CODIAGNOSABILITY OF NETWORKED DISCRETE EVENT SYSTEMS SUBJECT TO COMMUNICATION DELAYS AND INTERMITTENT LOSS OF OBSERVATION. Carlos Eduardo Viana Nunes

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1 CODIAGNOSABILITY OF NETWORKED DISCRETE EVENT SYSTEMS SUBJECT TO COMMUNICATION DELAYS AND INTERMITTENT LOSS OF OBSERVATION Crlos Edurdo Vin Nunes Tese de Doutordo presentd o Progrm de Pós-grdução em Engenhri Elétri, COPPE, d Universidde Federl do Rio de Jneiro, omo prte dos requisitos neessários à otenção do título de Doutor em Engenhri Elétri. Orientdores: João Crlos dos Sntos Bsilio Mros Viente de Brito Moreir Rio de Jneiro Outuro de 2016

2 CODIAGNOSABILITY OF NETWORKED DISCRETE EVENT SYSTEMS SUBJECT TO COMMUNICATION DELAYS AND INTERMITTENT LOSS OF OBSERVATION Crlos Edurdo Vin Nunes TESE SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA (COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE DOUTOR EM CIÊNCIAS EM ENGENHARIA ELÉTRICA. Exmind por: Prof. João Crlos dos Sntos Bsilio, Ph.D. Prof. Mros Viente de Brito Moreir, D.S. Prof. Ptríi Nsimento Pen, D.S. Prof. José Edurdo Rieiro Cury, Doteur d Ett Prof. Antônio Edurdo Crrilho d Cunh, D.Eng. RIO DE JANEIRO, RJ BRASIL OUTUBRO DE 2016

3 Nunes, Crlos Edurdo Vin Codignosility of Networked Disrete Event Systems sujet to ommunition delys nd intermittent loss of oservtion/crlos Edurdo Vin Nunes. Rio de Jneiro: UFRJ/COPPE, XII, 102 p.: il.; 29, 7m. Orientdores: João Crlos dos Sntos Bsilio Mros Viente de Brito Moreir Tese (doutordo) UFRJ/COPPE/Progrm de Engenhri Elétri, Referênis Biliográfis: p Disrete Events Systems. 2. Communition network. 3. Filure dignosis. I. Bsilio, João Crlos dos Sntos et l.. II. Universidde Federl do Rio de Jneiro, COPPE, Progrm de Engenhri Elétri. III. Título. iii

4 Stisftion lies in the effort, not in the ttinment, full effort is full vitory. Mhtm Gndhi iv

5 Aknowledgments First, I thnk God, retor of ll things, y the infinite love deposited in ll humnity nd y the opportunity to e on Erth in order to lern nd evolve. I thnk Jesus, y his tehings of love nd humility tht guide ll humnity to the pth of good. I thnk my prents Avny nd Hélio, y their unonditionl support nd euse they provide me ll love nd edution neessry to live with dignity. I thnk my rothers Hélio Jr, André, nd Dniel, y their support nd wonderful hppy moments tht they lwys provide me. I thnk my wife Crin, wonderful womn tht help me whenever I need. Thnk you der y trust nd love tht you lwys hve for me. I thnk deeply my dvisors João Crlos Bsilio nd Mros Viente Moreir for trusting me in exeution fo this thesis nd y ptiene nd dedition to teh me. A speil thnk to my friend Mros Viniius, tht provided me his time to disuss mny spets of this thesis nd y his help whenever I needed. I nnot forget the friends tht, in some wy, help me during ll dotorl ourse: Cristino Crvlho, Dyro Brhon, Gustvo Vin, Ingrid Antunes, Felipe Crl, Lilin Kwkmi, Leonrdo Bermeo nd Félix Gmrr. Thnks everyone. To CNPq y finnil support. Crlos Edurdo Vin Nunes v

6 Resumo d Tese presentd à COPPE/UFRJ omo prte dos requisitos neessários pr otenção do gru de Doutor em Ciênis (D.S.) CODIAGNOSTICABILIDADE DE SISTEMAS A EVENTOS DISCRETOS EM REDE SUJEITOS A ATRASOS DE COMUNICAÇÃO E PERDA INTERMITENTE DE OBSERVAÇÃO Crlos Edurdo Vin Nunes Outuro/2016 Orientdores: João Crlos dos Sntos Bsilio Mros Viente de Brito Moreir Progrm: Engenhri Elétri No dignóstio de flhs de Sistems Eventos Disretos distriuídos, é usulmente onsiderdo que n omunição entre os dispositivos não há perds nem trsos n omunição d oorrêni de eventos pr os dignostidores. No entnto, os nis de omunição reis são sujeitos trsos e perds intermitentes de potes que podem levr o dignostidor oservr eventos for d ordem de oorrêni, proporionndo um inorreto dignostio d flh. Neste trlho, investigmos odignostiilidde de um sistem em rede om trsos de omunição e perds intermitentes de oservção. Introduzimos definição de odignostiilidde em rede ontr trsos de omunição e perds intermitentes de oservção, presentmos um ondição neessári e sufiiente pr odignostiilidde em rede e propomos um lgoritmo pr verifição dest propriedde. Plvrs-hve: Sistems eventos disretos, dignóstio de flhs, omunição em rede. vi

7 Astrt of Thesis presented to COPPE/UFRJ s prtil fulfillment of the requirements for the degree of Dotor of Siene (D.S.) CODIAGNOSABILITY OF NETWORKED DISCRETE EVENT SYSTEMS SUBJECT TO COMMUNICATION DELAYS AND INTERMITTENT LOSS OF OBSERVATION Crlos Edurdo Vin Nunes Otoer/2016 Advisors: João Crlos dos Sntos Bsilio Mros Viente de Brito Moreir Deprtment: Eletril Engineering In filure dignosis of networked Disrete Event Systems, it is usully ssumed tht ommunition mong devies is lossless nd without dely in the ommunition of event ourrenes to the dignosers. However, rel ommunition hnnels re sujet to trnsporttion delys nd intermittent loss of pkets whih n mke the dignosers oserve events out of their order of ourrene, leding to n inorret fult dignosis. In this work, we investigte the odignosility of networked systems with ommunition delys nd intermittent losses of oservtion. We introdue the definition of network odignosility ginst ommunition delys nd intermittent losses of oservtion, present neessry nd suffiient ondition for network odignosility nd propose n lgorithm for the verifition of this property. Keywords: Disrete event systems, filure dignosis, network ommunition. vii

8 Contents List of Figures List of Tles x xii 1 Introdution 1 2 Disrete Event Systems: theory nd fundmentls Modeling of disrete event systems Lnguges Lnguge of Disrete Event Systems Opertions on lnguges Automt Deterministi utomt Nondeterministi utomt Oserver utomt Opertions on utomt Filure dignosis Algorithms of filure dignosis Filure dignosis of DES Dignoser Centrlized dignosis Deentrlized dignosis Codignosility verifition Finl remrks viii

9 3 Communition Networks Sujet to Delys nd Losses Communition networks OSI model Ciruit swithing nd pket swithing Components of network Delys nd losses Speifition of ommunition network Finl remrks Codignosility of Networked Disrete Event Systems Prolem formultion Model of the plnt sujet to ommunition delys Modeling of intermittent loss of events Model of the plnt sujet to ommunition delys nd intermittent loss of oservtions Definition of network odignosility of disrete-event systems Verifition of network odignosility of disrete-event systems Complexity nlysis of Algorithm Conluding Remrks Conlusion nd Future Works 92 Biliogrphi Referenes 94 ix

10 List of Figures 2.1 Automton G Nondeterministi utomton Nondeterministi utomton with ɛ-trnsition Automton G Oserver utomton of G Automton H Coessile prt of utomton H Automton T im(h) Comp[T rim(h)] A generl dignosis frmework [1] Clssifition of dignosis lgorithms [1] Lel utomton A l for uilding dignoser () Automton G () Prllel omposition etween G nd A l, () G d = Os(G A l ) Codignosis struture Automton G of exmple Automton A N l Automton G N Automton G F Automton G R Automton G R Verifier utomton G V OSI model Resume nd pplition of OSI model x

11 3.3 Types of delys of network ommunition Convoy nlogy Peer-to-peer topology Bus topology Ring topology Str topology Tree topology Simplex mode Hlf duplex mode Full duplex mode Network deentrlized dignosis rhiteture Automton G Network odignosis sheme of exmple Constrution of utomton D 1 step-y-step Automton D () Automton G 1 ; () Automton G Automton H of Exmple Automton H dil () utomton G 1; ()utomton G () utomton Ḡ 1; ()utomton Ḡ () Automton G 1,ρ, () Automton G 1,F () Automton G 2,ρ, () Automton G 2,F () Pth of V 1 with yli pth l 1 emedded, () pth of V 2 with yli pth l 2 emedded Pth of G V with n emedded yli pth l tht violtes the network odignosility of L () Automton Ḡ 1,ρ, () Automton Ḡ 1,F () Automton Ḡ 2,ρ, () Automton Ḡ 2,F () Pth of V 1 with yli pth l 1 emedded, () pth of V 2 with yli pth l 2 emedded Prt of verifier ḠV xi

12 List of Tles 2.1 Sttes nd events of the DES omposed of mhines M 1, M 2 nd root xii

13 Chpter 1 Introdution Industril systems re eoming more omplex with the dvne of tehnology, thus, filure dignosis in omponents of these systems eomes omplex tsk to e solved y using only the experiene nd knowledge of the opertor of the system. In this ontext, the improvement of utomti filure dignosis systems eomes n importnt re to e developed. The importne of reserh in the re of filure dignosis is refleted in the numer of works pulished in interntionl onferenes. In [2], it is presented the sttistis of pulished works in WODES (Workshop on Disrete Event Systems) nd DCDS (Workshop on Dependle Control of Disrete Systems) etween nd , respetively. In WODES, round 12% of pulished works re relted with filure dignosis nd in DCDS this sttisti is 22%. Furthermore, there re other ontriutions to fult dignosis of DES tht hve een pulished in ontrol journls, suh s Automti, IEEE Trnstions on Automti Control, Control Engineering Prtie, nd others. Sine the first pulitions tht ddressed the prolem of filure dignosis in Disrete Event Systems (DES) [3, 4, 5], in whih the fundmentl onepts of dignosility of DES were presented, mny issues relted to this prolem were presented in the literture suh s: (i) filure predition, (ii) seletion of sensors nd dynmi tivtion nd, (iii) roust dignosis. In (i), the prolem is to predit the ourrene of filure events sed on the oservtion of events nd then prevent these filure event ourrene [6, 7, 8, 9, 10, 11, 12, 13]. In (ii), the prolem is to ensure the dignosility of the lnguge of the system with respet to set of oservle 1

14 events with smllest rdinlity, thus, minimizing the ost of the dignosis system ssoited with the use of sensors [14, 15, 16, 17, 18, 19, 20] nd, finlly, in (iii), the prolem is to detet the ourrene of unoservle fult events using set of sensors tht themselves re sujet to filures, suh s, intermittent or permnent mlfuntion [21, 22, 23, 24, 25, 26, 27, 28, 29]. The prolem of filure dignosis ws lso onsidered in [30, 31, 32, 33, 34, 35, 36], with severl pplitions in [37, 38, 39, 40, 41, 42], where it is ssumed tht ommunition etween the sensors of the system nd the dignoser is perfetly relile nd the sensors work perfetly, i.e., there is neither loss of oservtion nor delys in the ommunition of the events to the dignosers. It is relisti to ssume tht if dignoser nd plnt ommunite vi dedited ommunition link, lso lled point-to-point link [43, 44] sine there is wire for eh sensor or tutor point tht onnets them to the entrl ontrol omputer. This kind of ommunition is omplex nd expensive nd the whole system is diffiult to mintin nd dignose due to the lrge numer of onnetors nd les [45]. Thus, due to the omplexity of the plnts, the dignosers re often implemented in distriuted wy nd, onsequently, with the development of network tehnology, there is trend in industries to implement ommunition systems y using shred (wired or wireless) ommunition networks. In shred network, ommunition prolems suh s loss of oservtion of events or ommunition delys re unvoidle [44]. In [46], [26] nd [27], the prolem of loss of oservtion is ddressed onsidering the mlfuntioning of sensors of the plnt. In [46], it hs een developed proilisti methodology for filure dignosis in finite stte mhines sed on sequene of unrelile oservtions, nd in [26, 27], it hs een developed deterministi methodology for filure dignosis in Disrete Event Systems. In these works, the prolem of ommunition delys etween sensors nd dignosers ws not onsidered. In networked filure dignosis systems, ommunition dely n mke the dignoser to reeive signls out of the originl order of ourrene, nd the dignoser n erroneously detet the ourrene of filure event. One wy to solve this prolem is to insert loks nd time stmps in the ommunition protool. Thus, the protool will e le to reorgnize the events y using the times inserted in the 2

15 pk of trnsmission of eh event. However, in order to dd time informtion in the ommunition protool, it is neessry to synhronize the loks of the devies, whih is not simple tsk, sine eh devie must hve extly the sme time. Another prolem is mintenne euse the synhroniztion proess must e exeuted periodilly on the whole plnt, whih inreses the ost of implementtion [47]. In the literture there re some works out DES with network ommunition ddressing the prolem of ommunition delys, nd most of them re relted to supervisory ontrol, [48, 49, 50, 51, 52, 53]. The prolem of filure dignosis of DES with ommunition dely is ddressed in [47, 54, 55, 56]. The prolem of deentrlized supervisory ontrol with ommunition delys is presented in [48]. Two types of delys re defined: unounded dely nd ounded dely y onstnt k N (k-ounded dely). In the unounded dely pproh, the plnt n exeute ny numer of events etween the trnsmission of the ourred event nd its reeption y the supervisor. On the other hnd, in the k-ounded dely pproh, the plnt n exeute t most k events etween the trnsmission nd reeption of n event y the supervisor. In [48], it is ssumed tht the ommunition is lossless, tht is, ll messges re eventully delivered within finite dely nd there is no hnge of order in the oservtion y the supervisor. The prolem of entrlized supervisory ontrol with ommunition delys nd prtil oservtions is ddressed in [49], where nonloking supervisor is designed to reh speifition even if there re ommunition delys etween plnt nd supervisor. It is importnt to remrk tht there is no hnge of order mong events in [49]. The prolem of deentrlized supervisory ontrol with ommunition delys sed on onjuntive nd permissive struture is ddressed in [50] ssuming tht unontrollle events n our unexpetedly efore ny ontrol tion tkes ple on the plnt, in order to find onditions of existene of nonloking deentrlized supervisor. The notion of dely o-oservility for given speifition is lso presented, nd it is shown tht o-oservility is the key ondition for the existene of deentrlized supervisor tht n reh the speifition. A strtegy of ontrol for system with network ommunition is proposed in [51], whose gol is to ompute supervisor tht is tolernt to delys nd events oservtion losses. In [51], it is ssumed tht supervisor nd plnt re networked, 3

16 nd there re two hnnels onneting oth: oservtion hnnel nd ontrol hnnel, nd, in oth hnnels, ommunition delys nd loss of informtion n our. The ommunition delys re ssumed to e k-ounded, s in [48], nd there is no hnge of order of event oservtion. As onsequene, n event tht hs ourred in the plnt my not e seen y the supervisor until the ourrene of the next event in the plnt, nd the supervisor my not e ple of disling this event. In [52], the work presented in [51] is ontinued, ssume the sme ssumptions s in [51]. Bsed on these ssumptions, the uthors show how to solve the prolem of entrlized ontrol of Disrete Event Systems with network ommunition. In [53] the prolem ddressed in [52] is extended to deentrlized supervisory ontrol of networked systems. Control poliies re defined for ll lol supervisor in order to stisfy the design speifitions. In [47], the prolem of deentrlized filure dignosis with ommunition delys is ddressed onsidering protools 1 nd 2 of [30], nd ssuming tht there re ommunition delys etween lol dignosers nd the oordintor, whih is responsile to reorder the events nd to infer the ourrene of filure events. It is lso ssumed in [47] tht eh ommunition hnnel tht onnets dignoser nd the oordintor is FIFO (Fist-in-First-out), so tht, only events tht ome from different dignosers n e oserved in order different from tht exeuted in the plnt. In [54], the prolem of distriuted filure dignosis in DES sujet to ommunition delys is presented. It is ssumed tht, the dignosis system does not hve oordintor nd, eh lol dignoser sends its oservtion to other lol dignosers. As in [48], the oservtion dely of n event σ tht ours in the plnt is ounded. In [54], it is ssumed tht the ommunition dely tkes ple etween lol dignosers. Moreover, it is ssumed tht oth ommunition hnnels tht onnet the lol dignosers hve the sme mximum dely, the ommunition is lossless nd first-in-first-out (FIFO), i.e, there is no hnge of order on the event oservtions tht re trnsmitted through the sme hnnel. The prolem of deentrlized filure dignosis with ommunition delys is ddressed in [55] onsidering protool 3 of [30], i.e., (i) there is no ommunition etween the lol dignosers; (ii) eh lol dignoser infers the ourrene of the 4

17 filure event sed on its own oservtions, nd; (iii) the filure event is dignosed when t lest one of the lol dignosers identifies its ourrene. It is ssumed in [55] tht there exist ommunition delys etween mesurement sites nd lol dignosers, resulting in n out of order oservtion, y lol dignosers, of the events exeuted y the system. The prolem of loss of oservtion ws not ddressed in [55]. In this work, we extend the prolem onsidered in [55] to lso tke into ount loss of oservtion. Sine the ommunition hnnels re independent, the delys generted y eh hnnel n e different. In this work, we ssume tht the delys re k-ounded s presented in [48]. Bsed on tht, we present the definition of network odignosility ginst ommunition delys nd loss of oservtion. Moreover, we propose n utomton model tht desries ll possile delys tht the system is sujet nd, onsequently, ll possile orders of oservtion y the lol dignosers. We lso propose n lgorithm for the verifition of network odignosility ginst ommunition delys nd loss of oservtion, nd we show tht the pproh presented in [54] n e onsidered s prtiulr se of the prolem formulted in this work. This thesis is orgnized s follows. In hpter 2, the fundmentl onepts of DES modeled y utomt inluding the notion of filure dignosis re presented. We lso review the verifier lgorithm proposed in [57]. In hpter 3, we present the min resons of delys nd losses in ommunition networks. In hpter 4, we introdue the definition of network odignosility ginst ommunition delys nd loss of oservtion, present neessry nd suffiient ondition for network odignosility nd propose n lgorithm for its verifition. The finl remrks nd suggestions of future reserh works re presented in hpter 5. 5

18 Chpter 2 Disrete Event Systems: theory nd fundmentls Disrete Event Systems (DES) re dynmi systems with disrete sttes spe where the trnsitions etween sttes re mde y the ourrene, in generl synhronous, of disrete events. The ft tht the sttes of the systems re disrete implies tht it n ssume symoli vlues, for exmple, (on, off), (green, yellow, red), or numeri vlues tht elong to sets N or Z, or e formed y suset of enumerle elements of R. Events n e ssoited to speifi tions (for exmple, someone presses utton, n irrft tkes off, et), or e the result of severl onditions tht re stisfied (e.g, n ojet rehes point of prodution line, liquid tht rehes determined height). Although it is possile to model ny physil system s DES ording to some level of strtion tht we n onsider, some systems re nturlly disrete nd their evolution re determined y the ourrene of events. This hpter ims to present the si definitions nd notions out DES nd intends to introdue the min onepts on DES for novie reders. The theory presented is sed on [58]. This hpter is strutured s follows: in Setion 2.1, we present the min formlisms for DES. In Setion 2.2, some definitions nd onepts relted to DES re ddressed. In Setion 2.4, we introdue the si onepts out filure nd filure dignosis in DES. Finlly, in Setion 2.5 the fundmentl onepts out entrlized dignosers, deentrlized dignosers nd the verifier proposed in [57] re reviewed. 6

19 2.1 Modeling of disrete event systems The model of DES must e ple of reproduing, within some prespeified tolerne, the ehvior of the system. In ontinuous vrile dynmi systems (CVDS), the sttes re expressed s funtions of time, while in DES, the system ehvior is desried in terms of tres of events. All tres tht n e generted y given DES desrie the lnguge of this system, whih is defined over set of events (lphet) of the system. Thus, when we onsider the evolution of sttes of DES, the min onern is the tre ssoited with visited sttes nd with the events tht use the orresponding stte trnsitions, i.e. the model of DES onsists silly of two elements: stte nd trnsition. The DES models tke into ount some hrteristis in order to represent determined system: The model greement with relity regrding the purpose of the model. The omplexity of the model. In the literture severl formlisms to model DES re presented, whose hoie is relted to the ury nd omplexity. The model formlisms more frequently used in the literture re: Automt; Petri nets; Hyrid utomt. The utomton modeling hs the following dvntges: severl prolems found in industries n e modeled with it, the omputtionl implementtion is simple nd, lgorithms nd opertions mong utomt re well known. On the other hnd, simultneous tsks re etter modeled y Petri nets. In order to illustrte the modeling of system, onsider the following exmple [59]. Exmple 2.1 Consider mnufturing ell omposed of two mhines (M 1 nd M 2 ) nd root whih trnsports prts from M 1 to M 2. Mhine M 1 7

20 reeives rw prts, nd, proesses them. After eing proessed, they re olleted y the root. In se the root is usy, the mhine M 1 holds the prt until the root eomes ville. In se nother prt rrives while mhine M 1 is proessing some prt, it rejets the prt. When the root tkes prt from M 1, it trnsports it to M 2. When the prt rrives t M 2, the root will deliver it to M 2, only if M 2 is free. Otherwise, the root holds the prt until M 2 eomes free. After the root delivers the prt to M 2, it returns to M 1. Finlly, mhine M 2 reeives the prt from the root nd proesses it. Tle 2.1 desries the sttes nd events of mhines M 1, M 2 nd the root. Notie tht, events e 1 (prt delivered to root) nd 2 (prt delivered to M 2 ) elong to two susystems: mhine M 1 nd root, nd root nd mhine M 2, respetively. It is importnt to notie tht, in order to event e 1 to our, mhine M 1 must e in stte H 1 nd the root in stte I; in order to event 2 to our, the root must e in stte H nd the mhine M 2 must e in stte I 2. For other sttes of the system, i.e., those tht re in only one of the susystems, the ourrene is determined only y the urrent stte of the susystem; for exmple, the ourrene of event t 1 (end of proessing) depends only on mhine M 1 to e in stte P 1, independently of whih sttes the root nd mhine M 2 re. 2.2 Lnguges Lnguge of Disrete Event Systems One of the forml wys to study the logil ehvior of DES is sed on the theories of lnguge nd utomt. The strting point is the ft tht ny DES hs n ssoited event set Σ. The event set Σ is the lphet nd the tres re the words of lnguge. We will ssume tht Σ is finite. The length of word is the numer of events it ontins. We denote the length of tre s y s. The word tht does not ontin events is lled the empty tre, nd is denoted y ɛ. The length of the empty tre ɛ is zero. Definition 2.1 (Lnguge) A lnguge defined over n event set Σ is set of tres (words) with finite length formed y events of Σ. 8

21 Tle 2.1: Sttes nd events of the DES omposed of mhines M 1, M 2 nd root. Element Sttes Events M 1 ville: I 1 Arrivl of prt to M 1 : 1 Mhine M 1 M 1 proessing: P 1 End of proessing: t 1 M 1 holding proessed prt: H 1 Prt delivered to root: e 1 X 1 = {I 1, P 1, H 1 } E 1 = { 1, t 1, e 1 } Root ville: I Prt delivered to root: e 1 Root Trnsporting M 1 M 2 : T 12 Arrivl t M 2 : 2 Witing t M 2 : H Prt delivered to M 2 : 2 Returning to M 1 : R Root returned to M 1 : r 1 X r = {I, T 12, H, R} E r = {e 1, 2, 2, r 1 } Mhine M 2 M 2 proessing: P 2 End of proessing: t 2 M 2 ville: I 2 Prt delivered to M 2 : 2 X 2 = {I 2, P 2 } E 2 = { 2, t 2 } Exmple 2.2 Let Σ = {,, g} e set of events. Lnguge L 1, defined s L 1 = {ɛ,, }, onsists of only three tres. Lnguge L 2 tht ontins ll possile tres of length 3 tht strt with event, n lso e listed, nmely L 2 = {,, g,, g,, gg, g, g}. Let us denote y Σ the set of ll finite tres formed with events σ Σ, inluding the empty tre ɛ. Σ is lso referred to s the Kleene-losure of Σ. Notie tht the set Σ is ountly infinite sine it ontin tres of ritrrily long length. For exmple, if Σ = {,, }, then Σ = {ɛ,,,,,,,,,,,,,...}. A lnguge L defined over n event set Σ is, therefore, suset of Σ. The key opertion to onstrut tres, is the ontention. The tre in Exmple 2.2 is the ontention of tre with event ; onsequently, tre is the ontention of events nd. The empty tre ɛ is the identity element of ontention, i.e., ɛu = uɛ = u for every tre u. 9

22 Some terminology out tres n e onsidered. If tuv = s with t, u, v Σ, then t is prefix of s, u is sutre of s nd v is suffix of s Opertions on lnguges The usul set opertions, suh s union, intersetion, differene, nd omplement with respet to Σ, re pplile to lnguges sine lnguges re sets. Besides these opertions, four other opertions n e defined for lnguges: ontention, prefix-losure, Kleene-losure nd nturl projetion. Contention: Let L, L Σ, then L L := {s Σ : (s = s s ) (s L ) (s L )}. In words, tre is in L L if it n e written s the ontention of tre in L with nother tre in L. Prefix-losure: Let L Σ, then L := {s Σ : ( t Σ )[st L]}. In words, the prefix losure L L is formed y ll prefixes of ll tres in L. Kleene losure: Let L Σ, then L := {ɛ} L LL LLL... This is the sme opertion s tht defined for set Σ, exept, now, tht it is pplied to set L whose elements my e tres of finite length. Nturl projetion: The nturl projetion, or simply projetion, is mpping from set of events, Σ l, to smller set of events, Σ s, where Σ s Σ l nd, is denoted y P ; susript is usully dded to speify either Σ s or oth Σ l nd Σ s, for the ske of lrity, when deling with multiple sets. We extended the projetion opertion to tres s follows: P : Σ l Σ s s P (s) 10

23 with the following properties P (ɛ) = ɛ σ, if σ Σ s P (σ) = ɛ, if σ Σ l \ Σ s P (sσ) = P (s)p (σ), s Σ l, σ Σ l. Aording to the previous definition, the projetion opertion erses the events of s Σ l tht do not elong to the smller event set, Σ s. This opertion n e used to otin the oserved lnguge of system. We n lso work with the orresponding inverse mpping, lled inverse projetion, whih is defined s follows: P 1 : Σ s 2 Σ l s P 1 (s) = {t Σ : P (t) = s} The projetion P n e extended to lnguge L Σ l the tres in L. Thus, y pplying it to ll P (L) = {t Σ s : ( s L)[P (s) = t]}. The inverse projetion n lso e extended to lnguge L s Σ s s follows: [58]. P 1 (L s ) = {s Σ l : ( t L s )[P (s) = t]}. In order to illustrte the notion of projetion, onsider the following exmple Exmple 2.3 Consider set Σ = {,, } nd the two proper susets Σ 1 = {, } nd Σ 2 = {, } of Σ. Consider, in ddition, the following lnguge: L = {,,,, } Σ Let the projetions P i : Σ Σ i, i = 1, 2. Thus, P 1 (L) = {ɛ,,, } P 2 (L) = {,,, } P 1 1 ({ɛ}) = {} P 1 1 ({}) = {} {}{} P 1 2 ({}) = {} {}{} {}{} 11

24 2.3 Automt An utomton is devie tht is ple of representing lnguge ording to well defined rules. The forml definition of n utomton is presented in the sequel [58] Deterministi utomt The deterministi utomton is formlly defined s follows: Definition 2.2 (Deterministi utomt) A deterministi utomton, denoted y G, is six-tuple G = (X, Σ, f, Γ, x 0, X m ), where X is the set of sttes, Σ is the finite set of events, f : X Σ X is the trnsition funtion, where f(x, σ) = y mens tht there is trnsition leled y event σ Σ from stte x X to stte y X, Γ : X 2 Σ is the tive event funtion, i.e., Γ(x) is the set of events σ Σ for whih f(x, σ) is defined, x 0 is the initil stte, nd X m X is the set of mrked sttes. Automt re grphilly represented y stte trnsitions digrms. In these digrms, the sttes re irles tht re onneted y rs leled with events. The mrked sttes re identified y two onentri irles nd, in generl, re relted with the onlusion of tsk. The initil stte is indited y n rrow pointing to it. Exmple 2.4 Let G = (X, Σ, f, Γ, x 0, X m ) e the utomton shown in the Figure 2.1. Thus, X = {1, 2, 3, 4, 5}, Σ = {,,, d}, x 0 = 1, X m = {3}, the trnsition funtion f is defined s f(1, ) = 2, f(1, ) = 4, f(2, ) = 3, f(3, ) = 3, f(4, d) = 5 nd f(5, ) = 5, the tive event funtion is given y: Γ(1) = {, }, Γ(2) = {}, Γ(3) = {}, Γ(4) = {d}, nd Γ(5) = {}. An utomton is ple of representing lnguges. Two kinds of lnguges n e ssoited with the ehvior of n utomton G: the generted lnguge nd the mrked lnguge. The generted lnguge, denoted y L(G), is formed y ll tres tht n e exeuted y G, strting t the initil stte. The mrked lnguge, 12

25 G d Figure 2.1: Automton G. denoted y L m (G), is suset of the generted lnguge L(G) nd onsists of ll tres tht finish in mrked stte in the stte trnsition digrm of G. Definition 2.3 (Generted lnguge nd mrked lnguge). The generted lnguge of G = (X, Σ, f, Γ, x 0, X m ) is defined s: L(G) = {s Σ : f(x 0, s)!}. where f(x 0, s)! denotes tht f(x 0, s) is defined, i.e., y X suh tht f(x 0, s) = y. The mrked lnguge of G is defined s: L m (G) = {s L(G) : f(x 0, s) X m }. Notie tht the definition of the generted lnguge implies tht ɛ L(G). The lnguge generted y utomton G, shown in Figure 2.1, is L(G) = {ɛ,,, n, d n }, where n {0, 1, 2,...}. Sine X m = {3}, the mrked lnguge is L m = Nondeterministi utomt A nondeterministi utomton is defined y G nd = (X, Σ, f nd, Γ, X 0, X m ), where X is the stte set, Σ is the set of events, f nd : X Σ 2 X, where 2 X is the set of ll susets of X, Γ is the set of fesile events, X 0 X nd X m is the set of mrked sttes. Notie tht, differently from deterministi utomt, utomton G nd n hve more thn one initil stte nd the odomin of the trnsition funtion f nd is suset of X, not single stte. In order to illustrte nondeterministi utomton, onsider the following exmple. Exmple 2.5 Let the nondeterministi utomton, G nd = (X, Σ, f nd, Γ, X 0, ), shown in Figure 2.2. Notie tht, the trnsition funtion ssumes vlues in 2 X, 13

26 2 G nd 1 3 Figure 2.2: Nondeterministi utomton. for x X. For instne, f nd (1, ) = {2, 3}, f nd (2, ) = {1, 3} nd f nd (3, ) = {1, 2}. Thus, this type of onfigurtion suggests unertinty in the dynmi evolution of the system. Automt with ɛ-trnsitions Differently from the deterministi nd nondeterministi utomt seen efore, in n utomton with ɛ-trnsitions, sttes n hnge spontneously without deteting ny event. This lss of utomt re importnt when we wnt to model prolems in the plnt, for exmple, filure nd missing of sensors. Sine, one of the hrteristis of these utomt is the unertinty of dynmi evolution of the system, they re onsidered lso nondeterministi utomt. An utomton with ɛ-trnsitions, or simply, utomton-ɛ is defined s the sextuple G ɛ = (X, Σ {ɛ}, f ɛ, Γ, X 0, X m ), where eh prmeter of G ɛ is similr to prmeters of nondeterministi utomton G nd. Notie tht, the trnsition funtion f ɛ is defined s f ɛ = X Σ {ɛ} 2 X. Exmple 2.6 Consider utomton G ɛ shown in Figure 2.3. Notie tht, this utomton does not generte tre, i.e., there does not exist x suh tht x = f ɛ (x 0, ). However, utomton G ɛ genertes tre ɛɛ whih, when oserved, is equivlent to tre. In order to define the generted nd mrked lnguge y utomton G ɛ, let us introdue the notion of ɛ-reh. The ɛ-reh of stte x, denoted s ɛr(x), is defined s the set of ll sttes rehed from stte x y following only trnsitions leled y 14

27 G ε ε ε 4 5 Figure 2.3: Nondeterministi utomton with ɛ-trnsition. ɛ. By definition, x ɛr(x). The definition of ɛ-reh n e nturlly extended to set of sttes Y s follows ɛr(y ) = ɛr(y) (2.1) y Y Consider, gin, the ɛ-utomton shown in Figure 2.3. The ɛ-reh of sttes in G ɛ re s follows: ɛr(1) = {1, 4}, ɛr(2) = {2, 5}, ɛr(3) = {3}, ɛr(4) = {4}, ɛr(5) = {5}. The extended trnsition funtion f ɛ is defined in reursive wy s follows. First we set f ɛ (x, ɛ) = ɛr(x). (2.2) Seond, for w Σ nd σ Σ, we set f ɛ (x, wσ) = ɛr[{k : k f ɛ (y, σ) for some stte y f ɛ (x, w)]. We n now hrterize the generted nd mrked lnguges of utomton G ɛ. The lnguge generted y G ɛ is defined s: L(G ɛ ) = {w Σ : ( x X 0 )[ f ɛ (x, w)!]}, (2.3) nd the lnguge mrked y utomton G ɛ is defined s: L m (G ɛ ) = {w L(G ɛ )( x X 0 )[ f ɛ (x, w) X m ]}. (2.4) As sid erlier, n importnt pplition of nondeterministi utomt with ɛ- trnsitions is in the modeling of filures in the plnt nd loss of informtion in sensors, so tht some events eome unoservle. However, system with unoservle events n e modeled using deterministi utomt lled Oserver, whih will e desried in the next susetion. 15

28 2.3.3 Oserver utomt Suppose tht Σ is prtitioned s Σ = Σ o Σ uo, where Σ o is the set of oservle events nd Σ uo is the set of unoservle events. An event is oservle when its ourrene n e registered nd ommunited to the oserver. The unoservle events re those tht nnot e oserved y sensors (inluding the filure events) or, even though there re sensors to register it, these events nnot e seen euse of the distriuted nture of the system. When Σ = Σ o Σ uo, the utomton is lled utomton with unoservle events. The dynmi ehvior of n utomton with unoservle events n e desried y deterministi utomton lled oserver utomton, whose set of events is formed y oservle events only. The oserver for G, is denoted y Os(G), nd it is defined s follows: Os(G) = (X os, Σ o, f os, Γ os, x 0os, X mos ), where X os 2 X nd X mos = {B X os : B X m }. In order to define x 0os, Γ os, nd f os, it is neessry to introdue the onept of unoservle reh of stte x X, denoted s UR(x, Σ o ): UR(x, Σ o ) = {y X : ( t Σ uo)[f(x, t) = y]} The unoservle reh n e extended to set B 2 X s follows: UR(B) = UR(x, Σ o ) x B Thus, x 0,os = UR(x 0, Σ o ), nd for ll x os X os, Γ os (x os ) = x x os Γ(x), f os (x os, σ) = x (x os ) (f(x,σ)!) UR[f(x, σ), Σ o], if σ Γ os (x o,os ), or, undefined, otherwise. In order to illustrte n oserver utomton, onsider the following exmple [59]. Exmple 2.7 Let us onsider utomton G shown in Figure 2.4. Suppose tht the event set Σ = {,, } is prtitioned in Σ o = {, } nd Σ uo = {}. Thus, when the utomton strts, it is not possile to know if it is in the initil stte x 0 = 0 or if it hs hnged to stte x = 1, euse the ourrene of event nnot e registered. Thus, the initil stte of Os(G, Σ o ) shown in Figure 2.5 is {0, 1}. In 16

29 G Figure 2.4: Automton G. Os(G) {0, 1} {3, 1} {2, 3, 1} Figure 2.5: Oserver utomton of G. se event ours, we n see tht the utomton rehes stte {3}, ut, sine event is unoservle, it n lso hnge to stte {1} without relizing the ourrene of event. Therefore, the ourrene of event in the oserver utomton leds to stte {3, 1}, from the initil stte. There re lso trnsitions leled with event tht reh stte {2, 3, 1} from initil stte {0, 1} nd stte {3, 1}. Finlly, when the ourrene of event is registered, the oserver utomton will keep in stte {2, 3, 1}. However, if event ours, the oserver will return to the initil stte. Let us onsider the projetion P o : Σ Σ o. Then, y onstrution, oserver Os(G) hs the following properties: Os(G) is deterministi utomton. L(Os(G)) = P o [L(G)]. L m (Os(G)) = P o [L m (G)]. 17

30 Let B(t) X e the stte of Os(G) tht is rehed fter tre t P o [L(G)], i.e., B(t) = f os (x 0,os, t). Then, stte x B(t) iff x is rehle in G y tre in Po 1 (t) L(G) Opertions on utomt In order to nlyze DES modeled y n utomton, we n use set of opertions on single utomton. These opertions modify ppropritely the stte trnsition digrm ording to some lnguge opertion tht we wish to perform. We lso need to define opertions tht llow us to omine, or ompose, two or more utomt, so tht models of omplete systems n e uilt from models of individul system omponents. The opertions tht modify single utomton re lled unry opertions. They lter stte trnsition digrm of n utomton, ut the set of events Σ remins unhnged. The min unry opertions re: Aessile prt, Coessile prt, Trim opertion nd Complement. The omposition opertions re opertions tht omine more thn one utomton. The min omposition opertions re: prllel nd produt omposition. Aessile prt: If we delete from G ll the sttes tht re not rehle from x 0 y some tre in L(G) nd their relted trnsitions, without ffeting the lnguges generted nd mrked y G, then we tke the essile prt of utomton G. We will denote this opertion y A(G), where A stnds for tking the essile prt, nd is defined s follows: A(G) = (X, Σ, f, x 0, X,m ) where X = {x X : ( s Σ )[f(x 0, s) = x]}, X,m = X m X, nd f = f X Σ X. The nottion f X Σ X mens tht we re restriting f to the smller domin of the essile sttes X. Coessile prt: A stte x X of G is sid to e oessile if there exists pth in the stte trnsition digrm of G from stte x to mrked stte. We tke the oessile prt y deleting ll sttes of G tht re not oessile 18

31 nd their relted trnsitions. This opertion is denoted y CoA(G), where CoA stnds for tking the oessile prt, nd is defined s follows: CoA(G) = (X o, Σ, f o, x 0,o, X m ), where X o = {x X : ( s Σ )[f(x, s) X m ]}, x 0,o = x 0, if x 0 X o, undefined, otherwise, nd f o = f Xo Σ Xo. The CoA opertion my shrink L(G), sine we my delete sttes tht re essile from x 0. However, the CoA opertion does not ffet L m (G), sine deleted stte nnot e on ny pth from x 0 to X m. If G = CoA(G), then G is sid to e oessile; in this se, L(G) = L m (G). Trim opertion: An utomton tht is oth essile nd oessile is sid to e trim. We define the Trim opertion to e T rim(g) = CoA[A(G)] = A[CoA(G)]. Complement: Consider deterministi utomton G = (X, Σ, f, Γ, x 0, X m ) whose mrked lnguge is L m (G). The omplement of G os n utomton G omp suh tht L m (G omp ) = Σ \ L m (G), where G omp = Comp(G). The opertion Comp( ) is denoted s Complement. Automton G omp is otined s follows. First, omplete the trnsition funtion f of G. Thus, G will eome omplete utomton whose generted lnguge is L(G) = Σ. In order to do it, let us denote the new trnsition funtion y f t. Then, new stte x d is dded to X. After tht, ll undefined f(x, σ) in G re then ssigned to x d. Thus, f(x, σ), if σ Γ(x) f t (x, σ) = x d, otherwise The new utomton G t = (X {x d }, Σ, f t, x 0, X m ) is suh tht L(G t ) = Σ nd L m (G t ) = L(G). After we otin utomton G t, we hnge the mrking sttus of ll sttes of G t, i.e., we mrk ll unmrked sttes (inluding x d ) nd remove the mrking of the mrked sttes. Thus, Comp(G) = (X {x d }, Σ, f tot, x 0, (X x d ) \ X m ). 19

32 H Figure 2.6: Automton H CoA(H) Figure 2.7: Coessile prt of utomton H The following exmple illustrte the unry opertions [58]. Exmple 2.8 Let utomton H shown in Figure 2.6. In order to otin its oessile prt, we need to delete ll sttes from whih it is not possile to reh mrked stte 3. Thus, sttes 4, 5 nd 6 re deleted, leding to the utomton shown in Figure 2.7. Automt T rim(h) nd Comp[T rim(h)] re shown in Figures 2.8 nd 2.9. There re two omposition opertions lled produt nd prllel omposition. In order to desrie these opertions, onsider the following two utomt: H 1 = (X 1, Σ 1, f 1, Γ 1, x 01, X m1 ) nd H 2 = (X 2, Σ 2, f 2, Γ 2, x 02, X m2 ). Produt omposition: The produt opertion etween utomt H 1 nd T rim(h) Figure 2.8: Automton T im(h) 20

33 Comp[T rim(h)] 1 2 3,, x d,, Figure 2.9: Comp[T rim(h)] H 2 results in the utomton H 1 H 2 = A{X 1 X 2, Σ 1 Σ 2, f 1 2, Γ 1 2, (x 01, x 02 ), X m1 X m2 } where f 1 2 ((x 1, x 2 ), σ) = nd (f 1 (x 1, σ), f 2 (x 2, σ)), if σ Γ 1 (x 1 ) Γ 2 (x 2 ), undefined, otherwise. Γ 1 2 ((x 1, x 2 )) = Γ 1 (x 1 ) Γ 2 (x 2 ) In the produt, the trnsitions of the two utomt must lwys e synhronized on ommon events, i.e., events in Σ 1 Σ 2. Thus, it orresponds to intersetion of the generted nd mrked lnguges: L(H 1 H 2 ) = L(H 1 ) L(H 2 ), L m (H 1 H 2 ) = L m (H 1 ) L m (H 2 ). Properties of the produt 1. Produt is ommuttive up to reordering of the stte omponents in the omposed sttes. 2. Produt is ssoitive nd it n e defined s G 1 G 2 G 3 = (G 1 G 2 ) G 3 = G 1 (G 2 G 3 ). Prllel omposition: This omposition opertion is, usully, used to onnet different omponents in order to model unique system whose omponents work in synhrony. In generl, when system is omposed y different omponents tht intert with eh other, the event set of eh omponent 21

34 ontin events tht elong solely to itself, lled s privte events, nd ommon events tht re shred with eh other. Thus, the prllel opertion is suitle to model entire system from individul omponents. The prllel omposition etween utomt H 1 nd H 2 results in the following utomton H 1 H 2 = A(X 1 X 2, Σ 1 Σ 2, f 1 2, Γ 1 2, (x 01, x 02 ), X m1 X m2 ) where (f 1 (x 1, σ), f 2 (x 2, σ)), if σ Γ 1 (x 1 ) Γ 2 (x 2 ), (f 1 (x 1, σ), x 2 ), if σ Γ 1 (x 1 ) \ Σ 2, f 1 2 ((x 1, x 2 ), σ) = (x 1, f 2 (x 2, σ)), if σ Γ 2 (x 2 ) \ Σ 1, not defined, otherwise. In the prllel omposition, events in Σ 1 Σ 2, n only e exeuted if oth utomt exeute them t the sme time, so tht, the utomt re synhronized on the ommon events. On the other hnd, the privte events, i.e., events in (E 2 \ E 1 ) (E 1 \ E 2 ), n e exeuted whenever possile. In order to hrterize the generted nd mrked lnguges of G 1 G 2 with respet to G 1 nd G 2, we use the opertion of lnguge projetion introdued erlier. Let Σ 1 Σ 2 e the lrger set of events nd let Σ 1 or Σ 2 e the smller sets of events. Thus, we n onsider two projetions P i : (Σ 1 Σ 2 ) Σ i for i = 1, 2. Now, we n hrterize the lnguges resulting from prllel omposition: 1. L(G 1 G 2 ) = P 1 1 [L(G 1 )] P 1 2 [L(G 2 )], 2. L m (G 1 G 2 ) = P 1 1 [L m (G 1 )] P 1 2 [L m (G 2 )]. Properties of prllel omposition 1. P i [L(G 1 G 2 )] L(G i ), for i = 1, 2. The oupling of the two utomt y ommon events my prevent some of the tres in their individul generted lnguges to our, due to the onstrints imposed in the definition of prllel omposition regrding these ommon events. 22

35 Feedk Controller Controller Mlfuntion Proess Disturne Sensor Filure u Atutor Dynmi Plnt Sensors y Atutor Filure Struturl Filure Dignosti System Figure 2.10: A generl dignosis frmework [1]. 2. Prllel omposition is ommuttive up to reordering of the stte omponents in omposed sttes. 3. Prllel omposition is ssoitive: G 1 G 2 G 3 = (G 1 G 2 ) G 3 = G 1 (G 2 G 3 ). 2.4 Filure dignosis Filure is term tht defines devition from norml ehvior opertion of system. The filure tretment, lled filure dignosis, is n importnt prolem in engineering nd onsists of deteting nd isolting the filure with s muh detil s possile, suh s the ple where the filure ourred nd its dimension. In Figure 2.10, the omponents of generl filure dignosis frmework is desried. The figure shows ontrolled proess system nd indites the different soures of filures. In [1] the uthors lssify the filures in three lsses: Prmeter hnges in model. In omplex plnt, there re severl proesses ourring under the seleted level of detil of the model. Usully, the proesses whih re not detiled in the model, re typilly gthered in some prmeters of the model. Frequently, prmeter filures rise when there is disturne in the proess from the environment. Struturl hnges. Struturl hnges re hnges in the physil struture of the proess. They our due to filures in equipments. Struturl mlfuntions 23

36 result in hnge of hrteristis of the plnt, nd onsequently, in the informtion flow of severl vriles. In order to hndle suh filure in dignosis system, it is neessry to remove the urrent model equtions nd reple them with other equtions whih re ple to desrie the urrent sitution of the proess. Mlfuntioning sensors nd tutors. Errors nd mlfuntioning usully our in severl devies, ut, minly in tutors nd sensors. Some of the devies provide feedk signls whih re very importnt for the monitoring nd ontrol of the plnt. A filure in one of these instruments n use, in the plnt stte vriles, devition eyond sfe eptle limits, unless the filure is deteted nd orreted in time. The filure dignosis systems re very importnt in supervisor systems nd in the filure mngement of proesses. They re responsile for monitoring the ehvior of omplex plnt nd for providing informtion out norml opertion onditions of its omponents. In order to design good filure dignosis system, set of desirle hrteristis hve to e tken into ount. In the sequel, some of these hrteristis re listed [1]: Quik detetion nd dignosis. An importnt hrteristis of dignosis system is its pity to respond quikly in deteting nd isolting proess filures. Isolility. Isolility is the ility of the dignosis system to distinguish filure mong different types of filures. Roustness. It is importnt tht the dignosis system is roust to noises nd unertinties, so tht, when disturne ours, the dignoser does not onfuse it with filure nd send flse lrm. Adptility. Proesses, in generl, hnge nd evolve due to hnges in externl inputs or struturl hnges due to the feedk loop. Proess operting onditions n hnge not only due to disturnes ut lso due to hnge in environmentl onditions suh s vritions in prodution quntities with 24

37 hnging demnds, hnges in the qulity of rw mteril et. Thus the dignosis system must e dptle to hnges. Modeling requirements. The mount of modeling required for the development of dignosis lssifier is n importnt issue. For fst nd esy deployment of rel-time dignosis lssifiers, the modeling effort must e s miniml s possile. Due to these hrteristis, mny lgorithms hve een developed to filure dignosis prolem. In the next susetion, the lssifition of some lgorithms is shown Algorithms of filure dignosis A previous knowledge of the proess is neessry in order to uild dignosis system. It is provided y severl hrteristis nd reltions mong oserved symptoms nd filures. These knowledge n e quired with experienes in the proess nd is usully referred to s knowledge sed models or proess history [1]. The knowledge sed models n e lssified s qulittive or quntittive. In quntittive models, the plnt is expressed in terms of mthemtil funtionl reltionships etween the inputs nd outputs of the system. In ontrst, in qulittive model, these reltionships re expressed in terms of qulittive funtions in different units in proess. Other methods n e developed from the extrtion of the historil system dt. These extrtions or strtions of hrteristis n e qulittive or quntittive. In quntittive hrteristis, the strtions n ehve s sttistil or not sttistil [1]. In ontrst to the model-sed pprohes, where priori knowledge out the model (either quntittive or qulittive) of the proess is ssumed, in proess history sed methods, only the vilility of lrge mount of historil proess dt is ssumed. This lssifition n e seen in Figure The quntittive methods sed models require two steps: (i) verifition of inonsistenies nd residul r (differene mong vlues of severl funtions of outputs nd their desirle vlues under/without filure onditions) etween rel nd desirle ehvior nd; (ii) hoie of deision rules for filure dignosis. Prmeters 25

38 Dignosis methods Quntittive Model-Bsed Qulittive Model-Bsed Proess history sed Oservers Prity spe EKF Cusl models Astrtion hierrhy Qulittive Quntittive Digrphs Filure trees Qulittive physis Struturl Funtionl Expert QTA Sttistil Neurl systems networks PCA/PLS Sttistil lssifiers Figure 2.11: Clssifition of dignosis lgorithms [1]. nd sttes estimtors re used in this kind of strtegy. The residuls generted re verified nd deision funtions re performed sed on the residuls nd deision rules. The quntittive sed models lso involve nlysis of use nd effet on the system ehvior. The min disdvntges of this method is the use of too mny ssumptions nd its high omputtionl effort; on the other hnd, the ft tht it tries to imitte the humn nlysis is gret dvntge. Methods sed in proess history usully require gret del of dt. They re divided in qulittive nd quntittive methods. The qulittive methods re sed on speilized systems nd involve n expliit mpping of known symptoms. The quntittive methods, on the other hnd, use neurl networks nd sttistil methods. Trditionlly, the most widely used methods for filure dignosis re sed on proess models. As presented erlier, suh methods try to detet ll devitions etween the output of the system nd the expeted output model, supposing tht this devition is relted to filure. In this ontext, we use qulittive method sed on models (filure trees) in this thesis. The method is sed on disrete event systems s presented in [3] nd [59] Filure dignosis of DES In this setion we will present the min onepts of filure dignosis systems of DESs. As seen efore, when we dd, in the model G, unoservle events, they n represent filure event or not. Thus, it is possile to tke into ount the norml ehvior of the system, desried y oserved nd unoserved events tht re not ssoited with filures in the system or norml ehviors, tht is desried y 26