Component Reliability in Fault Diagnosis Decision-Making based on Dynamic Bayesian Networks

Transcription

1 Compoet Reliability i Fault Diagosis Decisio-Makig based o Dyamic Bayesia Networks Philippe Weber, Didier Theilliol, Christophe Aubru To cite this versio: Philippe Weber, Didier Theilliol, Christophe Aubru. Compoet Reliability i Fault Diagosis Decisio-Makig based o Dyamic Bayesia Networks. Proceedigs of the Istitutio of Mechaical Egieers Part O Joural of Risk ad Reliability, 28, 222 (2), pp <.243/7486XJRR96>. <hal-34373> HAL Id: hal Submitted o 2 Nov 28 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Compoet Reliability i Fault Diagosis Decisio-Makig based o Dyamic Bayesia Networks Philippe WEBER, Didier THEILLIOL, Christophe AUBRUN Cetre de Recherche e Automatique de Nacy - CNRS UMR Nacy Uiversité Faculté des Scieces et Techiques - BP Vadoeuvre Cedex Frace Phoe: Fax: philippe.weber,didier.theilliol,[email protected] Abstract: The decisio-makig i fault diagosis methods geerally relies o the aalysis of fault sigature vectors. This paper presets a ew approach of decisio-makig i the case of the sigature vectors for various idetical or similar faults. The mai cotributio of the paper cosists i the fusio betwee the reliability ad the evaluatio of the residuals i order to icrease the fault isolatio efficiecy. The decisio-makig, formalised as a bayesia etwork, is established with a priori kowledge o fault sigatures, false alarm ad missig detectio probability, o lie compoet state estimatio computed by a bayesia fusio of the compoet reliability ad measuremets. The effectiveess ad performaces of the method are illustrated o a heatig water process corrupted by various faults. Keywords: Model-based fault diagosis, Bayesia Networks, Reliability, Markov chais, Decisio-makig.. INTRODUCTION Moder cotrol systems are becomig more ad more complex. Cosequetly, there is a growig demad for fault diagosis to icrease the reliability of such systems. I this cotext, fault diagosis domai has gaied icreasig cosideratio. A fault is cosidered as malfuctio i the actual system that teds to degrade the overall system performaces. I this paper, our attetio is focused o model-based fault diagosis which makes use of the aalytical relatioships betwee measured variables. A short historical survey o fault diagosis ca be foud i [] ad various approaches have bee reported i [2]. The fault diagosis procedure cosists i three stages:

3 Residuals geeratio: it cosists i associatig a model-observatio pair to evaluate differece with respect to the ormal operatig coditios; Residuals evaluatio: the residuals are compared to some predefied threshold accordig to a test ad, at this stage, symptoms are produced; Decisio makig: the role of the decisio makig is to decide accordig to the symptoms, which elemets are faulty, that is isolatio. This requires the desig of residuals that are close to zero i the fault-free situatios while clearly deviatig from zero i the presece of faults. These residuals possess the ability to discrimiate betwee all possible modes of faults, which explai the use of the term decisio-makig. Classically, decisio-makig is realized accordig to a elemetary logic. Nevertheless, whe multiple faults or false alarms occur, the faults caot be isolated [3]. Some specific mathematics algorithms ca improve the efficiecy of the decisio-makig, for istace: [4], [5], have proposed methods which icrease the robustess of residual by decouplig the effects of faults from each other ad from the modellig errors ad ucertaities. [6] have developed a method based o adaptive threshold approach to reduce the sesitivity of the residuals evaluatio agaist false alarms. However, i ay case, the biary data produced by residuals evaluatio are poor i iformatio. Moreover, the degree of isolability based o Hammig distace as suggested [7] is very low. Cosequetly, some other kowledge related to the residuals should be cosidered for isolatio. [8] ad [9] have combied qualitative ad quatitative kowledge to improve the fault diagosis efficiecy. I the spirit of [], fault isolatio performace may be icreased by itegratig extra iformatio i the diagosis process. Thus, reliability, classically computed by meas of stochastic process model as Markov Chais (MC), defies the a priori behavior of the probability distributio over the omial states ad faulty states of the system. Also, this additioal iformatio is seldom used to improve decisio-makig i model-based fault diagosis []. The aim of the paper is to propose a ew approach which icreases the performace of the decisio-makig i fault diagosis by takig ito accout a priori kowledge o the system state through a dyamic Bayesia etwork.

4 The paper is orgaized as follows: Sectio 2 presets the priciple of the decisio-makig i model-based fault diagosis ad fault isolatio problem is stated. Sectio 3 describes the method which achieves the decisiomakig based o the Bayesia etwork (BN) iferece. The, the proposed approach used to merge both the fault diagosis ad the dyamic reliability is preseted. The proposed approach is illustrated through a simulatio example i Sectio 4. Fially, coclusios ad perspectives are give i last Sectio. 2. PROBLEM STATEMENT 2. Symptoms geeratio The first step produces idicators sesitive to the faults. These idicators are defied as Residuals, oted r. A residual is defied as the differece betwee a measuremet ad the correspodig referece value estimated with a model of the fault-free system (Figure ). Iputs System Outputs Residuals geerator Residuals r (k) Figure : Illustratio of residual geerator. While a sigle residual is sufficiet to detect a fault, a set of residuals is required for fault isolatio. Several methods have bee proposed i the literature to geerate structured residuals ad to perform the fault diagosis []. Oe of the most popular is the observer-based desig [2], [3]. The causal kowledge-based addressed i [4] deals with the complexity of large scale systems. Aother way to geerate structured residuals is to develop models havig a appropriate structure. The structured residuals may be geerated by parity equatios from the iput-output model, or balace equatios (chap. 6 ad 9, [4]). The obective is to decouple the faulty effects from each residual. Each residual is desiged so that it is sesitive to differet faults or subsets of faults. Usually, the secod step of the diagostic procedure, Residuals evaluatio, is based o the assumptio that if a fault occurs, the statistical characteristic of the residuals is modified. The residuals evaluatio ivolves statistical testig such as limit checkig test, geeralized likelihood ratio test, tred aalysis test. The output vector of the

5 statistical test, called coherece vector U, ca be built accordig to a test applied to the set of J residuals: U = [u, u, u J ] T where u represets the status of the residuals: u is equal to whe the residual sigal is closed to zero i some sese ad equal to otherwise. u is called the symptom associated to the residual r. Ufortuately, the residual is corrupted by oise, which affects the decisio-makig. The efficiecy of the detectio is related to the false alarm ad missig detectio probability. I the residual evaluatio, the problem may be formulated as a hypotheses testig problem. Let us recall the mai defiitio: H : the residual is ot affected by a fault; H : the residual is corrupted by fault. f ( r( k), H ) f r( k), H ) ( risk β risk α r (k) Hits H Reect H Figure 2: Illustratio of the error of the first kid α ad error of the secod kid β. Figure 2 gives a illustratio of hypotheses testig problem with Gaussia desity fuctios. The desity fuctios f r( k), H ) ad f r( k), H ) characterize the probability that the residual r (k) respect the hypothesis ( ( H or H. The surfaces represetig the probability of false alarm correspod to the risk α defiig as the coditioal probability: ( reect H H is true) α = () p ad the probability of missig detectio correspod to the surface risk β defiig the coditioal probability: ( ot reect H H is true) β = (2) p

6 More details ca be foud i the referece book of Basseville ad Nikiforov [5] or i the secod chapter of the book of Youg ad Calvert [6]. The risk values α ad β are calculated a priori if the desity fuctios f r( k), H ) are kow. However, this ( i is rarely the case i practice. If the desity fuctios are ot available, the a estimatio of the probabilities is computed by mea of a frequetist approach based o data icludig a posteriori detectio results. 2.2 Icidece matrix Several approaches have bee proposed to geerate structured residuals ad cosequetly to geerate the icidece matrix [2]. Let us cosider the followig example where three differet faults (F, F 2 ad F 3 ) ca be isolated by desigig three symptoms u, u 2 ad u 3 (Table ). Table : Icidece matrix example. F F 2 F 3 u u 2 u 3 I Table, a "" deotes that a symptom u is sesitive to a fault (F, F 2 or F 3 ), while a "" deotes isesitivity to a fault. This table is called a icidece matrix ad ca be cosidered as a a priori kowledge. Each colum of the icidece matrix represets a fault sigature: the vector [ ] T correspods to the sigature of the faulty elemet F. I this paper, icidece matrix is aotated D with differet elemets D(,), where is the umber of elemets suspected to be faulty (=...N) ad is the umber of residuals (=...J).

7 2.3 Fault Isolatio Usually, a very simple logic aalysis betwee each fault sigature F ad each coherece vector U is used to isolate the faulty compoet. I practical cases, false alarms occur ad corrupt the decisio logic. The coherece vector ca be differet from all sigatures. Therefore, the goal of the decisio-makig is to miimize the false alarms ad missig detectio rates due to the effects of modellig errors ad ukow disturbaces that affect residuals. Moreover, i spite of the residuals geeratio ad evaluatio robustess, a simple logic rule is ot efficiet eough to isolate faults whe simultaeous multiple faults occur [7]. This is ustified by the fact that if D(,)=, the u caot brig ay iformatio about the occurrece of fault F, sice the residual r might be differet from due to oise or modellig errors or aother fault F k (with D(k,)=) affectig the system. Thus, ew decisio-makig method is ecessary. Moreover, a ew source of iformatio should be itegrated i fault diagosis. [8] has recetly proposed to itroduce reliability aalysis for fault diagosis purpose. System reliability aalysis allows determiig the degradatio degree of the system compoets. The paper aims at developig a method which itegrates a dyamic reliability estimatio of the system compoet as preseted i the ext sectio. I this ew approach the reliability of the compoets is computed accordig to the states of the residuals (with the bayesia approach) ad ot oly a a priori estimatio of the reliability. This is the reaso why a dyamic computatio of the reliability is used. 3. DECISION-MAKING PROCESS DESIGN 3. Bayesia etwork equivalet to icidece matrix a) Bayesia etwork (BN): Bayesia etwork are probabilistic etworks based o graph theory. They are directed acyclic graphs used to represet ucertai kowledge i Artificial Itelligece [9]. Each ode represets a discrete variable defied

8 over several states ad the arcs idicate direct probabilistic relatios betwee the odes. A discrete radom variable X is represeted by a ode with a fiite umber of mutually exclusive states defied as { s,... s } ( m [,...,M ]) : M s m. A set of states is S. A probability distributio over these states is defied as a vector p () with p ( = ) is the margial probability of beig i states 3, odes i ad are liked by a arc where i is cosidered as a paret of s m. As illustrated i the graph depicted at Figure. s m i Figure 3: Elemetary Bayesia etwork. A coditioal probability distributio quatifies the probabilistic depedecy betwee ad its paret i ad is defied through a Coditioal Probability Table (CPT). Therefore, the odes i ad S : i M ad { } : s,... s L sets { } i s,... s i are defied over the S. The CPT of is the calculated by the coditioal probabilities p ( i ) over each state accordig to its parets i states as preseted i Table 2. Table 2: Coditioal Probability Table for. i L s s s i i p = = s i s s i M i p = = sl i sm Cocerig the root odes, i.e. those without paret, the CPT cotais oly a row icludig the a priori probability of each state.

9 Various iferece algorithms ca be used to compute margial probabilities for each uobserved ode give iformatio o the states of a set of observed odes. The most classical oe relies o the use of a uctio tree. Iferece i BN the allows to take ito accout ay state variable observatio (a evet) such that it updates the probabilities of the other variables. Without observatio, the computatio is based o a priori probabilities. Whe observatios are give, this kowledge is itegrated ito the etwork ad all the probabilities are updated. Kowledge is formalised as evidece. A hard evidece of the radom variable X idicates that the state of the ode is oe of the states S { s,... s } : M. For istace, X is i state s : p ( s ) = ad p ( ) =. Moreover, whe this kowledge is ucertai, soft evidece ca be used to defie the distributio over. = = s m b) Bayesia etwork model as a icidece matrix: The relatioship betwee symptoms ad faults are represeted by a graph. Obviously a fault ca be cosidered as the cause of the residual deviatio. Therefore, some coectios ca be established from the fault to the symptoms i order to defie the relatio of causality betwee fault occurrece ad the symptom states. Whereupo, a Bayesia etwork ca defie directly a icidece matrix D(,). Let us cosider two icidece matrices that represet two differet cases of possible isolability coditios (Table 3). Table 3: Two icidece matrices. a) b) D(,) F F F N D(,) F F F N u u u u u J u J As depicted i Figure 4, each above icidece matrix is represeted as elemetary graph with their appropriate arc which correspods to the lik betwee fault sigature ad coherece vectors.

10 a) b) u F u F u F u F u J F N u J F N Figure 4: Two structured Bayesia etworks. The fault occurrece probability is modelled as a radom variable F associated to each fault. F is described by two states {ot occurred, occurred}. Moreover, the symptoms are represeted also as radom variable u defied over the set of two states: {ot detected, detected} with p(u =detected), if the fault affects the system ad the residual r is detected differet from. The probability distributio over the symptom states depeds o the false alarms ad missig detectios. Usig Bayesia etwork model, a CPT is determied to model the relatio betwee variables. I order to compute the probability distributio of symptoms u, a CPT is defied accordig to the fault F paret of u. For istace, whe oly oe symptom is associated to oe fault, as preseted at Figure 4a, the the CPT has the structure preseted i Table 4: Table 4: CPT for ode u i Figure 3a. u F ot detected detected ot occurred -α α occurred β - β where the probability β of missig detectio ad the probability α of false alarms both for the fault F are defied such as:

11 α = p u β = p u ( = detected F = ot occurred) ( = ot detected F = occurred) (3) Therefore, the probability distributio over the states of the causes (fault occurrece) is based o the residual evaluatio result. The ode u is defied as hard evidece. If chages i the residual are detected, the: p(u = detected) = ad p(u = ot detected) = (4) Otherwise: p(u = detected) = ad p(u = ot detected) = (5) The Bayes theorem is used to compute p ( ), for istace, relatively to Figure 4a, ( ) F u p F u is equal to: p( F ) p( u F ) p ( F u ) = (6) p( u ) where ( ) p F u is the a posteriori distributio of probability over the fault states accordig to the states of the symptoms. ( ) F p is the a priori distributio of probability over the fault states ad ( ) distributio over the symptoms. p u F are the coditioal Geerally, several faults are associated to oe symptom, as illustrated i Figure 4b. I this case, the CPT is more difficult to determie ad it possesses a differet structure from Table 4. For istace, whe oe symptom is associated to few faults, as preseted at Figure 4b where few are equal to two, the the CPT is defied as the Table 5.

12 Table 5: CPT for ode u i Figure 3b. u F F N ot detected detected ot occurred ot occurred -α α occurred β N -β N occurred ot occurred β -β occurred β,n - β,n The probability that a false alarm exists, whe few faults are the cause of the residual deviatio, is estimated idepedetly to the faults. O the cotrary, the missig detectio depeds o the fault, as a fault F has ot the same impact o the residual as a fault follows: F N. Let us cosider that false alarms ad missig detectios estimated as p p p p ( u = detected F = ot o., FN = ot o. ) = α ( u = ot d. F = occurred, FN = ot o. ) = β ( u = ot d. F = ot o., FN = occurred) = β N ( u = ot d. F = occurred, FN = occurred ) = β, N (7) where ot d. (resp. ot o.) deotes ot detected (resp. ot occurred ). The defiitio of false alarms ad missig detectios are represeted Figure 5.

13 f r( k), H ) ( risk α r (k) Hits H Reect H f ( r( k), H) risk β r (k) f N ( N r( k), H ) risk β N r (k) f ( r( k), H,, N N ) risk β,n r (k) Figure 5: Illustratio of the error of the first kid α ad error of the secod kid β ad β N. As preseted i Table 5, ( u F ) ( u F ),F N p is geerated accordig to the icidece matrix defied i Figure 4b.,F N p is defied accordig to the missig detectio rate β (resp. β N ) ad the false alarm rate α for the fault F (resp. F N ). The Bayes theorem is applied i the BN iferece algorithm to determie p ( F u, ) symptom u ad u such as: N u from the states of the

14 p( F N p( FN, u, u ) p( u FN ) p( u FN ) p( FN ) u, u ) = = (8) p( u, u ) p( u ) p( u ) where p ( F u, ) N u symptoms ad ( ) F N is the a posteriori distributio of probability over the fault states accordig to the states of the p is the a priori distributio of probability over the fault states ad ( u ) the coditioal distributios over the symptoms. p. ( ) F N p u F N are 3.2 Dyamic model of reliability: a dyamic Bayesia etwork solutio [2] I order to model dyamic behaviour of the system performaces degradatio, dyamic Bayesia etwork has bee cosidered. Let us recall some fudametal Markov Chai (MC) model. I the framework of decisio-makig, a discrete radom variable X with two states {up, dow} is cosidered. These states represet respectively the operatioal ad failure state of the compoet. Associated to a discrete radom variable X, a matrix P X defies the probabilistic state trasitios betwee (up) ad (dow): P X ϕdu = ϕdu (9) Where ϕ du represets the failure probability of the compoet betwee sample k- ad k ( X = dow X up) ϕ. du = p k k = I reliability aalysis, λ represets the failure rate of the compoet with ϕ λ k where k represets the du time iterval betwee (k-) ad (k). It ca be oticed that for a costat failure rate, the Mea Time To Failure (MTTF) is equal to λ. Based o this elemetary defiitio, a discrete-time Markov chai is defied whe the iitial state probability vector is specified as ( X ) = [ p( X = up) p( X dow) ]. p = The trasiet aalysis of the MC based o the Chapma-Kolmogorov equatio [22] provides a expressio for p ( X k ) with k k ) = p( X ) ( P X for k=,2, p ( X )

15 Uder dyamic cosideratio i a Bayesia etwork, the state of the variable X i (to model the i th compoet of a system) is represeted at sample k by a ode i (k) with a fiite umber of states. p( i ( k)) deotes the probability distributio over these states at time step k. The dyamic Bayesia etworks allow to represet radom variables ad their impacts o the future distributio of other variables [2]. Startig from a observed situatio at sample k=, the probability distributio p( i ( k)) over M states for the compoet Xi associated to the ode i is computed by the dyamic Bayesia etwork iferece. Ideed, it is possible to compute the probability distributio of ay variable X i at sample k based o the probabilities defied at sample k- as show i the elemetary etwork preseted i Figure 6. i ( k ) (k) i Figure 6: Dyamic Bayesia etwork for X i. The first slice cotais the odes correspodig to the curret time step k-, the secod oe cotais the odes of the followig time step k. Observatios, itroduced as hard evidece or probability distributios, are oly realised i the curret time slice. The time icremet is carried out by settig the computed margial probabilities of the ode at sample k as observatios for its correspodig ode i the previous time slice. The CPT i dyamic Bayesia etwork is equivalet to P X (Table 6). Table 6: CPT for ode i (k). (k) i i ( k ) up dow up -ϕ du ϕ du dow

16 3.3. Fusio of the Icidece Matrix ad the dyamic Bayesia etwork model of reliability: a solutio for icreasig effectiveess of decisio-makig As preseted previously, the dyamic Bayesia etwork models the compoet reliability which takes ito accout the time degradatio of compoets. The represetatio of icidece matrix as a graph, defied at the begiig of the paper, provides a formalism, which is used to realize the fusio betwee fault diagosis ad reliability model. The decisio-makig is established after fusio of iformatio obtaied from both residual aalyses ad reliability estimatio. Therefore, based o the Bayesia etwork represetatio, image of the icidece matrix, the dyamic evolutio of the compoet reliability is take ito accout o each ode F i as follows: Figure 7. i ( k ) (k) i u F i u J Figure 7: Fault Diagosis Scheme with DBN for decisio-makig dedicated to a failure F i. This relatioship ivolves the defiitio of a CPT. The CPT of F i is very simple to defie if the compoet reliability is modelled with the states (up) or (dow) which is a commo case i fault diagosis. The idetity matrix is used to compute the distributio o F i as preseted i the Table 7. Table 7: CPT of the ode F i. F i i (k) ot occurred occurred up dow

17 However, if more tha 2 states are used to defie (k), the CPT is defied to determie the occurrece of the i fault accordig to the state of degradatio of the compoet. Usually, each fuctioig states of i (k) leads to the o-faulty state: (ot occurred), ad each faulty states leads to the occurrece of the fault (Table 8) Table 8: CPT of the ode F i with a degradatio state. F i i (k) ot occurred occurred up degradatio dow dow2 It should be possible that fault occurrece is due to the failure of several compoets. Let s cosider two compoets, the ode (k) ad k (k) modelled the state of these compoets. The fault is occurred whe the compoets are i specific states. The the combiatio of the compoet states is merge i F i for istace with AND or OR gate accordig to the serial or parallel structure (Table 9) (or aother combiatio of the compoets states). Table 9: CPT of the ode F i with OR gate. (k) k (k) ot occurred occurred up up up dow dow up dow dow F i Based o the elemetary example, illustrated i Figure 6, the computatio of p( F u, u, ( )) i is performed J i k thaks to the iferece algorithm i dyamic Bayesia etwork a simplificatio of the formulatio is preseted by:

18 p ( ) ( Fi, u, uj, i ( k) ) F u u, ( k) = p i ( u, u, ( k) ) ( Fi i ( k) ) p( u Fi ) p( uj Fi ) p( F ( k) ) p( u F ) p( u F ) p, J i = () p J i F i i i i J i I this formulatio the probability of failure occurrece is deduced from the reliability of the compoet i (k) ad the residuals states u i (k) ad u J (k). Moreover, the back propagatio (from F i (k) to (k) ) allows to verify the coherece whe computig the reliability of the compoet i (k) accordig to the states of the residuals. i 4. ILLUSTRATION EXAMPLE 4.. Process descriptio ad fault diagosis To illustrate our approach, a simulatio example is cosidered: a heatig water process. The process, preseted i Figure 8, is composed of a tak with a sectio S equipped with two heatig resistors R ad R 2. The iputs are the water flow rate Qi, the water temperature Ti ad the heater electric power P. The outputs are the water flow rate Qo ad the temperature T which is maitaied aroud a operatig poit. The temperature of the water Ti is assumed to be costat. The obective of the thermal process is to provide a costat water flow rate at a give temperature. Qi Ti V T sesor Qo sesor P H sesor R R 2 H Qo T Figure 8: Heatig water process.

19 Usig the hydraulic ad thermal equatios, the system ca be described by the followig equatios: ( t) dh S dt dt dt = q ( t) q( t) ( t) P( t) = ρcsh( t) i ( T( t) Ti ) qi ( t) Sh( t) (3) where ρ C represets a costat thermal variable ad T i is equal to 2 C. Based o previous equatio, a discrete state space represetatio of the system aroud a operatig coditio ( h op =.6m, T op =5 C) is determied as follows: x y ( k + ) = A x( k) + B u( k) ( k) = x( k) d d (6) where the output vector y is equal to [ h] T T ad the iput vector u defies [ q P] T the samplig period is fixed to 36s i order to respect the closed-loop time costats. i. It should be oted that Moreover, the measured output flow rate ca be determied by usig the Torriceli-rule as: q o ( k) = η h( k) (9) where η defies the outflow coefficiet. I this study, oly sesor failures are cosidered: level sesor H, output temperature sesor T ad output flow rate sesor Qo. As idicated previously, several methods have bee proposed i the literature to geerate structured residuals ad to perform the fault diagosis []. However i this example, due to the property of matrix A (which is diagoal), structured residuals ca be geerated directly with a covetioal observer: each residual is sesitive to oe fault. The based o the state space represetatio, a covetioal Lueberger y( k) ŷ( k) observer is cosidered to geerate the residuals vector, such that: xˆ ŷ ( k + ) = A xˆ ( k) + B u( k) + K( y( k) ŷ( k) ) ( k) = xˆ ( k) d d (24)

20 [ ] T T Based o the residuals vector [ r ( k) r ( k) ] = T ( k) Tˆ ( k) h( k) ĥ( k) [ ( k) u ( k) ] T 2 evaluatio, the symptoms vector u 2 is performed i order to detect fault occurrece o H level sesor or T temperature sesor. Moreover, accordig to the physical equatio betwee output flow rate Qo ad liquid level H, a residual ( k) = [ q ( k) qˆ ( k) ] r3 is calculated. o o Based o the residual evaluatio of ( k ) [ r ( k) r ( k) r ( k) ] T defied i the followig table: r = 2 3, the associated fault icidece matrix is Table : Icidece matrix. H Qo T u u 2 u Dyamic BN desig for decisio-makig The proposed approach has bee desiged with the help of the software BayesiaLab ( The icidece matrix, defied i Table, leads to a DBN model preseted i Figure 9. Figure 9: Graphical model of the decisio-makig with DBN.

21 For all faults of the system, the probability of missig detectio is assumed to be fixed to.2 ad the probability of false alarms is equal to.5. The probability of missig detectio with two simultaeous faults is fixed to.. Cosequetly as preseted i 3. (see Table 4) the CPT of u, is defied Table, ad also the CPT of u 3 (see Table 5) is defied Table 2. Table : CPT of the ode u. u Fault T ot detected detected ot occurred 95 5 occurred 2 98 Table 2: CPT of the ode u 3. u 3 Fault H Fault Q ot detected detected ot occurred 95 5 ot occurred occurred 2 98 occurred ot occurred 2 98 occurred 99 I order to defie the dyamic reliability model, Figure to Figure 2 preset the MC ad the Mea Time To Failure (MTTF) which is used to determie the failure rates λ. This failure rate quatifies the trasitio betwee the states of 3 cosiderig faulty compoets ad associated probabilistic state matrix P X defied i eq. (9). The Markov Chais of the compoets are supposed to be idepedet. It should be oted that two states {up, dow} are cosidered for sesors Qo ad H, but oe more state (dgd) is cosidered for sesor T which correspods to a degraded state of the compoet. As defied i 3.2., the CPT used to simulate the MC for the sesor H (resp. T) reliability described i Figure (resp. Figure 2) is preseted i Table 3 (resp. Table 4). The state (dgd) is a fuctioig state so o fault occurred i this state as implemeted i the CPT preseted i Table 5.

22 λ MTTF =45 h λ =.22-4 up dow Figure : Reliability MC model for sesor Qo. λ MTTF =5 h λ =2-4 up dow Figure : Reliability MC model for sesor H. MTTF =8 h λ =.25-4 λ λ 2 MTTF 2 =3 h λ 2=3.3-4 MTTF 3 =45 h up dgd λ 3 dow λ 3=.22-4 Figure 2: Reliability MC model for sesor T. Table 3: CPT of the ode H_sesor(k). H_sesor(k) H_sesor(k-) up dow up dow Table 4: CPT of the ode T_sesor(k). T_sesor(k) T_sesor(k-) up dgd dow up dgd dow Table 5: CPT of the ode Fault_T. T_sesor(k) T_sesor(k) up dow up dgd dow

23 4.3 Results ad commets Based o the icidece matrix (see Table ) ad uder ay assumptios of the umber of faults, the if the coherece vector issued from the residual evaluatio at sample k is equal to [ ] T or to [ ] T, for example, the fault idicators I geerated by a logic test is as i Table 6. Table 6: Fault idicators. U I H I T I Qo [ ] T [ ] T Because H ad T fault sigatures are differet, ad Qo fault sigature is icluded i H fault sigature, the fault isolatio is ot easy to perform: - whe coherece vector is equal to [ ] T, the decisio-makig provides a fault isolatio o sesor Qo (I Qo =) the a maiteace actio is performed to repair this sesor; - If the coherece vector is equal to [ ] T, the three sesors are suspected to be dow with the same possibility ((I H = I T =I Qo =). However, based o our approach, it could be possible to optimize the maiteace actio. I order to illustrate the performace ad the limitatio of the proposed method, various faults scearios have bee cosidered: Sceario A) A false alarm occurs at sample k=6 which appears as a outlier o sesor T. Sceario B) A bias o the sesor Qo is assumed to occur at sample k=9 ad to repair at sample k=4. Sceario C) The system is i a fault-free case. Sceario D) T ad H sesors faults are supposed to occur simultaeously at sample k=26. The dyamic behavior of the structured residuals vector has bee illustrated i the presece of the various faults scearios Figure 3. Sceario A) ad Sceario B) The residual sesitive to a fault is affected. The other residuals are close to zero. Sceario C) The system is i a fault-free case. All residuals are close to zero. Sceario D) Accordig to the fault icidece matrix issued from the structured residuals (see Table ), all residuals are differet from zero at sample k=26 whe the sesor faults occur simultaeously.

24 A B C D r r r Figure 3: Residuals behavior. u u2.5 A B C D u Samples Figure 4: Symptoms sceario. These residuals are evaluated usig the statistical test ad are detected isolated correctly as show i Figure 4: Sceario A) A false alarm occurs at sample k=6 which appears as a outlier o the first symptom which switch to durig oe sample. Sceario B) Accordig to the structured residuals defied i the icidece matrix (see Table ), oly the third symptom switched to. Few samples after the third symptom switch to due to a maiteace actio.

25 Sceario C) Durig this period, o fault occurs. Symptoms are equal to. Sceario D) T ad H sesors faults are supposed to occur simultaeously. Based o their fault sigatures, all symptoms switched simultaeously to. The failure probabilities for the three sesors are preseted at Figure 5 without takig ito accout the dyamic reliability of compoets. Figure 6 is devoted to the illustratio method through the failure probabilities evolutio icludig the dyamic reliability of compoets. Sceario A) The outlier geerates a false alarm, the CPT for T ca oly reduce the value of failure probability to.955 computed based o the Baye s theorem accordig to the false alarm probability α T =.5 ad the missig detectio β T =.2 with a priori probability distributio: p(fault T= occured) =.5 ad p(fault T = ot occured) =.5 (see Figure 5). However, as show at Figure 6, the decisio-makig is based o reliability of the compoet. Thaks to the Bayes theorem, the iferece algorithm i the BN computes the reliability of the compoet takig ito accout the Markov Chai model with failure rate parameters ad the olie iformatio based o the residual evaluatio (symptoms). Therefore, if all the symptoms are kow ad close to the the reliability of the compoet is close to. However, the fault is suspected to have occurred if the residuals deviate from zero. Aother advatage, of takig reliability ito accout, lies i the fact that if the residuals states are ot computed or if o ambiguity appears i the fault sigature, ad the the kowledge relative to the reliability of the compoet is domiatig. I this sceario, the itegratio of reliability ad the symptoms result i similar way a slidig widow ad therefore aihilate the false alarm. Sceario B) The two BN methods isolate the fault. It could be oted that a time delay is observed for the secod oe due to reliability cosideratio. After tree samplig periods, the probability of fault occurrece I Q is equal to oe. The probability of the I H fault occurrece is icreased. This fault geerates deviatio o residual u 3 but accordig to the iferece i the BN, this fault is cosidered as ot coheret with the states of the other symptoms. Therefore, the probability of the I H fault occurrece stays close to. Sceario C) Whe a maiteace actio is realized, the decisio-makig is back to a fault free case. The probability that the fault I Q occurred is re-iitialized to zero.

26 .973 IH IQ IT Samples Figure 5: Failure probabilities for the three sesors with BN. I.5 close to IQ.5 close to Reset IT.5 close to Sample s Figure 6: Failure probabilities for the three sesors with DBN ad reliability. Sceario D) This sceario highlights the proposed approach. Without reliability cosideratio, it is ot possible to geerate a suitable decisio-makig. For multiple faults, all fault sigatures ca be suspected: the symptom u 3

27 is explaied by the failure o sesor H, the the Qo failure probability is computed based o the Baye s theorem, equal to.556 this value traslate the ucertaity (see Figure 5). However, accordig to the DBN, the the Qo failure probability icreases by takig ito accout the reliability of compoets (Figure 6). A priori the fault I Q has ot occurred. Nevertheless, after log delay, the fault is suspected due to the degradatio of the compoet modeled i the DBN from the failure rate of the compoet. With the proposed method, it is possible to pla a maiteace actio without visitig the Q sesor at the first place due the low level of failure probability. The, the maiteace actio ca be focused to the others, T ad H sesors, showig a higher level of failure probability. 5. CONCLUSION This paper presets a ew strategy to icrease the performace of the decisio-makig i model-based fault diagosis. The developed approach cosists i takig ito accout i fault diagosis scheme a priori kowledge o the faulty or o faulty system by a Markov chais modellig. Thaks to the Bayes theorem, the iferece algorithm i the BN computes the reliability of the compoet takig ito accout the a priori Markov Chai model with failure rate parameters ad the o-lie iformatio based o the residual evaluatio (symptoms). Therefore, if all the symptoms are kow ad close to the the reliability of the compoet is close to. For complex systems, the problem of the decisio-makig whe various fault sigatures vectors are idetical or similar ca be allayed by usig a suitable dyamic Bayesia etwork. The simulatio example, a heatig water process, has highlighted the performaces of the method. The desig of the dyamic Bayesia etwork requires the false alarms ad miss-detectio probabilities of the residual evaluatio parameters that are ot always possible to assess. Nevertheless, the results obtaied i this paper are ecouragig ad allow us to advocate the method i order to optimize the maiteace actios. Therefore, for a system which is liable to various occurrig faults simultaeously or which is defied through a icidece matrix with similar fault sigatures, the fault probabilities, provided by the method will eable to pla the maiteace actios. I future works we are iterestig to aalyse the sesitivity of the decisio makig to the parameter as false alarms ad missig detectio.

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