o allow b< a and a may be fired, yielding o he presence of wo okens in he place p ab and breaking he safeness condiion Since ( a, b) CausalR, b SP( a) and hus here mus be a leas one pah from p ab o p ba which does no conain b Similarly, here mus be a leas one pah from p ba o p ab which does no conain a Since ( a, b) TC, a b a should no be enabled and hus, here mus be a leas one SE circui o which a belongs, bu b does no belong The resuling ne violaes he freechoice condiions (observe Fig 5) However, he obained model would be sill capable o reproduce he sequence S I is well known ha in pracice, he sequence w is no complee, since in he general case, he observed sysems do no show all heir possible behavior during a finie ime of daa collecion In fac, i is no possible o assure ha he whole behavior of a sysem has been observed The consideraion of Proposiion 5 is hen very resricive, since i demands he observaion of all possible behavior; i could lead o he consrucion of incorrec models in case of incomplee sequences Then, some less consraining rules o find concurrence mus be considered Nex, we presen several properies which allow us o idenify couples of ransiions which mus be concurren in he idenified ne N Firs, we will inroduce he noion of Sequenial Independence, which is a characerisic of concurren ransiions Laer, he proposiions o find concurrency will be inroduced Definiion 2 Two ransiions a and b are Sequenially Independen if a SP( b) and b SP( a) From he SP ses of Example 2 (see Definiion 8) we compue he se of Sequenially Independen ransiions: {(, 5), (, 6), (, 7), ( 2, 5), ( 2, 6), ( 2, 7), ( 3, 4), ( 3, 5), ( 3, 6)} Observe he ne in Fig 6 which is composed by wo independen -componens X and X 2 wih suppors <X >= { a, i} and <X 2>= { b, k} respecively In a sequence belonging o he language of such a ne, ransiions belonging o differen - componens are sequenially independen In fac, SP ses of his ne correspond exacly o -componens of he ne p ab a i b k a b Fig 6 A ne wih wo -componens p ba Fig 5 Srucure where (a, b) Seq and (b, a) Seq bu (a,b) ConcR Suppose now ha ( a, b) ConcR Tha means ha hey can be boh enabled simulaneously and one can be fired wihou needing he firing of he oher one, and hus a SP( b) and b SP( a) Also, since here canno be any place from a o b nor from b o a, neiher he subsequence a b a, nor he subsequence b a b can be enabled, and hus ( a, b) CausalR and ( b, a) CausalR Noice ha our mehodology allows compuing also non free-choice nes Only in he case where he sysem includes a behavior like he one shown in Fig 5, he ransiions a and b would be wrongly considered as concurren and he exisence of links from a o p ab and from b o p ba would be missed Proposiion 6 Le a and b be wo ransiions in S which have been observed consecuively in boh orders ( a< b and b< a) If: a) ( a, b) CausalR and ( b, a) CausalR, b) and SP( a) > and SP( b) >, hen a b Proof Suppose ha a and b are no concurren Wihou loss of generaliy, we suppose here is a place p ab from a o b Since b< a has been observed, here mus be also a place p ba from b o a (and as consequence N conains a wo-ransiion cycle); oherwise, a could be enabled simulaneously wih b o allow b< a and a may be fired, yielding o he presence of wo okens in he place p ab and breaking he safeness condiion Since b SP( a), here mus be a leas one pah leading from p ab o p ba no including b Since SP( a) >, here mus be a leas one circui including a and no including p ab, p ba nor b Since a SP( b), here mus be a leas one pah leading from

13 p ba o p ab no including a Consider he firs ransiion x of his pah The free-choice condiions are no saisfied, since x and a share p ba as inpu place, bu a has a leas one differen inpu place We may observe ha for Example 2, ( 3, 4) are sequenially independen (see Definiion 2), however, SP( 4) = and hus we canno infer any concurrence When SP( j) is a singleon, i means ha i belongs o several elemenary circuis and hen Proposiion 6 does no allow o find concurren ransiions o j u if j is included in he SP of oher ransiions, we may find some concurrence relaions, as shown in he nex proposiion Proposiion 7 Le a and b be wo ransiions in S ha have been observed consecuively in boh orders ( a < b and b < a) If a and b a) are Sequenially Independen and b) here exiss a ransiion k such ha a k ( a SP( ) and b k ( b SP( ) hen a b Proof Suppose ha i does no hold ha a b Wihou loss of generaliy, we suppose ha here is a place from a o b Since a SP( and b SP(, afer he firing of k, boh a and b mus be fired before he nex firing of k Since b < a may happen, he place from a o b mus be marked However a < b may occur oo, leading o he presence of wo okens in he same place afer he firing of a, and making he ne no safe Fig 7 shows an example of he case characerized by Proposiion 7 I is he general case of ransiions belonging o concurren hreads ( a, c and b, d, e, f respecively), which are evenually synchronized by one ransiion ( If we make several firings o build a ransiion sequence, evenually he SP ses would become: SP( = { k, a, c, b, d, f}, SP( a) = SP( c) = { k, a, c}, SP( b) = SP( f) = { k, b, d, f}, SP( e) = { e, d}, SP( d) = { d} Even if SP( d) is a singleon, he synchronizaion poin k help us o find by Proposiion 7 several concurren relaionships: a b, a d, a f, c b, c d, and c f In Example 2, ( 3, 4) are Sequenially Independen (see Definiion 2), and we have deermined ha 3 and 4 (see Definiion 8), hus we can conclude ha 3 4 k a b Fig 7 Concurren hreads synchronized by a ransiion d c e f been observed consecuively in boh orders ( a < b and b < a) If a and b are: a) Sequenially Independen, and b) k such ha k SP( b), k SP( a), and c) ( a, Seq hen a b Proof Suppose here is a place p ab from a o b Since b< a has also been observed, here mus be also a place p ba from b o a; oherwise, a should be enabled simulaneously wih b o allow b< a and a may be fired, yielding o he presence of wo okens in p ab Since here exiss k such ha k SP( b), hen here mus be a SE circui conaining boh b and k If such a circui conains places p ba or p ba, i is no possible o fire a< k and hus such a circui mus conain anoher inpu place p kb of b and anoher oupu place p bk of b Now, o accomplish ha b SP( a), here mus be a leas one pah leading from p ab o some inpu place of a no including b Consider he firs ransiion x of his pah In order o respec he free-choice condiions, p kb should be an inpu place of x, making he occurrence of a< k impossible Definiion 3 The Inverse Sysemaical Precedence se of a ransiion SP - : T 2 T conains he ransiions which are dependen of a common ransiion o re-enable heir firing: PS ( j ) = { k k j and j PS( k )} (3) Proposiion 9 Le a and b be wo ransiions which have been observed consecuively in boh orders ( a < b and b < a) If a and b are: Sequenially Independen, and SP - ( a), j SP - ( a), j b, hen a b Proof Suppose here is a place p ab from a o b Since b< a has also been observed, here mus be also a place p ba from b o a; oherwise, a should be enabled simulaneously wih b o allow b< a and hus a may be fired, yielding o he presence of wo okens in he place from a o b Since b SP( a), here mus be a leas one pah leading from p ab o p ba no including b Similarly, here mus be a leas one pah leading from p ba o p ab no including a Since SP - ( a), here is a leas one ransiion j concurren o b such ha j a and here mus be a SE circui including a and j Such a circui canno conain p ab nor p ba oherwise j may be able o fire wihou need of firing a Consider he inpu place p x of a in his pah The freechoice condiions are no saisfied beween p x and p ba: hey share a as oupu ransiion, bu p ba has a leas anoher oupu ransiion An example where Proposiion 9 can be used is shown in Fig 8 SP - ( a) ={ j, j2} and j b, j2 b are deermined by Proposiion 6 Consequenly, a b Remark Compuing CasualR and ConcR can be execued in polynomial ime on he size of S If concurren ransiions do no belong o synchronized hreads, condiions of he nex proposiions help us o find a subse of concurren ransiions which do no depend from anoher ransiion k Proposiion 8 Le be wo ransiions a and b which have

14 j2 a j b The same place could be used o relae several consecuive ransiions If a ransiion k has been observed followed by wo ransiions ai, aj in S ( k< ai and k< aj), here are wo cases o represen such observaions ino he PN model: he case of selecion, where hey are represened wih he same place [ k, ai aj] (Fig 2a) or he case of concurrence, where hey are represened wih differen places [ k, ai] [ k, aj] (Fig 2b) k k Fig 8 Concurrence beween ransiions whose SP is a singleon D uilding he non-observable PN We will use now he compued daa from sequence S o infer inernal evoluions of he sysem We will make an analysis of causal and concurrency relaions ha have been found beween consecuive ransiions in order o compue non-observable places of he ne Definiion 4 The se Seq = (Seq - CausalR) - ConcR conains he se of ransiion pairs ( a, b) which have been observed consecuively, bu are no in a causal relaion or in a concurrency relaion Unil now, we have compued for Example 2 ha Seq = {(, 2), ( 2, 3), ( 3, 4), ( 4, ), ( 2, 4), ( 4, 3), ( 3, 5), ( 5, 6), ( 6, 7), ( 7, 4), ( 4, 5), ( 3, )}, CausalR={(, 2), ( 2, 3), ( 4, ), ( 4, 5), ( 5, 6), ( 6, 7), ( 7, 4), ( 2, 4), ( 3, )} and ConcR = {( 3, 4)( 4, 3)} Thus, Seq = {( 3, 5)} This means ha here is a relaionship which has no been explained If Seq, hen here are wo possibiliies for he remaining ransiion pairs ( a, b) in Seq : b) They are boh inpu and oupu ransiions of a place wih several inpu and oupu ransiions c) They are concurren, bu w (hus, S) is no complee enough o find such a relaionship Since our goal is o approximae as much as possible he language generaed by idenified IPN, o he observed sequence S, we assume ha if we have observed wo ransiions consecuively ( a< b) bu by none of he previous proposiions we have deermined ha hey are concurren, hus he firing of a has enabled b This is made in order o preserve in he PN he firing order observed in S Then, a place will be added from a o b; his denoed by [ a, b] When i is found ha [ a, c] and [ b, c], and he involved ransiions are relaed by a single place, his is represened as [ a b, c] In general, a place p can be denoed as [ a a2 al, b b2 bh], where ai are he inpu ransiions of p and bi are he oupu ransiions of p, and l= p, h= p, as illusraed in Fig 9 a a2 al b b2 bh Fig 9 A PN place p = [a a2 al, b b2 bh] a) b) Fig 2 Selecion and parallelism represenaion a)shows he case where ai, aj are no concurren and have no been observed consecuively whereas b) shows he case where ai, aj are concurren or have been observed consecuively In a generalized form, for every se k< a,, k< aw of nonconcurren consecuive ransiion pairs wih he same firs ransiion k, we can hus merge all k< a,, k< ax whose second ransiions a aw are non-concurren nor consecuive and represen hem ino a single place [ k, a aw], as illusraed in Fig 2 a ai aj a2 aw b bx Once we have made he firs merging, all places [ k, a aw], [ k2, a aw],, [ kz, a aw] whose inpu ransiions are non-concurren nor consecuive and whose oupu ransiions are he same, can be merged ino a single place as illusraed in Fig 22 Remark uilding he non-observable PN can be execued in polynomial ime on he size of Seq k k2 kz k k2 kz E Iniial marking Once he srucure of he ne is buil, he iniial marking can be compued by allowing he firing of S All ransiions are processed, from he las ransiion ill he firs one The processing of a ransiion is as follows: If is oupu places are unmarked, he okens in such places are reired, Tokens are added o is unmarked inpu places Example 2 (Con) y considering he couples of consecuive non-concurren ransiions in Seq (which in his example is only ( 3, 5) see Definiion 4), he places: [, 2] k ai aj b2 c c2 cy Fig 2 A PN place p = [a a2 al, b b2 bh] a a2 aw Fig 22 Selecion and concurrence beween pre-ransiions

15 [ 2, 3] [ 3, 5] [ 4, 5] [ 5, 6] [ 6, 7] and [ 2 7, 4] are compued The PN srucure and he compued iniial marking is shown in Fig 23 p 2 p 3 p 4 5 p p p Fig 23 (G nobs, M) he non-observable IPN of Example p 6 F Token flow verificaion As saed before, wih he proposed mechanisms in las secion, he sequence w may no have shown enough combinaions which allow us o deermine concurrence If he sequence w were complee, all he concurren and sequenial behavior could be found and represened, according o Proposiion 6 However, since we know ha w could no be complee, in order o approximae he language of he idenified IPN o S as much as we can, we have considered ha if wo ransiions have no been declared as concurren, hey mus be in a sequenial relaionship u if he ransiions are acually concurren, he sequenial consideraion could lead us o links or places in he buil model which resric oo much he behavior of he sysem and don allow he firing of S Now, we presen some noions ha will help us o verify if added places unil now do no inerfere in he correc reproducion of S Proposiion If he IPN model has been correcly build, : k a b c 5 : k2 a b c every compued non-observable place p in N mus fulfill he place inpu/oupu flow equaion: A + Occ ( i ) = Occ( i ) ± (6) 2 : a 6 : a2 i p i p A C where Occ( is he number of occurrences of k in S Proof Equaion follows sraighforward from he IPN ransiion enabling and firing condiions and from he fac ha (G nobs, M ) mus be safe Proposiion If here exiss a place p such ha p =, hen j p, k SP( j), SP( j) where k is he inpu ransiion of p Also, if here exiss a place p such ha p =, hen j p, k SP( j), SP( j) where k is he oupu ransiion of p Proof If p =, for he re-enabling of j, p mus be marked and he only way o do so is he firing of k, and hus k SP( j) Similarly, if p =, for he re-enabling of j, p mus be unmarked and he only way o do so is he firing of k, hus k SP( j) Correcion rule If he inpu/oupu flow equaion or he condiions in Proposiion are no saisfied by some place, he arcs relaing ransiions which are no in CausalR are removed If here are no CausalR represened in such a place, i is deleed Example 2 (Con) In he model of Figure 23, we verify he inpu/oupu flow equaion for each place From Example 2, we can compue Occ( )=2, Occ( 2)=, Occ( 3)=, Occ( 4)=2, Occ( 5)=9, Occ( 6)=9, and Occ( 7)=9 We check also he condiion of Proposiion p : Occ( ) = Occ( 2) (±), SP( 2), 2 SP( ) p 2: Occ( 2) = Occ( 3) (±), 2 SP( 3), 3 SP( 2) p 3: Occ( 3) Occ( ) + Occ( 5) (±), 3 SP( ), 3 SP( 5) p 4: Occ( 4) = Occ( ) + Occ( 5) (±), 4 SP( ), 4 SP( 5) p 5: Occ( 5) = Occ( 6) (±), 5 SP( 6), 6 SP( 5) p 6: Occ( 6) = Occ( 7) (±), 6 SP( 7), 7 SP( 6) p 7: Occ( 2) + Occ( 7) = Occ( 4) (±), 4 SP( 2), 4 SP( 7) As can be observed, p 3 is a wrong place, since Occ( 3) Occ( )+ Occ( 5)± Since ( 3, 5) Seq ; his means ha he sequence is no complee, and hus he causal relaionship we assumed beween 3 and 5 is wrong In order o fix his, we can delee he arc going from place p 3 o ransiion 5 Afer his correcion, all of he condiions from Proposiion and Proposiion are saisfied Finally, he idenified IPN of he soring sysem described in Example is obained by merging he observable model in Fig 2 and he non-observable model from Fig 23 afer applying he places correcion We can also delee nonobservable implici places Then he IPN shown in Fig 24, which reproduces w, is he final resul of he model merging In he supplemenary file [26] several addiional examples regarding he mehod for idenifying a non observable model from a sequence S are included 3 : b 4 : a 7 : c Fig 24 Final IPN model for he process in Example G Feaures of he mehod G Reproducibiliy of S Proposiion 2 The PN model (G nobs, M ) buil wih he previous procedures summarized in Algorihm 3 reproduces he sequence S Proof Regard ha we have compued he following ses: Seq conaining all he consecuive ransiion couples in S If we represen ino a ne all couples in Seq, he ne will be able o reproduce S, CausalR conaining ransiion couples ( a, b) Seq ha mus be relaed by a place,

16 ConcR conaining ransiion couples ( a, b) Seq, ha mus no be relaed by any place If he se Seq = (Seq - CausalR) - ConcR =, i means ha all ransiion couples ( a, b) Seq are correcly represened in N and hus he sequence S is reproducible If Seq, i means ha here are sill ransiion couples ha canno be disinguished as concurren or sequenial Thus, by merging several couples in Seq, all couples in Seq are considered as sequenial by creaing places wih several inpu and oupu ransiions If hey are acually sequenial, all he verificaion rules are saisfied Oherwise, hey are acually concurren and hey are correced using he described procedure Once hey are correced, i only remains places relaing sequenial ransiions and hus he sequence S is reproducible G2 Performance Given ha all of he procedures of Algorihm 3 are execued in polynomial ime on S, he consrucion of (G nobs, M ) is efficienly performed Noe also ha he applicaion of Algorihm 3 o a sequence S yields always he same PN model, due o ha all he consrucive seps in he procedures are deerminisically performed, ie here are no random selecions on he inpu and inermediae daa VI METHOD IMPLEMENTATION AND APPLICATION ased on he presened algorihms, a sofware ool has been developed o auomae he IPN model synhesis The archiecure of he ool is shown in Fig 25 Mnemonics, κ User inerface Opions Inpu reader Inpu file I/O vecors Idenificaion Algorihm Fig 25 Sofware archiecure do file Drawer The user inerface allows capuring he inpu/oupu sequence and shows he obained model graphically Following inpu daa is provided o he ool: he name of a ex file conaining he I/O sequence (wih one line per I/O vecor), he names of he inpu and oupu signals, and he desired name for he oupu file Addiionally i is specified he order in which inpus and oupus appear in he x file (since depending on daa collecion procedure, order could change) and he index numbers of he signals o ake ino accoun if a mask is going o be applied (some inpus or oupus could be ignored like indicaor lighs or push-buons) Laer, an inpu reader componen processes he inpu file and ransforms he inpu/oupu sequence ino a vecor IPN sequence These vecors are delivered o a componen called Algorihm in which he idenificaion procedure is implemened The oupu of his componen is an XML file ha can be opened wih he Plaform Independen Peri ne Edior (PIPE [25]), which is an edior for visualizaion and analysis of Peri nes The presened idenificaion ool has been esed on several examples of diverse size and complexiy A small size case sudy regarding an acual manufacuring sysem is described in he supplemenary files o his aricle [26] in which he use of such a sofware ool is illusraed VII CONCLUSION The proposed idenificaion mehod discovers he acual inpu-oupu relaion of PLC conrolled discree even sysems The echnique allows building a concise IPN model in which he ransiions are labeled wih sufficien condiions on he inpus which represen boh he inpu changed and he inpus execuion conex The obained srucure is remarkably more clear and expressive han ha synhesized wih a previous mehod The echnique copes wih complex auomaed DES because i akes ino accoun echnological characerisics of acual conrolled sysems, and because i is based on efficien algorihms This feaure is no sill addressed in curren lieraure on he maer, in which several feaures considered in he curren saed problem have no been deal The algorihms issued from he presen mehod have been implemened as a sofware ool and esed on experimenal case sudies which are very close o acual indusrial discree even processes The performed ess reveal he efficiency of he mehods when daa including housands of inpu-oupu vecors are processed in few seconds Due o his is a black-box approach, he obained models represen he observed behavior; consequenly, when he observaion has been made for a long ime, he idenified IPN approximaes closely he acual behavior Aferwards his model can be compleed using available knowledge on he process REFERENCES [] EM Gold, Language Idenificaion in he Limi, Informaion and Conrol, vol, pp , 967 [2] D Angluin, Queries and Concep Learning, Machine Learning, vol 2, no 4, pp , 988 [3] K Hiraishi, Consrucion of Safe Peri Nes by Presening Firing Sequences, Lecures Noes in Compuer Sciences, vol 66, pp , 992 [4] AP Esrada-Vargas, E López-Mellado, J-J Lesage A Comparaive Analysis of Recen Idenificaion Approaches for Discree-Even Sysems, Mahemaical Problems in Engineering, Vol 2, 2 pages, 2 [5] M P Cabasino, P Darondeau, M P Fani, C Seazu, Model idenificaion and synhesis of discree-even sysems, Conemporary Issues in Sysem Science and Engineering, IEEE/Wiley Press ook Series, M Zhou, H-X Lim M Weijnen (Eds), 23 [6] MP Cabasino, A Giua, and C Seazu, Idenificaion of Peri Nes from Knowledge of Their Language, Discree Even Dynamic Sysems, vol 7, no 4, pp , 27

17 [7] M P Cabasino, A Giua, C Seazu, Linear Programming Techniques for he Idenificaion of Place/Transiion Nes, in Proc IEEE In Conf on Decision & Conrol, pp 54 52, Cancun, Mexico, Dec 28 [8] M Meda-Campaña, and E López-Mellado, Idenificaion of Concurren Discree Even Sysems Using Peri Nes, in Proc 7h IMACS World Congress, pp 7, Paris, France, Jul 25 [9] M Meda, A Ramírez, E López, Asympoic Idenificaion for DES, in Proc IEEE Conf on Decision and Conrol, Sydney, Ausralia, pp , Dec 2 [] S Klein, L Liz, J-J Lesage, Faul deecion of Discree Even Sysems using an idenificaion approach, in Proc 6h IFAC World Congress, pp 6, Praha, Czech Republic, Jul 25 [] M Roh, J-J Lesage, L Liz, lack-box idenificaion of discree even sysems wih opimal pariioning of concurren subsysems, in Proc American Conrol Conference, pp , alimore, Maryland, USA, Jun 2 [2] M P Cabasino, A Giua, C N Hadjicosis, and C Seazu, Faul Model Idenificaion and Synhesis in Peri Nes, Discree Even Dynamic Sysems, vol 24, no 3, pp , 24 [3] S Ould El Medhi, E Leclercq, D Lefebvre, Peri nes design and idenificaion for he diagnosis of discree even sysems, in Proc 26 IAR Annual Meeing, Nancy, France, Nov 26 [4] M Dooli, M P Fani, and A M Mangini, Real ime idenificaion of discree even sysems using Peri nes, Auomaica, vol 44, no 5, pp 29 29, May 28 [5] M Dooli, M P Fani, A M Mangini, and W Ukovich, Idenificaion of he unobservable behaviour of indusrial auomaion sysems by Peri nes, Conrol Engineering Pracice, vol 9, no 9, pp , Sep 2 [6] S Ould El Mehdi, R ekrar, N Messai, E Leclercq, D Lefebvre, Riera, Design and Idenificaion of Sochasic and Deerminisic Sochasic Peri Nes, IEEE Trans on Sysems, Man and Cyberneics, Par A, vol 42, no 4, pp [7] J E Cook, Z Du, C Liu, A L Wolf, Discovering models of behavior for concurren workflows, Compuers in Indusry, vol 53, no3, pp , 24 [8] W van der Aals, T Weijers, L Maruser, Workflow Mining: Discovering Process Models from Even Logs, IEEE Trans on Knowledge and Daa Engineering, vol 6, no 9, Sep 24 [9] A P Esrada-Vargas, J-J Lesage, E López-Mellado, A Sepwise Mehod for Idenificaion of Conrolled Discree Manufacuring Sysems, In Journal of Compuer Inegraed Manufacuring, vol 28, no 2, pp 87-99, 24 [2] A P Esrada-Vargas, E López-Mellado, J-J Lesage, Inpu-Oupu Idenificaion of Conrolled Discree Manufacuring Sysems, In Journal of Sysems Science, vol 45, no 3, pp , 24 [2] A P Esrada-Vargas, J-J Lesage, E López-Mellado, Idenificaion of Indusrial Auomaion Sysems: uilding Compac and Expressive Peri Ne Models from Observable ehavior, in Proc American Conrol Conference, pp 695 6, Monréal, Canada, Jun 22 [22] A P Esrada-Vargas, E López-Mellado, J-J Lesage, Idenificaion of Parially Observable Discree Even Manufacuring Sysems in Proc IEEE Inernaional Conference on Emmerging Technologies and Facory Auomaion, pp-7, Cagliari, Ialy, Sep 23 [23] R David and H Alla, Peri Nes for Modeling of Dynamic Sysems A Survey, Auomaica, vol 3, no 2, pp 75 22, 994 [24] M Roh, J-J Lesage, and L Liz, Idenificaion of Discree Even Sysems, implemenaion issues and model compleeness, in Proc 7h In Conf on Informaics in Conrol Auomaion and Roboics, pp 73 8, Funchal, Porugal, Jun 2 [25] PIPE 2: Plaform Independen Peri ne Edior 2, hp://pipe2sourceforgene/ [26] Supplemenary file Examples and Case sudy: hp://wwwgdlcinvesavmx/lack-box-iden-supplzip iographies Ana-Paula Esrada-Vargas received he Sc degree in compuer engineering from he Universidad de Guadalajara, Guadalajara, Mexico, in 27, and he MSc degree from CINVESTAV, Guadalajara, Mexico, in 29 She obained he PhD degree from boh CINVESTAV in Guadalajara and he ENS de Cachan, in Cachan, France in 23 Currenly, she belongs o he Oracle Semanic Technologies eam in Mexico Developmen Cener Her research ineress include idenificaion of Discree Even Sysems, formal modelling and analysis using Peri nes, as well as Semanic Web echnologies Erneso López-Mellado received he Sc degree in elecrical engineering from he Insiuo Tecnologico de Cd Madero, México, in 977, he MSc degree from he CINVESTAV, México Ciy, México, in 979, and he Doceur-Ingénieur degree in auomaion from he Universiy of Toulouse, France, in 986 Currenly, he is Professor of Compuer Sciences a CINVESTAV Unidad Guadalajara, Guadalajara, Mexico His research ineress include discree even sysems, and disribued inelligen sysems Jean-Jacques Lesage received he PhD degree from he Ecole Cenrale de Paris and he Habiliaion à diriger des recherches from he Universiy Nancy in 989 and 994 respecively He is currenly Professor of Auomaic Conrol a he Ecole Normale Supérieure de Cachan, France, where he was head of he Auomaed Producion Research Laboraory during eigh years His research ineress are in he field of formal mehods and models for synhesis, analysis and diagnosis of Discree Even Sysems (DES), and applicaions o manufacuring sysems, nework auomaed sysems, energy producion, and more recenly ambien assised living