2005 lEEE lnternational Conference on Systems, Man and Cybemetics Waikoloa, Hawaii October 10-12, 2005
Distributed Fault Diagnosis Using Petri Net Reduced Models J. Arámburo-Lizárraga, E. López-Mellado, A. Ramirez-Trevüío CINVESTAV Unidad Guadalajara. Prol. LópezMateos Sur 590,45090 Guadalajara, Jal. Mexico
[email protected] Abstract - This paper 1.7 concerned with the distrihuted diagnosis of Discrete Event Systems (DES) using reduced Interpreted Petri Nets (IPN) modeL7. Two contrihutions are presented extending a previous work ,fiom the authors. First, the condition of event detectahility is relaxed over parts of the model where the faults are not expected; thus the diagnoser handles LI reduced model. Second. a method ,jor splitting the glohal moakl into communicating modules is proposed; ihis k a d s to the design o f a set of distrihuted diagnosers. Keywords: Reduced diagnosers, Distributed diagnosis, Discrete Event Systems, Petri nets.
1
Introduction
Opporhme fault detection and isolation ailows to minimise risks to humans and damages to equipment during the operation of large and wmplex systems; it wnstitutes an important first stage of a fault recovery process within an autonomous control system. Failure diagnosis of automated discrete event systems (DES) received special attention in the literature through several approaches and methods. Regarding the model based approach severa1 methods using finite automata (FA) [2], [3] and Petri Nets (PN) [4], [5], [6], [7], [l] have been proposed to address this problem. In these works it is studied the diagnosability property and fault detection schemes based on a centralised approach using the global model of the DES. Distributed systems appeared in industry in some natural ways (computer and telecommunication networks, manufacturing process control and power systems, etc.) [15]. The challenges arising from the constmction of distributed systems are the heterogeneity of its components, openness, which ailow to add or replace components, secunty, scalability, failure handling, concwency of components, and cleamess [16]. Recently, fault diagnosis of DES has been addressed through a distributed approach allowing to break down the complexity when dealing with large models [8], [9], [lo], [ l l ] , [12], [13]. In [8] it is proposed a decentralised and modular approach to perform failure diagnosis based on Sampath's results [¿l. O. Contant et al. [lo] and Y. Pencolé [9] propose incrementai algorithms to perform diagnosability analysis based on [2] in a distributed way; they
consider systems whose components evolve by the occurrence of events; the parailel composition leads to a complete system model intractable. In [ l l ] S. Genc and S. Laforhme proposed a method that handles the reachability graph of the PN model in order to perform the analysis similar to [S]; based on design considerations the system model is partitioned into two labelled PN and it is demonstrated that the distributed diagnosis is equivalent to the centralised diagnosis. In our approach we use the interpreted PN (IPN) model of the system including the possible faults that may occur. In a previous work we proPosed a structural charactensation for diagnosability that avoid the reachability analysis and an on-line diagnoser scbeme [ 6 ] . In this paper we extend that work by proposing a relaxation of the condition of event detectability over parts of the model where the faults are not expected; thus the diagnoser handles a reduced model. Also, a method for designing a set of distributed diagnosers by splitting the global model into communicating modules is proposed. This paper is organised as follows. In section 2 basic definitions of PN and IPN are provided. Section 3 summarises the concepts and results for centralised diagnosis. Section 4 presents a methodology for designing reduced diagnosm. Finally, section 5 presents the methodology to split a global models into a set of comrnunicating local models.
2
Background
This section presents the basic concepts and notation of PN and IPN used in this paper. Definition 1: A Petri Net structure G is a bipartite digraph represented by the 4-tuple G=(P,TJ.0) where: P = {pl.p2,....p.,) and T = {ti.t2,...,t,) are finite sets of vertices calied respectively places and transitions, 1 10): P x T + Z ' is a function re~resentine " the weighkd arcs going from places to transitions (transitions to places), where L; 'is the set of nonnegative integers. \
,
Pictorially, places are represented by circles, transitions are represented by rectang1es7 and ares are depicted as mows. The symbol 'tj ($9 denotes the set of al1 places P; such that
[email protected])#o (o@i,tj#o). Analogously, @;9 denota the set of al1 transitions ~ c that h o@;.ti>#o (Ik.fi)fo).
>;
The pre-incidence matrix of G is C - = [ c i ] where cij = I ( p i , t j ) ; the post-incidente matrix kf G is C: = [C; 1 where c i = q p ,,t ); the incidente matrix of Gis C = C + - C - .
Definition 5: Let
U
= titj ...tk ... be a finng transition
sequence. The Parikh vector ;:T -r ( Z ) m of o maps every t E T to the number of occurrences of t in u. According to functions ?and . q,transitions and places A marking function M : P-rZ + represents the number of an IPN (Q.M,))can be classified as follows. of tokens (depicted as dots) residing inside each place. The Dejnition 6: If h(t;)#~the transition t, is said to be marking of a PN is usually expressed as an n-entry vector. manipulated. Otherwise it is nonmanipulated. A place pie P Definition 2: A Petri Net system or Petri Net (PN) is is said to be measurable if the i-th column vector of q is the pair N=(G.M,)),where G is a PN structure and M,, is an not null, i.e. q (*,i)#O.Othenvise it is nonmeasurable. initial token distribution. The following definitions relate the input and output in a PN system, a transition ti is enabled at marking symbol sequences with the fuing hnsition sequences and Mk if Vp; E P. Mk(pj)? I(p,t,);an énabled transition ti c& the generated marking sequences. These concepts are be fued reaching a new marking M k , , which can be useful in the study of the diagnosability property. computed as M k l/ = Mk + Cvk,where vk(i)=O,ifj, vkCi)=l, Definition 7: A sequence of input-output symbols of this equation is called the PN state equation. The (Q.M,))is a sequence o = (a,j,y,))(a,,y~)...(a~,~~), where aj E reachability set of a PN is the set of al1 possible reachable Z U { E ) and a i l Iis the current input of the IPN when the marking fiom M,)finng only enabled transitions; this set is output changes from y, to y,, 1. It is assumed that a,)= E , yo= denoted by R(G,Mn). d M , )and (a, ,,y, ,) belongs to the sequence when: This work uses Interpreted Petri Nets (IPN) [14] an (a,y,) belongs to the sequence, extension to PN that allo; to associate input and output Y U I # Y ; ,and signals to PN models. there exists no yj # y¡, y j # y i l , occurring a f t a the Definition 3: An IPN (Q. M,)) is an interpreted Petri occurrence of y, and before the occurrence of y,, Net structure Q=(G,Z,A.q) with an initial marking h. Definition 8: Let (Q,M,j) be an IPN. The set G isaPNstructure. A(Q,M,))={olo is a sequence of input-output symbols) C = {a,.a2.....a,)is the input alphabet of the net, where denotes the set of al1 sequences of input-output symbols of a, is an input symbol. (Q.M0).The set of al1 input-output sequences of length ?. : T + ~ u { E )is a labelling function of transitions with greater or equal than k will be denoted by A~(Q.M,)), i.e. the following constraint: V$tk E T, j#k, if Vp;I(pi,t,)= A'(Q.w) = A(Q.M,))II~I?~). I(p,tk) # o and both a-($)# E, V t k )# E, then V t j )# Vtk). Definition 9: Let o = (aoyO)(a,,yI) ...(an,yn) be a In this case E represents an internai system event. sequence of input-output symbols, then the finng transition 4 : R(Q.M,))-r( Z ')' is an output function, that associ- sequence o E £(Q,M,))whose finng actually generates o is ates to each marking in R(Q.Mn)an output vector. Here denoted by o,. The set of al1 possible finng transition sequences that could generate the word o is defined as q is the number of outputs. ) finng of o produces o ) . In this work q is a q xn matrix. Each column of g, is n ( o ) = { o I o ~ £ ( Q . MA, ~the Definition 10 :The prefix of a sequence s is another an elementary or null vector. If the output symbol i is present (tumed on) every time that M@,)? 1, then q (i.j)-1, sequence S' such that theie exists a sequence S" @lfilling that S=S'S". The set of alI prefixes of s is denoted by s . othenvise di,j)=O. Definition 11 : The set of al1 input-output sequences A transition tj E T of an IPN is enabled at marking Mk if Vp, E P, Mk(pj)1 I(p,t,).If ?.($) = a, # E is present and $ leading to an ending marking in the IPN (markings enabling no transition or only self-loop transitions) is is enabled, then $ must fire. If ?.(tJ)= E and t, is enabled then denoted by AB(Q.&), i.e., hB(Q.Mn) = { ~ E A ( Q , M J ) I $ can be fued. When an enabled transition $ is fired in a ) that Mo -%MJ and 4enables no marking Mk,then a new marking Mk( ,is reached. This fact 3 o ~ n ( w such transition, or when M , -% then C(*,ti)=O). is represented as Mk -bM&+, ; Mk, can be computed Definition 12: An IPN (Q,Mo) = (N,X,@,hD.q) using the state equation: descnbed by the state equation (1) is event-detectable iff Mki I = Mk + C V ~ (1) the fuing of any pair of transition t,, t, of (Q,M,j) can be Yk = dMk) distinguished fiom each other by the observation of the where C and vk are defined as in PN and yk E (Z ')q is the sequences of input-output symbols. k-th output vector of the IPN. The following lemma [14] gives a polynornial characDejnition 4: A finng transition sequence of an IPN tensation of event-detectable IPN. (Q,Mll) is a transition sequence o = trtJ...tk ... S U C ~that Lemma 13: Let (Q.M,))= (N,X,@,?.,D,q) be an IPN Mo +MI - b . . . ~ . r +... The set £(Q.Mn) of descnbed by the state equation (1). (Q.M,,) is evental1 fuing transition sequences is called the fuing lan- detectable iff al1 giC columns are not null and different fiom guage£(Q.M,))= { o = t , t , ..f k ... A M o - - % ~ i ... each other. Mr A... }.
,
,.
3
Centralised Diagnosis
The main results on diagnosabiiity and diagnoser desing in a centralized approach presented in [6] are outlined below.
3.1
Diagnosability
considered. In ( e N ,M: ), the set of places is P Y =P and the set of transitions is y=T- f . Now, the following result provides sufficient structural conditions for determining the input-output diagnosability of an IPN model. Theorm 16: Let (Q.M,j) be a binary IPN ohtained with the proposed modeling methodology, such that ( e N ,M : ) is live, strongly connected and event detectable. Let (X,,...Y ),. be the set of al1 T-semiflows of (Q,M,,). If V p; t Pv,p; n f #O the following conditions hold l . Vr, 3j XA)? 1, where $ t (py )'_ f .
The characterisation of input-output diagnosable IPN is based on the partition of R(Q,Mo) into normal and faulty markings; al1 the faulty markings must be distinguishable fiom other reachable markings representing normal functioning. R(Q,M,,) is composed o f a) the set of the 2. Vt, E ).- f , .(tk)=( pfl ) and l(tk)7 t faulty markings F = {M 1 3pk E PF S U C ~that M ~ ~ ) > o , then the IPN (Q,Mo) is input-output diagnosable. MtR(Q.M,j)l and, b) the set of the normal states N = ProoJ Included in [6] R(Q,M") - F. De$nirion 14: An IPN given by (Q,Mo) is said to be 3.4 Diagnosability test input - output diagnosable in k < a, steps if any marking Determining when an IPN is input-output diagnosable M, E F is distinguishable from any other Mk t R(Q,M,) is reduced to the following tests[6]. using any word ~ E A ~ ( Q , M u [ ) AB(Q,M,). 1.Binanty of (Q.Mo). It is fulfilled because permissive and synchronic compositions of binary modules lead to binary 3.2 System modeling IPN. Defnition 15: The sets of nodes are partitioned into 2.Liveness of modules can be tested efficiently; however faulty nodes ( p . places coding faulty states. and f , the property is not preserved in the composed IPN model transitions leading to faulty states) and normal functioning when it result from the arbitrary application of the composition operators. Thus some constraints in the nodes (Pv and T'); so P = f u P' and T = ~ u T ' . pfl application of synchronous and permissive compositions denotes a place in PVof M[ ). Since PN P then N could guarantee that iiveness is preserved. Such constraints P , also belongs to (Q,Mo). The set of nsky places of may foiiow the rules introduced by Koh and Dicesare in (Q,Mo) is = ' f . The post-nsk transition set of (Q,Mo) is [ 171 for module composition. P = P nY . 3.Event detectability and strongly connectedness are Figure 1 presents an IPN model of a system obtained determined in polynomial time as well as detecting places with the modeling methodology presented in [S]. Notice P? E Pv such that 7. O . Then condition 2 of that this model has three faulty states, represented by places previous theorem is efficiently tested. p3,p8,p13.Function 1is defined as h(tl)=a, h(t,)=b, h(t4)=c 4.Finally, condition 1 can be venfied in polynomial time. In h(t7)=d, h(t,)=g, h(tll)=h, h(t14)=m, for others transitions this case we need to check that there exists not T-semiflow W = E . Mea~urablepkces are P,, ~ 4P,,, P,, PI,, Pl4r P I ~PIH, , that does not include transitions in ( p f l E PN such that p2o. p21 and p = ~ P Z . P ~ , P~=(t2.t7,t14}. I~},
6:
(eN
bfln +
( p f l n 7.f 8. Thus we need to check that the following linear prograrnming prohlems have no solutions. Vp> ~ s u c h t h a pt > P , i3X s.t. CX-O, XCi)=O,
vi, 3.5 Figure 1. Global system model
3.3
Characterisation of diagnosabiiity
The embedded normal behavior IPN ( e N ,M/ ) is the IPN included in (Q,M,,) when f and f = 'fare not
b;).
b ~ r fnf O .
-7.
A centralised scheme for diagnosis
Diagnosability tests given above, provide the basis for designing an on-line diagnoser (see figure 2). Such diagnoser can detect a fault and locate faulty markings reached by an IPN. The proposed scheme for diagnosis [5],[6] handles (QN,M/) which must evolve similarly to the system; the outputs of both the system and the model are compared and, when there is a difference, a procedure is started wmputing the faulty marking.
fired. When a difference between the outputs is detected then the diagnosis procedure operates as descnbed above.
5
Figure 2.On-line diagnosis scheme
4
Reduced Diagnosers
The condition of event detectability may be relaxed on rj e 'p and rj e p.;these transitions are not involved dunng the detection of faults when (QN, M:) is handled. We profit of this fact to obtain a reduced model M:" ) containing the pertinent parts of M:)
(eRM,
(eN,
Building a Distributed Diagnoser
The distributed scheme for on-line diagnosis is depicted in figure 4; it is a straightforward extension of the centraiised scheme of figure 2. In this scheme there exist r "local" diagnosers corresponding to the module in which the system model is partitioned. Every diagnoser is composed of the embedded reduced normal behavior model corresponding to a module, and a diagnosis procedure; it handles subset of inputs and outputs regarding the module behavior. Since the local models may handle common events the diagnosers communicate for implementing synchronisation and inform of eventual faults arise within the wrresponding part of a diagnoser module. The functioning is the same but the precise - of the diagnosers behavior is determined by the corresponding local model.
regarding the modelied faults in (Q.M,).Consequently, the diagnoser monitors the system only when faults may arise. The procedure performing this task is presented below .
* n i
lwd
4
m
b,~")
Al~orithm17: Reduced model Input: (Q,M,) Output: ( Q R M . MOM)
vp, E pR then
(QRM
,MOM) is defmed as the subnet induced by:
PRM = ~ i } u P ; ' T
RM
= .pl u P,* u t r
l . . . . . .
, tris a transition that it fire
when p;" is rnarked. Include the input symbols associated with 7'? Include the output symbols associated with p" if M&¡) > O then M Y @ , ) O Using the previous procedure on the model of figure 1, the obtained reduced model is shown in figure 3. Notice
that in this example the number of places is reduced to a third of the complete.
Figure 3. Diagnoser's reduced model The diagnoser scheme of figure 2 handles the reduced model and compares the outputs when this model is "active" i.e. when the ñrst event detectable transition is
-
-............................................................................... -.
Figure 4. Distributed scheme for diagnosis
5.1
Local IPN models
In order to build a decentralised diagnoser, the IPN model of the system must be partitioned or distributed. A module or partition of (Q,Mo)is a subnet of (Q,M,,), where different modules can share common nodes, but there is no arcs between modules. Thus two cases arises. when modules share transitions or places. Then two net transformation operations are used to transform the modules. When there exists a synchronisation transition rj and we require to partition this transition to obtain two modules. then the transition rj is duplicated in r j , and rj-. Then places ph E 'rj and ph E rj* are also duplicated in ph, and ph .,. Al1 these duplicates nodes are connected in the same way that the original nodes. See figure 5 . When there exists a conflict place p, and we require to partition this place to obtain two modules. then the place p, is duplicated in p,. and pj-. these duplicates places are connected in the same way that the original place. See figure 6.
One set of duplicated places is lefi in one module and the other set is left to the other module. Incoming arcs to places are dashed if they are coming fiom the other module. These dashed arcs indicate the communication among modules. Notice that the new places are implicit ones, thus the IPN behavior is preserved. The following algonthm is proposed to get a model equivalent to the centralised one. Algorithm 18: Partitioning IPN Input: IPN N = (Q. Mí,) output: IPN N' = (Q :M,,*)such that f(QS,Mo'1 = f(Q ',MI,') and A(Q.Md = A(Q '.M"') (synchronisation transitions) or Vii E Vpi E P ~ " "(conflict places) do Duplicate t, in 1,. and t,,: places p k E '9 andpk E ti' are also duplicated in pk. and p k,,, or Duplicatep, i n p , andp,,., Build the incorning arcs, i.e., duplicates places (transitions) are comected in the same way that the original place (transition).
t, (p,)represents the chosen synchronisation transition (chosen conflict place) for partitioning the IPN.
The next step is to define the diagnoser modules and the interaction links that may be implemented in communication channels. Algorithm 19: Building diagnoser modules. Input: IPN N1=(Q',MI, ] Output: Di, i=l,..,r (number of modules). The distribiited diagnoser. Let Nu = {N, :...,N,') be the set of modules of net N: Build the set of embedded normal behavior models
N: = [Q~,M:),) . For each (Q",M:)¡ ( Q R M , MDRM)i
Moduk I
Module2
ModukI
Module2
Figure 6. Splitting a model by places The result is an IPN N1=(Q',M1,'). where the comections among modules are performed through dashed arcs. Since ail added nodes are implicit ones, then f(Q '.M" 7 = £(Q '.M,') and A(Q.Mo) = A(Q '.MI,'). The obtained IPN N'=(QV,MO')is another way to represent the behavior of the system modeled by the IPN N=(Q,M"). This new net N'=(Q7,Mn7will be used to design a distributed diagnoser.
the
reduced
model
using the algonthm 17.
Define k communication channels C C ~ of ' module N¡', i=l ,...,k,one for each outgoing dashed arc. p . In the resulting distributed scheme the finng of each transition can be detected, since N'=(Q'.L~~~,] is event detectable on 'pand p.. Thus, every time that the finng of one transition is detected and this transition has an outgoing dashed arc, the information of this firing is propagaíed by the appropnate comrnunication channels to the modules needing this information.
5.2
Figure 5 . Splitting a model by tmsitions
build
Local diagnoser procedure
Diagnosabiljty tests given ahove (see suhsection 3.4). provide the basis for designing a diagnoser. Such diagnoser can determine faulíy markings reached by an IPN. The diagnoser scheme herein used (see figure 4) contains a copy of the underlying normal reduced behavior of the IPN model, a communication procedure that has the function to send the common events for implementing synchronisatjon and inform of eventual faults ansing within the corresponding part of a diagnoser module and an algorithm to compute which faulty marking was reached. The underlying normal reduced behavior wiil be named a local diagnoser-model IPN and it is defined as follows. Dejnition 20: Let N1=(Q'.M0] an IPN modeling a system, where the condition of theorem 16 hold; let N1,=(Q'.Mo3 be any IPN module of N, where the net (QRM,MORM), is the embedded normal reduced behavior of (Q ', M0 3 and the firing rule is defined as follows: 0 If t, was fired in the system and $ E pw, then it is fued in (Q '.MI, 1. There exists one diagnoser-model for each module.
5.2.1
Pault detection
The error between the system output and the local ~) diagnoser-model output is Ek = p7(Mk) - y l ( ~ f where p'(Mk) is restricted to the corresponding measurable places in p .The following algonthm, devoted to detect which faulty marking was reached in N',=(Q ',Mr,3, is executed when Ek(,$ O.
Algorithm 21: Detecting Faulty Places Inputs: q(M/U) - the output of the local diagnoser-model. i ( t j ) - the input symbol of the system Ek,- diagnoser error 0utuuts: D - the isolated faultv dace 1 .Constants: deU) . . . - embedded normal behavior 2.Repeat 2.a. Read y, ( M ~ U ) 2.b. If q(MFU)fd M k . T )then computes q = u l ( ~ / ~ ) - d M(a~column . ~ ~ )of di(ePU)) 2.c. i = index of the column of d e Usuch ) that d e U ) ( * , i )= q. then ti was fired; p E 'ti 2.d. If EkfO then - vp E .ti M ~@)=O ~ . ~ - Vp E tieMPw@)=0 - vpF€ (*ti) n F, M X ~ ) =1 - Retum (uF)
References
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-
(eRM,
