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A Spectrum of Symbolic On-line Diagnosis Approaches

1. any compilation at all is space efficient but very slow Our second model incorporates a limited form of compilation arising from performing synchronisation off line The global model is the synchronous product of the n component models its state space is the Cartesian product of the state spaces of the components and its transitions are synchronised in that any shared event always occurs simul taneously in all components that define it Similarly to the component models it is symbolically represented as G bX bX b gt ao To Ts Te where b U _ b resp bX UT_ b is the union of the components local state resp primed variables State zo Axo is the initial state Also the BDDs To Ts Tf rep resenting the global observable shared and fault transitions are computed from the local transitions mainly by applying the operator Abstracted Model Diagnosing based on the global model is also not very ef ficient since only limited information about unobservable events has been compiled away We therefore add another model to our spectrum the abstracted model which is de rived from the global one by abstracting all unobservable non fault transitions and the order in which faults can occur Hence its states X are obtained from the global ones by re moving all states except the initial one that are not the start or target state of an observable transition T All sequences of unobservable events are replaced by
2. on line the results of a set of diagnosers compiled for small subsystems We plan to extend our sym bolic spectrum to decentralised approaches such as this one Another line of future work is to extend our framework to stochastic systems and compute probability distributions on diagnoses using for instance algebraic decision diagrams which are generalisation of BDDs to real valued functions over the booleans Finally integrating diagnosis and plan ning for repair or reconfiguration actions is one of the most significant challenges faced by the field of model based diagnosis Console amp Dressler 1999 Given the recent success of planning techniques based on symbolic model checking we believe that our framework will prove a good basis for addressing this challenge References Baroni P Lamperti G Pogliano P and Zanella M 1999 Diagnosis of large active systems Artificial Intelligence 110 1 135 183 Brusoni V Console L Terenziani P and Dupre D T 1998 A spectrum of definitions for temporal model based diagnosis Artificial Intelligence 102 1 39 79 Bryant R E 1986 Graph based algorithms for boolean function manipulation IEEE Trans on Computers C 35 8 677 691 Cimatti A Pecheur C and Cavada R 2003 Formal verifi cation of diagnosability via symbolic model checking In Proc IJCAI 03 363 369 Console L and Dressler O 1999 Model based diagnosis in the real world lessons learned and cha
3. a single transition la belled with the union of the corresponding faults which can be empty if the sequence consists only of shared events The set T of these new transitions is defined as I 112 O1 Ok 1 Ok x gt a 3 path x Lyre k 1 a c E X 01 0 EX and 1 o1 0n N Ep Figure shows an example of an abstracted model In the symbolic setting the abstracted model G 0 0 b bF o To Tr is encoded using the same boolean state variables as the global model the subset b of boolean variables represent ing the observable events and an additional p variables bF fol F C 2 1 There is a one to one correspondence between fault events and these variables and a fault transition label is encoded as a conjunction of literals over b whose signs depend on whether the corresponding fault belongs to the la bel Note that the abstracted states X C X are encoded over the same boolean variables as the global ones since their number is not significantly smaller than X oh needed for the fault transition labels Diagnoser Model The abstracted model still requires the on line computation of fault information which slows down on line diagnosis We therefore also consider a diagnoser model in which this entire information is compiled A diagnoser is a determin istic finite state machine whose transitions are only labelled with observable events and whose states are directly lab
4. of the models and algorithms using optimised automata data structures which facilitate the manipulation of individual states and transitions These two implemen tations enable us to present experimental evidence that the symbolic approach yields important gains in time or space Our experiments below were run on a 1 2 GHz Pentium IV with 512 Mb of memory We first use the largest ex ample in Schumann Pencol amp Thi baux 2004 which is derived from a telecommunication application It consists of 3 components a switch with 12 states and 18 transitions a primary control station of 13 states and 15 transitions and a backup control station of 19 states and 28 transitions 9 observable events 11 fault types and 8 other unobservable events We generated by simulation 100 arbitrary scenarios possible sequences of observations of 10000 observations each and used them as input to all models Figure 2 compares the time performance of the various on line diagnosis methods All symbolic models except the diagnoser are more efficient to use than their enumerative counterparts This should not come as a surprise BDDs are well suited to triggering transition sets and enable the consideration of all diagnosis tuples at once but do not gen erally pay off when only a single transition is involved as is the case with the diagnoser The differences in symbolic diagnosis times across the spectrum correlate with the extent to which the accumula
5. times over 10 scenarios of 1000 observations each Conclusion Related amp Future Work We have presented a spectrum of symbolic diagnosis ap proaches which differ in the amount of model compilation performed off line The underlying models range from the small component models that do not incorporate any compi lation to the diagnoser model in which the diagnosis infor mation is compiled for the entire observable behaviour of the system The abstracted model constitutes an interesting al ternative to the diagnoser it is considerably smaller but not much slower and so applies to a wider range of applications Thanks to the symbolic implementation we are able to handle large sets of transitions and diagnosis hypotheses at once This leads to a simple and efficient way of obtaining the correct and complete set of diagnoses In comparison to an enumerative implementation only the on line use of the symbolic diagnoser incurs a small time overhead In all other cases the run time of the symbolic approach is sig nificantly reduced and so are the space requirements of the larger models Therefore an enumerative approach is mainly useful for very small applications for which the com putation and storage of the large diagnoser is feasible There are only few other works presenting results ob tained with generic on line diagnosis software The UMDES Library http www eecs umich edu umdes provides an enumerative implementation of Sam p
6. tion of faults function AddFault described on page 4 is performed on line Even though a fault f can be simul taneously added to all fault labels Ly AddFault still re quires the individual consideration of fault labels in gen eral Ly Ly for fi fj which is the main bottleneck of the symbolic computation The component and global models yield similar diagnosis times because AddFault is applied the same number of times in both cases and sym bolic synchronization is very fast In contrast the abstracted model yields significantly faster diagnosis times because AddFault only needs to be applied once per observation With the diagnoser AddFault is never called Taken in conjunction with the diagnosis times the corre sponding model sizes see Table 1 illustrate the time space tradeoff of the methods across the spectrum and the supe riority of the symbolic approach Comparing the symbolic models resp the enumerative ones we can state that the faster the on line diagnosis based on a model the larger the G G G states Nr 17 7 1063 965 18474 transition Nr 34 2912 48958 120698 space symb Mb 0 01 0 2 0 6 Bs space enum Mb 0 01 0 2 2 7 123 9 Table 1 Model sizes model size For all models the symbolic representation is as small as or smaller than the enumerative one yet except for the diagnoser the symbolic run times are significantly bet ter Importantly the symbolic diagnoser is as
7. A Spectrum of Symbolic On line Diagnosis Approaches Anika Schumann National ICT Australia amp The Australian National University Canberra ACT 0200 Australia Anika Schumann anu edu au Abstract This paper deals with the monitoring and diagnosis of large discrete event systems The problem is to determine on line all faults and states that explain the flow of observations Model based diagnosis approaches that first compile the di agnosis information off line suffer from space explosion and those that operate on line without any prior compilation have poor time performance Our contribution is a broader spec trum of approaches that suits applications with diverse time and space requirements Approaches on this spectrum dif fer in the amount of reasoning and compilation performed off line and therefore in the way they resolve the tradeoff be tween the space occupied by the compiled information and the time taken to produce a diagnosis We tackle the space and time complexity of diagnosis by encoding all approaches in a symbolic framework based on binary decision diagrams This allows for the compact representation of the compiled diagnosis information and for its handling across many states at once rather than for each state individually Our experi ments demonstrate the diversity and scalability of our sym bolic methods spectrum as well as its superiority over the corresponding enumerative implementations Introduction Ther
8. alia amp The Australian National University Canberra ACT 0200 Australia Sylvie Thiebaux anu edu au The diagnoser approach Sampath et al 1996 is the archetype of compilation based techniques Off line it compiles all possible diagnoses into a finite state machine the diagnoser On line this machine is simply run to effi ciently retrieve the diagnoses explaining the current flow of observations Unfortunately diagnosers can be so large that they are not computable for all but the smallest applications On line simulation based approaches Baroni et al 1999 fall in the no compilation camp They directly compute the diagnosis from the behavioral model of the system by sim ulating possible trajectories Here the space requirements are reasonable but the simulation time can be excessive for large applications Clearly we need a more flexible resolution of the tradeoff between on line and off line computation that is between time and space Research in that direction includes the de centralised diagnoser approach Pencol amp Cordier 2005 which precomputes diagnosers for small subsystems only but needs to ensure consistency of the local diagnoses at run time Another recent line of work deals with the incremental on line compilation of diagnosis information and its reuse Lamperti amp Zanella 2006 In this paper we take an orthogonal approach to resolving the space time tradeoff We present a spectrum of methods which di
9. ath s diagnoser Sampath et al 1996 UMDES cannot compete with either our symbolic or enumerative implemen tations In fact one of our motivations to implement our own enumerative algorithms was that UMDES was unable to compute the diagnoser for the smallest of the examples given in Schumann Pencol amp Thi baux 2004 The idea of exploiting symbolic representations in the context of discrete event systems diagnosis is not new but it has traditionally been applied to different problems e g checking diagnosability Cimatti Pecheur amp Cavada 2003 Rintanen amp Grastien 2007 off line diagnosis using off the shelf model checkers Cordier amp Largou t 2001 or computing a symbolic diagnoser Marchand amp Roz 2002 Schumann Pencol amp Thi baux 2004 In contrast we ex ploit the power of the symbolic representation to design a range Of efficient on line diagnosis approaches Symbolic representations based on Decomposition Nega tion Normal Forms DNNFs have successfully been applied to diagnosing static systems see e g Darwiche 1998 For diagnosing dynamic systems however BDDs are better suited because the main operation namely the triggering of transitions can be performed in polynomial time in the size of the BDD while it would require exponential time in the size of the DNNF In Pencol amp Cordier 2005 the authors resolve the time space complexity tradeoff using a single approach which merges
10. b A bx A wbx and the set of states 2 25 by the DNF abs A bs A AbX V bs A bx A OX Transitions require the introduction of another set of state variables b b bx n x te called the primed vari ables which are used to represent the target states of the transitions Each transition can then be given as a con junction involving the state variables event variables and primed variables For instance in a FSM consisting of 6 states and 3 events the transition t x2 x5 would be given by t b AbX A bX A 20 ADP A bX Ab A bX N The transition relation i e a set T of transitions can then be given as a DNF which the BDD data structure will hopefully greatly reduce Spectrum of Symbolic Diagnosis Models This section formally defines four symbolic models on which we base our diagnosis algorithms These models are inspired from Schumann Pencol amp Thi baux 2004 Rather than merely using them as successive steps in the computation of a symbolic diagnoser as Schumann et al do we adapt them and build efficient on line diagnosis al gorithms upon each of them These algorithms and their evaluation are the main technical contributions of the paper The models differ in the extent to which information is compiled starting with the component models the simple representation of the system without any precomputation to the diagnoser model which compiles the diagnosis informa tion for every possible sequ
11. e is an increasing need for automated monitoring and supervision tools for large discrete event systems in areas as diverse as telecommunication power distribution manufac turing spatial exploration and web services Such tools aim at assisting the operator in charge of the system supervision with tasks that include diagnosis reconfiguration and con trol This paper is concerned with automated diagnosis and more specifically with the on line identification of the faults that explain the continual flow of observations received from the system Existing model based approaches typically fall into two categories In the first a significant amount of off line reasoning is performed to compile the system model into a larger model that embeds diagnosis information This information generated once and for all is then exploited on line to more efficiently produce the diagnosis from the actual observations In the second category no such compilation is performed and all the reasoning is done on line This work was supported by NICTA s SuperCom project NICTA is funded through the Australian Government s Backing Australia s Ability initiative in part via the ARC Copyright 2007 American Association for Artificial Intelli gence www aaai org All rights reserved Yannick Pencol LAAS CNRS University of Toulouse 7 avenue du Colonel Roche 31000 Toulouse France Yannick Pencole laas fr Sylvie Thi baux National ICT Austr
12. e strength of our approach and conclude with related and future work Symbolic Finite State Machines Ordered binary decision diagrams OBDDs or BDDs for short Bryant 1986 are a form of reduced decision graph that provide a compact canonical representation of boolean functions B B While the BDD representation still re quires exponential space in the number of boolean variables in the worst case the reductions often make the BDD of a function much smaller than its disjunctive normal form DNF Any boolean operation f x g on two BDDs f and g can be carried out in O f g at most where f denotes the number of nodes in the BDD f In our approach the finite state machines FSMs de scribing our diagnosis models are encoded symbolically by means of BDDs and all diagnosis algorithms are imple mented in terms of BDD operations This confers us the ability to compactly represent and efficiently manipulate sets of states and transitions To encode the set of states X and the set of events X of a FSM it is necessary to introduce Nr Q logs Q boolean variables for each set Q Thus the events labelling the transitions can be encoded with the boolean variables Oe Lie cass bN and the states with the variables be E ale ON xy The initial state of the FSM is then simply given by a boolean function represented by a BDD over these state variables For instance in a 6 state FSM the state x2 would be given by the conjunction a
13. elled by the diagnosis information that is consistent with the past observations This information consists of a sets of pairs x l denoting a state and a fault label of the abstracted model Let X be the set of diagnoser states and let o be the initial diagnoser state Let R denote the diagnoser state labelling function which associates a diagnoser state to the pairs in its label and verifies R 0 xo 0 The set T of diagnoser transitions then satisfies S a T iff G Ea I x 1 such that A x L x Tp and A x x T and Tul Figure 1 gives an example Symbolically the diagnoser G b bZ bX be bE 0 8 T is encoded using the additional variables b and b for representing diagnoser states in their role as start and tar get states of transitions The BDD encodes the diagnoser state labelling function R and is defined over the variables b Ub UDF z in G with x2 xl Fre py X5 Lol 56 s1 fl f1 2 sl ry fl f2 ol xl gt x3 x5 gt x6 1 6 fl 2 2 XLT 6 T x4 3 Figure 1 Global left abstracted top right and diagnoser models bottom right o 01 s si Ef f1 fo Symbolic On line Diagnosis On line diagnosis aims to detect faults while the system is working Given a sequence of observations it identifies all the faults and system
14. ence of observations We choose the encodings that make use of BDDs as few as possible while still allowing an efficient on line retrieval of diagnosis information Efficiency requires for instance that we parti tion the transition sets of our FSMs Since the focus of the paper is the use of the models for on line diagnosis we will only briefly allude to their off line computation We refer the reader to Schumann Pencol amp Thi baux 2004 for details of how this might be done Component Models As in Sampath et al 1996 the diagnosed system is com posed of a set of n individual components G with respec tive sets of states X and a global event set X The events are partitioned into observable X and unobservable events the latter of which is further partitioned into faults Ur and shared events The shared events are used to describe the communication between components Following the usual symbolic FSM representation de scribed above the symbolic components are Gi p bX b Wi Pos To Th where b bX and b are the Boolean variables that define the following BDDs zo to represent the initial state and Toi Tsi Tf to represent the observable shared and fault transitions Note that every transition in every component G is defined over the same global event variables but over local state variables that is over b U b U bX Global Model Diagnosing directly from the component models without
15. ffer by the degree of reasoning performed off line and by the nature and the size of the underlying compiled models These methods range from no compilation to full compilation of diagnosis information but are not limited to those extreme cases To increase efficiency all models are represented by sym bolic finite state machines using binary decision diagrams BDDs and all methods are implemented via symbolic op erations BDDs enable the compact encoding and the im plicit manipulation of sets of states and transitions On the one hand they allow us to reduce the space requirements of models with a high degree of compilation On the other hand they help reducing the diagnosis time of approaches with a low degree of compilation by avoiding the individual consideration of all possible diagnosis explanations Our experiments illustrate the diversity of space time re quirements of methods across the spectrum and clearly demonstrate the superiority of our symbolic methods over the equivalent enumerative ones This is not to be confused with the spectrum of diagnosis def initions presented in Brusoni ef al 1998 nor the spectrum of symbolic compilations in Darwiche amp Marquis 2002 The paper is organised as follows After a brief reminder of BDDs and symbolic finite state machines we present the successive models underlying the respective methods give an on line diagnosis algorithm for each of them experimen tally illustrate th
16. ic procedure To trig ger the transition step 1 we apply three BDD operations namely A Extract and Swap Applying the operation to the encodings of a start state an event o and the tran sition set T retrieves the transition that starts in lt and is labelled with o Next the operation Extract is used to ob tain only the target state lt of the transition We then Swap the encoding of lt over the primed variables for an encod ing over the non primed ones in order to determine its label and to trigger future transitions To determine the label of lt step 2 we first conjoin with the state labelling function and then abstract from the boolean variables representing diagnoser states Algorithm 1 DiagDiagnose G o c nont Extract b a Swap a b b 2 KU A info return Abstract 2 fo b On line diagnosis based on the abstracted model Using the abstracted model the retrieval of the diagnosis information given its predecessor in fo requires 1 computing states Xros that can be reached from those contained in in fo before observing the new event o 2 computing the fault labels representing the faults that have occurred on a path from the initial state to a state in XunObs gt and 3 triggering all transitions starting from states in awe and labelled The corresponding three symbolic computation steps are shown in Algorithm 2 Once the
17. llenges remaining In Proc IJCAI 99 Cordier M O and Largou t C 2001 Using model checking techniques for diagnosing discrete event systems In Proc DX 01 39 46 Darwiche A and Marquis P 2002 A knowledge compilation map JAIR 17 229 264 Darwiche A 1998 Model based diagnosis using structured sys tem descriptions JAIR 8 165 222 Lamperti G and Zanella M 2006 Flexible diagnosis of discrete event systems by similarity based reasoning techniques Artificial Intelligence 170 232 297 Marchand H and Roz L 2002 Diagnostic de pannes sur des syst mes v nements discrets une approche base de mod les symboliques In 13 me Congr s AFRIF AFIA de Reconnais sances des Formes et Intelligence Artificielle 191 200 Pencol Y and Cordier M O 2005 A formal framework for the decentralised diagnosis of large scale discrete event systems and its application to telecommunication networks Artificial In telligence 164 121 170 Rintanen J and Grastien A 2007 Diagnosability testing with satisfiability algorithms In Proc IJCAI 07 532 537 Sampath M Sengupta R Lafortune S Sinnamohideen K and Teneketzis D 1996 Failure diagnosis using discrete event models IEEE Trans on Control Systems Techn 4 2 105 124 Schumann A Pencol Y and Thi baux S 2004 Diagnosis of discrete event systems using binary decision diagrams In Proc DxX 04 197 202
18. rvable transitions T starting in a state x E X consistent with the previous diagnosis information X Extract in fo bX and e all observable transitions T labelled with the new obser vation o To 0 To To obtain the corresponding global transitions efficiently via the A operator a synchronous product is required In a syn chronous system when a transition is triggered in a compo nent G a transition is also triggered in every other compo nent Hence we add for every event o that can occur in G but not in G and every state x of a component model G a 1 transition x 2 x Now the relevant global transitions are computed as follows e Ty N1 Tu and similarly e To Nilo Using these two transition sets the new diagnosis informa tion is computed as in Algorithm 3 The only change needed is the replacement of o A To with T in line 9 of the algo rithm Experimental Evaluation We implemented our approach on top of the CUDD BDD package http vlsi colorado edu fabio time ins 90 81 38 I Symbolic Dia 80 gnosis i Enumerative Diagnosis 12 73 14 27 _ 397 0 89 0 01 Comp nent Global Abstfiicted Dia noser Figure 2 Average diagnosis times over 100 scenarios of 10000 observations each CUDD In order to evaluate the benefits of our symbolic framework we also implemented a traditional enumera tive version
19. s 67 transitions 1 fault 8 shared and 2 observable events The global model of a grid of size n x m closely approaches the 14 states bound E g the 2 x 2 grid has 143 85 26 000 states The exam ple is poorly diagnosable Every system state can be asso ciated with the 2 fault hypotheses and the observations do not allow discrimination between faults due to a masking phenomenon nodes reboot silently and reboot requests from other nodes are not observed Consequently there is a huge set of diagnoses that explains a given observation sequence Figure 3 compares the performance of the symbolic and enumerative approaches as the size of the grid increases note the logarithmic scale The gap between the two ap proaches increases by an order of magnitude with each ad dition of a new component For the three larger grids the enumerative approach failed to refine the diagnosis within an average of 10 sec the theoretical number of diagnoses for the 2 x 2 grid is 2 458 624 All other enumerative approaches are unsuitable as the enumerative global model could not be computed In contrast the symbolic approach was able to refine the same diagnosis in 0 079 sec time ins 100000 Ii Symbolic Diagnosis 14353 4 YPE a oe E Enumerative 40000 410 84 r a 1859 6 Diagnosis 1000 828 1 100 z9 19 6 10 z 0 1 idl 1x2 1x3 1x5 2x 2x2 3 Grid size Figure 3 Average component based diagnosis
20. s Xnew and Xtarg represent sets of labelled states that is sets of tuples x l Seman is computed us ing breadth first search lines 1 8 Initially Xunobs and Xnew are composed of the previous diagnosis information Zin fo lines 1 2 As long as there are still new diagnosis tu ples Xnew that have not been processed line 3 applicable unobservable transitions are triggered line 4 and any fault labelling them is added line 5 The tuples already closed are removed from the resulting tuples Xtarg to ensure the termination of the algorithm operator A in line 6 The new tuples are added to the set of closed ones operator V in line 7 Once Xunobs is obtained the new diagnosis infor mation is retrieved as in step 3 of Algorithm 2 line 9 Algorithm 3 GlobDiagnose Ty To finfo a 1 Xnew Linfo 2 XunObs Xnew 3 while sDef Xnew do 4 Trew Ty A Extract Xnew b 5 Xtarg AddFault Thew Xnew Xtarg Swap Xtarg b bX Xnew Xtarg XunoObs XunObs XunObs V Xnew end while o Extract Xunovs o To b UDE E return Swap i b b info On line diagnosis based on the component models In addition to the previous algorithm on line diagnosis based on the component models requires the computation of those of the global transitions that are needed to determine the new diagnosis information For every component G we only need to consider e all sequences of unobse
21. small as sa the size of the enumerative one Its size is rather comparable to that of the enumerative abstracted model yet it is an order of magnitude faster than the latter Focusing on the symbolic spectrum the abstracted model appears to provide a particularly interesting tradeoff It is 13 times smaller than the diagnoser but only 4 times slower Compared to the global model the percentage decrease of diagnosis time of the abstracted model is slightly higher than its percentage increase in size The advantage of the ab stracted model results from the efficiency of 1 the symbolic triggering of sets of transitions and 2 the update of fault labels by considering fault sets rather than by considering a sequence of individual faults The component model also presents an interesting trade off due to its very small size of only 8 kilobytes For large applications it appears to be the only option We show how the component based approach scales as the size of the sys tem increases using a grid of computer nodes inspired from the example in Rintanen amp Grastien 2007 All nodes have the same behaviour In normal mode each node performs its task sending an on message to a supervisor prior to starting and an off message upon completion When a node becomes faulty an automatic recovery system forces the node to re boot and to send his neighbours reboot requests which get propagated through the grid The model of a node has 14 state
22. states that are consistent with the oc currence of these events For each of the above models we give a procedure that uses symbolic reasoning to compute this diagnosis information as efficiently as possible Initially the system is in state x9 and no fault has oc curred so the diagnosis information is zo A Fo where Fy Aa b denotes the empty fault label Now each time an event is observed the diagnosis information fo is derived based on g one of the models and the previous diagnosis information info In this section we show how we can symbolically retrieve fo using the basic boolean operations and the following ones e IsDef bdd returns true iff bdd does not represent false e Extract bdd B deletes from bdd all occurrences of variables not in B e Abstract bdd B deletes from bdd all occurrences of variables in B e Swap bdd a1 ak b1 b renames in bdd variable a with b i 1 k and vice versa For the sake of readability the algorithms are presented in the following order from the diagnoser to the component based one On line diagnosis based on the diagnoser The precomputed diagnoser contains all the information to perform efficient on line diagnosis Given the previous di agnoser state and a new observation it is sufficient to 1 trigger the corresponding transition and 2 retrieve the fault information from its target state Algorithm 1 describes the symbol
23. unobservable transitions TunObs Starting in a state of inf are determined line 1 they contain all the new faults that could have occurred since the last observation These are added to the faults in in fo that have previously occurred using function AddFault Symbolically adding a fault f implies changing the value of the corresponding boolean variable bf from false to true It is done by abstracting bf from the fault label i e Abstract l bf and conjoining it with l i e LAbf This abstraction can be done simultaneously for all fault labels Ly to which f has to be added Finally the observable transitions are triggered and the new diagnosis information returned step 3 Algorithm 2 Abst Diagnose G Lin fo 0 if TATAN lt Ty A Extract in fo b 2 XunObs i Linfo V AddFault Tunovs info XunObs Swap Xunobs b b 3 info Extract Xunovs o A T gt b Ub return Swap b b On line diagnosis based on the global model Using the global model the symbolic computation of the diagnosis information is similar to that above except that states Xunobs now need to be computed based on transition sequences in G For this purpose we first combine shared and fault transitions into a single transition set T in which all events are defined over variables b Here shared transi tions are labelled with the empty fault label Fg Algorithm 3 describes the symbolic procedure All for mulas Kind

A Spectrum of Symbolic On-line Diagnosis Approaches

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