International Journal of Computers, Communications & Control Vol. I (2006), No. 4, pp. 61-72 Time Disturbances and Filtering of Sensors Signals in Tolerant Multi-product Job-shops with Time Constraints Nabil Jerbi, Simon Collart Dutilleul, Etienne Craye, Mohamed Benrejeb Abstract: This paper deals with supervision in critical time manufacturing job- shops without assembling tasks. Such systems have a robustness property to deal with time disturbances. A filtering mechanism of sensors signals integrating the robustness values is proposed. It provides the avoidance of control freezing if the time disturbance is in the robustness intervals. This constitutes an enhancement of the filtering mechanism since it makes it possible to continue the production in a degraded mode providing the guarantees of quality and safety. When a symptom of abnormal functioning is claimed by the filtering mechanism, it is imperative to localize the time disturbance occurrence. Based upon controlled P-time Petri nets as a modeling tool, a series of lemmas are quoted in order to build a theory dealing with the localization problem. Keywords: P-time Petri net, sensor signal, filtering, time disturbance, localization 1 Introduction This paper concerns critical time manufacturing job-shops. For each operation is associated a time interval. Its lower bound indicates the minimum time needed to execute the operation. The non respect of this value means that the operation was not achieved. The upper bound fixes the maximum time to not exceed otherwise the quality of the product is deteriorated. Such systems have a robustness property in order to maintain product quality when there are time disturbances [1], [2]. The robustness is defined as the ability of the system to preserve the specifications facing some expected or unexpected variations. So the robustness characterizes the capacity to deal with disturbances. The robustness is interpreted into different specializations. The passive robustness is based upon variations included in validity time intervals. There is no control loop modification to preserve the required specifications. On the other hand, active robustness uses observed time disturbances to modify the control loop in order to satisfy these specifications. Therefore, the robustness intervals must be integrated in the filtering mechanism of sensors signals. Furthermore, the observability of time disturbances occurrence is a fundamental data necessary for the control loop modification. It is also an important aspect of the maintaining task [3], [4]. When an abnormal functioning is claimed, it is important to know the initial occurrence of the disturbance. The localization problem is really difficult in robust systems since the rejection of disturbances may hide them [5]. The first part of this paper presents a filtering mechanism of sensors signals taking into account the robustness values. The second part considers the localization of time disturbances. It is necessary to perform this task when the disturbance value passes through the filter. Controlled P-time Petri nets are used for modeling the considered workshops. Afterward, the localization problem of time disturbances in critical time manufacturing systems is tackled. Some definitions and lemmas are quoted in order to build a theory dealing with such problems. 2 Robustness integration in the filtering of sensors signals At the occurrence of a dysfunction in a manufacturing workshop, it is crucial to react as soon as possible to maintain the productivity and to ensure the safety of the system. It has been recognized that Copyright c© 2006 by CCC Publications 62 Nabil Jerbi, Simon Collart Dutilleul, Etienne Craye, Mohamed Benrejeb the real time piloting, without human intervention, has a significant contribution regarding this type of problem [6], [7]. In the category of the workshops concerned by this article, the operations have temporal constraints which must be imperatively respected. The violation of these constraints can blame the health of the con- sumers. Thus, the detection of a constraint violation must automatically cause the stop of the production. On the other hand, when taking into account the system robustness, it is proven that this type of violation did not take place. In this case, we plan to maintain the production while describing it as degraded pro- duction [8]. Of course, the product is not degraded, but the production is degraded because the deliveries moments of the products are not those envisaged initially. It is this context which we propose to integrate in the generation of symptoms as it was presented in [9]. The finality of this section is not to contribute to the state of the art of the monitoring-diagnosis, but to show how the knowledge of the robustness could make the supervision more efficient. By considering this criterion, the filtering of sensors signals of [9] appears pedagogically interesting. 2.1 Symptoms generation The idea consists in modeling any operation from a temporal approach. At each operation Ai is asso- ciated a sensor signal CRi. At each sensor signal CRi is associated a temporal window [∆tm/CRi , ∆tM/CRi ] (Figure 1). CRi is valid only inside this window. ∆tm/CRi and ∆tM/CRi are defined relatively to the begin- ning of the operation Ai (Start-Event). The filtering principle is to position the temporal window of each sensor signal CRi when his Start-Vent was received. Two types of symptoms are distinguished [9]. Symptoms type I noted S1i : this class of symptoms corresponds to awaited sensor signal which is not received at ∆tM/CRi . The detection mechanism of this symptom type corresponds to the traditional mechanism of watchdog, but implemented in a separate way of the control. Symptoms type II noted S2i : it is generated by the occurrence of a sensor signal which is not expected. Two cases are considered. The first one corresponds to an action but its sensor signal occurs before the validation interval. The second case corresponds to the occurrence of a sensor signal in absence of any order which can create it. Ai (Start-Event) ∆tM/CRi Validation interval of CRi Time 0 ∆tm/CRi Figure 1: Operation associated model [9] 2.2 Robustness integration The tool used to represent the filtering mechanism is the interpreted T-time Petri net. Initially, we point out the definition of the interpreted T-time Petri net. After, we give the filtering mechanism of sensors signals integrating the two properties of passive and active robustness. Definition 1. A T-time Petri net is given by a pair < R; IS′ >, where R is a Petri net and IS′ : T → (Q+)×(Q+ ∪+∞) [10]. Time Disturbances and Filtering of Sensors Signals in Tolerant Multi-product Job-shops with Time Constraints 63 Definition 2. Interpreted T-time Petri net is a T-time Petri net including an operative part whose state is defined by a set of variables. This state is modified by the operations associated to the places. It determines the value of the conditions (predicates) which are associated to the transitions. The mechanism of watchdog is easily represented by an interpreted T-time Petri net. For example, figure 2 shows the detection of a normal state and an abnormal one. If the sensor signal arrives in [tm, tM[, the system is in a normal state. If the sensor signal arrives at the instant tM , the system is in an abnormal one. p2 t1 p1 t2 p3 [tm, tM[*CRi [tM, tM]*CRi Normal state Abnormal state Figure 2: Watchdog mechanism with interpreted T-time Petri net Within the framework of the robustness integration in the supervision of manufacturing systems with time constraints, we define, figure 3, five time intervals namely: I1i = [∆tm′′/CRi , ∆tm′/CRi [, I2i = [∆tm′/CRi , ∆tm/CRi [, I3i = [∆tm/CRi , ∆tM/CRi [, I4i = [∆tM/CRi , ∆tM′/CRi [ and I5i = [∆tM′/CRi , ∆tM′′/CRi [. The margin of passive robustness is available in (I2i ∪I4i) whereas the margin of active robustness is in (I1i ∪ I5i). From a functional point of view, there are three intervals of use in which it is possible to prove the validity: interval of normal functioning, interval of passive robustness and interval of active ro- bustness. In the case of an abnormal functioning, there is always duality of advance and delay scenarios. The adopted filtering mechanism is described by the interpreted T-time Petri net of the figure 4. Several cases can arise [11]. • If there are absence of order (not Ai) and presence of CRi, there are freezing of the control and generation of a symptom S2i (place p3). • If the sensor signal CRi arrives in the time interval [0, ∆tm′′/CRi [, there are freezing of the control and generation of a symptom S2i (place p3). • If the sensor signal CRi arrives in the time interval I1i = [∆tm′′/CRi , ∆tm′/CRi [, there are change of the control (active robustness to an advance) and memorizing a symptom S2i (place p4). • If the sensor signal CRi arrives in the time interval I2i = [∆tm′/CRi , ∆tm/CRi [, there is no change of the control (passive robustness to an advance) but only a memorizing of a symptom S2i (place p5). • If the sensor signal CRi arrives in the time interval I3i = [∆tm/CRi , ∆tM/CRi [, the behavior of the system is normal (place p6). 64 Nabil Jerbi, Simon Collart Dutilleul, Etienne Craye, Mohamed Benrejeb Ai (Start-Event) Time ∆tm/CRi ∆tM/CRi CRi (Normal functioning) ∆tm’/CRi ∆tm’’/CRi ∆tM’/CRi ∆tM’’/CRi 0 Normal functioning interval Passive robustness interval Active robustness interval Figure 3: Robustness integration in the operation associated model • At the instant ∆tM/CRi (transition t7), there is automatically memorizing of a symptom S1i (place p7). • If the sensor signal CRi arrives in the time interval I4i = [∆tM/CRi , ∆tM′/CRi [, it is a case of passive robustness to a delay (place p8). The symptom S 1 i is already memorized (place p7). • If the sensor signal CRi arrives in the time interval I5i = [∆tM′/CRi , ∆tM′′/CRi [, a change of the control is necessary (active robustness to a delay, place p9). • At the instant ∆tM′′/CRi (transition t10), there is freezing of the control (place p10). 3 Localization of time disturbances in a given topology When, for example, the filtering mechanism executes a control freezing, it is necessary to know where the initial disturbance was occurred. This task is performed on a model of the workshop which uses P-time Petri net in order to integrate the staying time constraints in its structure. This aspect is presented in the following section. 3.1 Controlled P-time Petri net The formal definition of a P-time Petri net is given by a pair < R; IS >, where [12]: • R is a marked Petri net, • IS : P → (Q+ ∪0)×(Q+ ∪+∞) pi → ISi = [ai, bi] with 0 ≤ ai ≤ bi. ISi defines the static interval of staying time of a mark in the place pi belonging to the set of places P (Q+ is the set of positive rational numbers). A mark in the place pi is taken into account in transition validation when it has stayed in pi at least a duration ai and no longer than bi. After the duration bi the token will be dead. Using [15], controlled P-time Petri net is defined as a quadruplet Rpc = (Rp, ϕ , U , U0) such that: • Rp is a P-time Petri net which describes the opened loop system, Time Disturbances and Filtering of Sensors Signals in Tolerant Multi-product Job-shops with Time Constraints 65 p1 p5 p7 p4 p9 p3 p10 p8 p6 p2 t1 t2 t3 t11 t13 t5 t7 t6 t9 t10 t14 t12 t16 t17 t15 t4 Normal functioning Passive robustness Active robustness Control freezing Generation S 2 i and active robustness Generation S 2 i and Control freezing Generation S 2 i and passive robustness Generation S 1 i Ai*CRi CRi [0, ∆tm’’/CRi[*CRi I2i*CRi I1i*CRi I3i*CRi [∆tM/CRi, ∆tM/CRi] [0, ∆tM’/CRi–∆tM/CRi[*CRi [∆tM’’/CRi–∆tM/CRi, ∆tM’’/CRi–∆tM/CRi] [∆tM’/CRi–∆tM/CRi, ∆tM’’/CRi–∆tM/CRi[*CRi t8 Figure 4: Robustness integration in the filtering mechanism of sensors signals 66 Nabil Jerbi, Simon Collart Dutilleul, Etienne Craye, Mohamed Benrejeb • ϕ is an application from the set of places (P) toward the set of operations (Γ): ϕ : P → Γ, • U is the external control of the set of transitions (T ) built on the predicates using the occurrence of internal or external observable events of the system: U : T → {0, 1}, • U0 is the initial value of the predicate vector. Let us denote by: • TO : the set of observable transitions, • TU O : the set of non observable transitions, • TS : the set of synchronization transitions, • TNS : the set of non synchronization transitions, • TP : the set of parallelism transitions, • t◦i (resp. ◦ti) : the output (resp. the input) places of the transition ti, • p◦i (resp. ◦ pi) : the output (resp. the input) transitions of the place pi, • qie : the expected sojourn time of the token in the place pi, • Ste(n) : the nnd expected firing instant of the transition t, • St(n) : the nnd effective firing instant of the transition t. 3.2 Functional decomposition A workshop in repetitive functioning mode is modeled by a Strongly Connected Event Graph (SCEG) [13]. Performances of a SCEG running in mono-periodic functioning mode are proved to be the same as when using the K-periodic functioning [13]. Consequently, a mono-periodic functioning is used in order to decrease the complexity of the supervisory problem [14]. In this case, for each transition t, Ste(n + 1) = Ste(n) + π0 where π0 is the period of the periodic functioning of the given discrete event system. In this paper, the scheduling task is supposed to be done. Therefore, the SCEG corresponding to the system is provided. Moreover, the setting of transitions firing instants is fixed too. As the sojourn times in places have not the same functional signification when they are included in the sequential process of a product or when they are associated to a free resource, a decomposition of the P-time Petri net model into four sets is made using [15]. The assumption of multi-product job-shops without assembling tasks as it was established in [16] is used: • RU is the set of places representing the used machines, • RN corresponds to the set of places representing the free machines which are shared between manufacturing circuits, • TransC is the set of places representing the loaded transport resources, • TransNC is the set of places representing the unloaded transport resources (or the interconnected buffers). Time Disturbances and Filtering of Sensors Signals in Tolerant Multi-product Job-shops with Time Constraints 67 Figure 5, shows a P-time Petri net (G) modeling a system composed by two sequential processes GO1 and GO2 with two shared machines (M1, M2), where: RU = {p2, p4, p11, p13, p15}, RN = {p6, p7, p8, p9}, TransC = {p1, p3, p10, p12, p14}, TransNC = {p5, p16}, GO1 = (t12, p10, t6, p11, t7, p12, t8, p13, t9, p14, t10, p15, t11) and GO2 = (t5, p1, t1, p2, t2, p3, t3, p4, t4). The intervals (ISi) and the expected staying times (qie) associated to the places (pi) are: IS1 = [30, 50], q1e = 38, IS2 = [5, 12], q2e = 7, IS3 = [10, 20], q3e = 15, IS4 = [5, 20], q4e = 10, IS5 = [1, +∞], q5e = 10, IS6 = [0, +∞], q6e = 5, IS7 = [0, +∞], q7e = 8, IS8 = [8, +∞], q8e = 13, IS9 = [8, +∞], q9e = 15, IS10 = [5, 15], q10e = 12, IS11 = [15, 20], q11e = 17, IS12 = [3, 7], q12e = 6, IS13 = [2, 20], q13e = 5, IS14 = [2, 7], q14e = 5, IS15 = [15, 20], q15e = 16, IS16 = [1, +∞] and q16e = 19. The initial expected firing instants of each transition are: St1e(1) = 15, St2e(1) = 22, St3e(1) = 37, St4e(1) = 7, St5e(1) = 17, St6e(1) = 12, St7e(1) = 29, St8e(1) = 35, St9e(1) = 0, St10e(1) = 5, St11e(1) = 21 and St12e(1) = 0. The repetitive functioning mode is characterized by the period π0 = 40. Definition 3. A mono-synchronized subpath is a path containing one and only one synchronization transition which is its last node. Definition 4. An elementary mono-synchronized subpath is a mono-synchronized subpath beginning with a place p such as ◦ p is a synchronization transition. In figure 5, there are eight elementary mono-synchronized subpaths constituting a partition of G: L p1 = (p13, t9, p14, t10, p15, t11, p16, t12, p10, t6), L p2 = (p13, t9, p9, t1), L p3 = (p2, t2, p3, t3), L p4 = (p2, t2, p8, t8), L p5 = (p4, t4, p5, t5, p1, t1), L p6 = (p4, t4, p6, t6), L p7 = (p11, t7, p7, t3) and L p8 = (p11, t7, p12, t8). Property 1. A place pmp belonging to a sequential process represents a shared machine if and only if p◦mp ∈ TP or ◦ pmp ∈ TS. Property 2. The first node of an elementary mono-synchronized subpath is a place belonging to RU and representing a shared machine. 3.3 Time disturbances localization Let us remember some definitions. Definition 5. A time disturbance is detectable if, when it occurs, there exists at least one transition t ∈ TO such as St(n) 6= Ste(n). Definition 6. A time disturbance is quantifiable if its value can be analytically known. Definition 7. A time disturbance is localizable when its occurrence node can be identified. Definition 8. A time disturbance is partially localizable when its occurrence node location can be proved to belong to a given subset of P. Definition 9. A time disturbance is observable when it is detectable, quantifiable and localizable. Definition 10. The time passive rejection capacity interval of a path L p is RC(L p) = [Ca(L p), Cr(L p)] where: Ca(L p) = ∑ pi∈(L p∩ (RN∪TransNC )) (qie −bi), (1) 68 Nabil Jerbi, Simon Collart Dutilleul, Etienne Craye, Mohamed Benrejeb A place belonging to TransC A place belonging to RU A place belonging to RN A place belonging to TransNC IS4=[2, 20] q4e=10 IS1=[30, 47] q1e=38 IS2=[5, 12] q2e=7 IS3=[10, 20] q3e=15 t4 p4 t2 p2 t1 p1 t5 t3 p3 IS5=[1, +∞] q5e=10 p5 M2 M1 IS10=[5, 15] q10e=12 t8 p12 t7 p11 t6 t10 p14 t9 t11 p15 p13 p10 t12 IS11=[15, 20] q11e=17 IS12=[3, 7] q12e=6 IS15=[15, 20] q15e=16 IS14=[2, 7] q14e=5 IS13=[2, 20] q13e=5 IS16=[1, +∞] q16e=19 p6 p7 p8 p9 IS6=[0, +∞] q6e=5 IS7=[0, +∞] q7e=8 IS8=[8, +∞] q8e=13 IS9=[8, +∞] q9e=15 M1 M2 M3 p16 Figure 5: An Hillion like model with functional decomposition Time Disturbances and Filtering of Sensors Signals in Tolerant Multi-product Job-shops with Time Constraints 69 Cr(L p) = ∑ pi∈(L p∩ (RN∪TransNC )) (qie −ai). (2) Ca(Lp) (resp. Cr(Lp)) is called the time passive rejection capacity for an advance (resp. a delay) time disturbance occurrence. Definition 11. Let δ a time disturbance and SN a set of nodes belonging to a P-time Petri net. δ ∈ SN (resp. δ /∈ SN) means that the occurrence of δ is (resp. is not) in a node of SN. Used notations: • Cse is the set of elementary mono-synchronized subpaths. • IN(L p) is the first node of the path L p. • OU T (L p) is the last node of the path L p. • L p(t∗,t) is the oriented subpath of L p beginning with t∗ and ending with t. • Mn−1(L p(t∗,t)) is the number of tokens in L p(t∗,t) after the completion of the cycle (n−1). • Given a time disturbance δ , δ rt (n) is the resulting residue quantified at the transition t which is fired at St(n). • EC(IN◦(L p),t) is the set of oriented paths connecting the node IN◦(L p) of the path L p to the transition t. • H(IN◦(L p),t) = min Li∈[EC(IN◦(L p),t)\L p(IN◦(L p),t)] (Cr(Li)) + δ rt (n). • H′(IN◦(L p),t) = min Li∈EC(IN◦(L p),t) (Cr(Li)) + δ rt (n). Lemma 12. Let L p ∈ Cse, t ∈ (L p∩TO ∩TNS), t∗ ∈ (L p∩TO) and δ a time disturbance having a residue δ rt (n) 6= 0 quantified at the transition t. The following results are established [17]: δ rt∗(n−Mn−1(L p(t∗,t))) = 0 =⇒ δ ∈ [L p(t∗,t)\{t∗}], (3) δ rt∗(n−Mn−1(L p(t∗,t))) 6= 0 =⇒ δ /∈ [L p(t∗,t)\{t∗}]. (4) This lemma discusses the case of two observable transitions, t and t∗, such that t is not a synchronization one. When a disturbance is detected at a downstream transition t and is not detected at t∗, it is generated between these two transitions. Otherwise, the disturbance occurrence is outside the restriction of the considered path that connects t∗ to t. Lemma 13. Let L p ∈ Cse, t ∈ (L p∩TO), t p ∈ (L p∩TP), IL p = {Li ∈ Cse/ OU T (Li) = ◦IN(L p)} and δ a time disturbance having a residue δ rt (n) > 0 quantified at the transition t. The following assertion is true [17]: δ rt p(n−Mn−1(L p(t p,t))) < H′(t p,t) =⇒ δ /∈ { ⋃ Li∈IL p { Li \{IN(Li), IN◦(Li)} } ⋃{◦t p, t p }} . (5) 70 Nabil Jerbi, Simon Collart Dutilleul, Etienne Craye, Mohamed Benrejeb In other words, when the residue of the disturbance at the parallelism transition t p does not justify the residue at the transition t, forcibly the disturbance has not crossed t p. Lemma 14. Let L p ∈ Cse, t ∈ (L p∩TO ∩TS), t∗ ∈ (L p∩TO) and δ a time disturbance having a residue δ rt (n) > 0 quantified at the transition t. The following results are established [17]: δ rt∗(n−Mn−1(L p(t∗,t))) = 0 =⇒ δ /∈ [L p(IN◦(L p),t∗)\{IN◦(L p)}], (6) { 0 ≤ Cr(L p(IN◦(L p),t∗)) < H(IN◦(L p),t) δ rt∗(n−Mn−1(L p(t∗,t))) = 0 =⇒ { δ /∈ [(L p\L p(t∗,t))∪{t∗}] δ rIN◦(L p)(n−Mn−1(L p(IN◦(L p),t))) < H(IN◦(L p),t) , (7) { δ rt∗(n−Mn−1(L p(t∗,t))) 6= 0 δ rt (n) + Cr(L p(t∗,t)) 6= δ rt∗(n−Mn−1(L p(t∗,t))) =⇒ δ /∈ [L p(IN◦(L p),t)\{IN◦(L p)}]. (8) The above lemma discusses the case of two observable transitions, t and t∗, such that t is a synchroniza- tion one. Several results are given. If the residue at the transition t∗ is equal to zero, the disturbance does not belong to the restriction of L p between its only parallelism transition IN◦(L p) and t∗. If the disturbance has crossed the parallelism transition of L p (IN◦(L p)) and if its residue at IN◦(L p) is greater than the passive rejection capacity of the restriction of L p between IN◦(L p) and t∗, the residue at t∗ must be different of zero. Otherwise, the disturbance has not crossed IN◦(L p). If the residue at t∗ is different of zero and if it does not justify the residue at the transition t, the occurrence of the disturbance is not in the restriction of L p between IN◦(L p) and t. Lemma 15. Let L p ∈ Cse, t p ∈ (L p∩TP ∩TU O), t ∈ (L p∩TO) and Cr(L p(t p,t)) the time passive rejec- tion capacity of L p between t p and t for delay occurrence. Let us call DIF(t p) the set of paths beginning with t p. Let us denote DIFn(t p) the restriction of DIF(t p) such that: ∀L p′ ∈ DIFn(t p), ∀t′ ∈ L p′, we have St′(n + mt′) < St(n) where mt′ = Mn−1(L p′(t p,t′))−Mn−1(L p(t p,t)). Now, let L p′ ∈ DIFn(t p), t∗ ∈ (L p′ ∩ TO) and Cr(L p′(t p,t∗)) the passive rejection capacity of L p′ be- tween t p and t∗. Given a delay time disturbance δ , the following results are true [17]:    (t /∈ TS)∧(δ rt (n) > 0) δ rt (n) + Cr(L p(t p,t))−Cr(L p′(t p,t∗)) > 0 δ rt∗(n + mt∗) = 0 =⇒ δ ∈ [L p(t p,t)\{t p}], (9) { (t /∈ TS)∧(δ rt (n) > 0) δ rt∗(n + mt∗) 6= 0 =⇒ δ /∈ [(L p(t p,t)∪ L p′(t p,t∗))\{t p}], (10)    (t ∈ TS)∧(δ rt (n) > 0) Cr(L p′(t p,t)) < H′(t p,t) δ rt∗(n + mt∗) = 0 =⇒ { δ /∈ {◦t p, t p} δ rt p(n−Mn−1(L p(t p,t))) < H′(t p,t) , (11) { (t ∈ TS)∧(δ rt (n) > 0) δ rt∗(n + mt∗) 6= 0 =⇒ δ /∈ [L p(t p,t)\{t p}]. (12) Time Disturbances and Filtering of Sensors Signals in Tolerant Multi-product Job-shops with Time Constraints 71 When t p is a non observable parallelism transition, the following assertion may be used: if a disturbance modifies the t p firing instant, it must be seen downstream of t p. Consequently, when the value of the residual effect of the disturbance is greater than the rejection capacity of a given path, a residual variation has to be observed. The different lemmas formulated constitute a tool aiming to define the set of nodes where the disturbance may occur and the subset where it is proved that it did not occur. Then the question of using the above lemmas in order to make them collaborate has to be tackled. In other words, it remains to establish an algorithm using these lemmas while testing all mono-synchronized subpaths of the given P-time Petri net model. 4 Conclusions This paper deals with supervision in critical time manufacturing job-shops. In such systems opera- tion times are included between a minimum and a maximum value. A filtering mechanism of sensors signals integrating the robustness values is described. It provides the avoidance of control freezing if the time disturbance is in the robustness intervals. Therefore, it makes it possible to continue the production in a degraded mode providing the guarantees of quality and safety. It should be noted that the knowledge of robustness intervals is a significant parameter in the proposed mechanism. The assumptions formu- lated in these lines are very restrictive. It is natural to consider different scenarios where the temporal specifications of the process are not fulfilled, nevertheless the production can continue. It is necessary to introduce a finer classification of abnormal functioning and their impact on the considered systems. In this context, fuzzy logic can be used. When a symptom of an abnormal functioning is claimed by the filtering mechanism, it is imperative to localize the time disturbance occurrence. Based upon controlled P-time Petri nets as a modeling tool, a series of lemmas are quoted in order to build a theory dealing with localization problem. This is quite useful for the maintenance task. In the near future, it is essential to develop an algorithm using the lemmas results and providing localization of time disturbances. References [1] S. Calvez, P. Aygalinc, and P. Bonhomme, Proactive/Reactive Approach for Maintenance Tasks in Time Critical Systems, IEEE International Conference on Emerging Technologies and Factory Au- tomation (ETFA’2005), Catane, Vol. 1, pp. 947-953, September 2005. [2] S. Collart Dutilleul, and E. Craye, Performance and tolerance evaluation, SAFEPROCESS’03, IFAC Symp. on Fault Detection, Supervision and Safety for Technical Processes, Washington, June 2003. [3] A. Boufaied, A. Subias, and M. Combacau, Chronicle modeling by Petri nets for distributed detection of process failures, IEEE Conference on Systems, Man, and Cybernetics (SMC’02), Hammamet, October 2002. [4] P. Declerck, and M. K. Didi Alaoui, Modelling and analysis of P-time event graphs in the (min, max, +) algebra, IEEE Conference on Systems, Man, and Cybernetics (SMC’04), The Hague, Vol. 2, pp. 1807-1812, October 2004. [5] N. Jerbi, S. Collart Dutilleul, E. Craye, and M. Benrejeb, Observability of Tolerant Multi-product Job-shops in Repetitive Functioning Mode, IMACS’05, Paris, July 2005. 72 Nabil Jerbi, Simon Collart Dutilleul, Etienne Craye, Mohamed Benrejeb [6] A. E. K. Sahraoui, Contribution à la surveillance et à la commande d’ateliers, Ph.D. Thesis, Univer- sité Paul Sabatier, Toulouse, 1987. [7] A. Toguyeni, E. Craye, and J. C. Gentina, A method of temporal analysis to perform on-line diagnosis in the context of Flexible Manufacturing System, IECON’90, Vol. 1, pp. 445-450, Pacific Grove- California, November 1990. [8] M. Nourelfath, Extension de la théorie de la supervision à la commande des systèmes à événements discrets: application à la sécurité opérationnelle des systèmes de production, Ph.D. Thesis, INSA de Lyon, France, July 1997. [9] A. Toguyeni, Surveillance et diagnostic en ligne dans les ateliers flexibles de l’industrie manufac- turière, Ph.D. Thesis, Université des Sciences et Technologies de Lille, November 1992. [10] M. Diaz, Les réseaux de Petri - Modèles fondamentaux, Ed. Hermès, Paris, 2001. [11] N. Jerbi, S. Collart Dutilleul, E. Craye, and M. Benrejeb, Intégrat0on de la robustesse dans la super- vision de systèmes manufacturiers à contraintes de temps, Conférence Internationale Francophone d’Automatique (CIFA’06), Bordeaux, May 2006. [12] W. Khansa, P. Aygalinc, and J. P. Denat, Structural analysis of P-Time Petri Nets, Computational Engineering in Systems Applications (CESA’96), Lille, pp. 127-136, July 1996. [13] S. Laftit, J. M. Proth, and X. Xie, Optimisation of invariant Criteria for Event Graph, IEEE Trans. on Automatic Control, Vol. 37, no. 5, pp. 547-555, May 1992. [14] S. Collart Dutilleul, J. P. Denat, and W. Khansa, Use of Periodic Controlled Petri Net for Discrete Event Dynamical System Control Synthesis, ECC’95, Rome, pp. 2060-2065, September 1995. [15] J. Long, and B. Descotes-Genon, Flow Optimization Method for Control Synthesis of Fexible Man- ufacturing Systems Modeled by Controlled Timed Petri Nets, IEEE International Conference on Robotics and Automation, Atlanta, Georgia, Vol. 1, pp. 598-603, May 1993. [16] H. P. Hillion, and J. M. Proth, Performance evaluation of job-shop systems using timed event graphs, IEEE Trans. on Automatic Control, Vol. 34, no. 1, pp. 3-9, 1989. [17] N. Jerbi, S. Collart Dutilleul, E. Craye, and M. Benrejeb, Localization of Time Disturbances in Tolerant Multiproduct Job-shops Without Assembling Tasks, Computational Engineering in Systems Applications (CESA’06), Beijing, pp. 45-50, October 2006. Nabil Jerbi1,2, Simon Collart Dutilleul1, Etienne Craye1, Mohamed Benrejeb2 1Ecole Centrale de Lille Laboratoire d’Automatique, Génie Informatique et Signal Cité Scientifique, BP 48, 59651 Villeneuve d’Ascq, France 2Ecole Nationale d’Ingénieurs de Tunis Unité de recherche LARA-Automatique BP 37, Le Belvédère, 1002 Tunis, Tunisie E-mail: nabil.jerbi@isetso.rnu.tn, simon.collart_dutilleul@ec-lille.fr, etienne.craye@ec-lille.fr, mohamed.benrejeb@enit.rnu.tn Received: November 11, 2006