Jana%20Flochov

1
On fault diagnosis of random free-choicePetri
nets

• Jana Flochová and René K. Boel
• Faculty of Informatics and Information Technology
  
• Slovak university of Technology,
• Bratislava, Slovakia
• EESA Department, Ghent University, Belgium

2
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples

3
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples
• Conclusions

4
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples
• Conclusions

5
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples
• Conclusions

6
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples
• Conclusions

7
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples
• Conclusions

8
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples
• Conclusions

9
Outline of the presentation

• Models, diagnosis of DES based on Petri net
  models
• Minimal context and explanations (Jiroveanu,
  Boel, Bordbar 2008)
• Probabilistic (random) free choice Petri nets
• Calculation of likelihood values for minimal
  explanations probabilities of failures
• Deterministic analysis of the past, probabilistic
  analysis of the future
• Examples
• Conclusions

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Models Petri Nets
4) M0 P ? N is the initial marking
lt, , ? denote precedence, conflict,
concurrency relations of nodes A free-choice
Petri net is a restricted class where every arc
from a place to a transition is either the unique
output arc from that place, or a unique input arc
to the transition.
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Models Petri Nets
An occurrence net O is a net O (B, E,?), with
the elements of B called conditions, those of E
called events, satisfying following
properties? ?x?B?E? ?x ? x (no node is in self
conflict) ?x?B?E? ?x lt x (is a partial order,
acyclic) ?x?B?E? ?y y lt x?lt ? (is
well-formed) ?b?B??b?? 1 (?b denotes the set
of input elements of b gt each place has at most
one input transition, no backward conflict). A
configuration C(Bc, Ec,) is a subset of O, which
is conflict free (no two nodes are in conflict),
causally upward-closed (if xlt1 x, and x?C, then
x?C), and min(C) ? min (O).
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Models Petri Nets
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Diagnosis based on PN problem statement

• We consider the following structural and
  functional assumptions
• The overall plant model is bounded (possibly
  well formed free-choice)
• The initial marking M0 is precisely known, the
  set of transitions T To?? Tuo
• The plant observation is represented by a subset
  of observable transitions
• The occurrence of an observable transition To is
  always reported correctly and without delays
• No design-error assumptions

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Diagnosis based on PN problem statement

• We consider the following structural and
  functional assumptions
• The overall plant model is bounded (possibly
  well formed free-choice)
• The initial marking M0 is precisely known, the
  set of transitions T To?? Tuo
• The plant observation is represented by a subset
  of observable transitions
• The occurrence of an observable transition To is
  always reported correctly and without delays
• No design-error assumptions

15
Diagnosis based on PN problem statement

• We consider the following structural and
  functional assumptions
• The overall plant model is bounded (possibly
  well formed free-choice)
• The initial marking M0 is precisely known, the
  set of transitions T To?? Tuo
• The plant observation is represented by a subset
  of observable transitions
• The occurrence of an observable transition To is
  always reported correctly and without delays
• No design-error assumptions

16
Diagnosis based on PN problem statement

• We consider the following structural and
  functional assumptions
• The overall plant model is bounded (possibly
  well formed free-choice)
• The initial marking M0 is precisely known, the
  set of transitions T To?? Tuo
• The plant observation is represented by a subset
  of observable transitions
• The occurrence of an observable transition To is
  always reported correctly and without delays
• No design-error assumptions

17
Diagnosis based on PN problem statement

• We consider the following structural and
  functional assumptions
• The overall plant model is bounded (possibly
  well formed free-choice)
• The initial marking M0 is precisely known, the
  set of transitions T To?? Tuo
• The plant observation is represented by a subset
  of observable transitions
• The occurrence of an observable transition To is
  always reported correctly and without delays
• No design-error assumptions

18
Diagnosis based on PN problem statement

• We consider the following structural and
  functional assumptions
• The overall plant model is bounded (possibly
  well formed free-choice)
• The initial marking M0 is precisely known, the
  set of transitions T To?? Tuo
• The plant observation is represented by a subset
  of observable transitions
• The occurrence of an observable transition To is
  always reported correctly and without delays
• No design-error assumptions

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Diagnosis based on PN problem statement

• Faults Tf are represented by a subset Tf ? Tuo
  of unobservable (silent transitions ( due e.g.
  limited sensor information )
• A fault or an unreliable sensor (when some
  messages may become lost) can be modelled
  provided that another unobservable transition is
  included in the model "in parallel" to the
  observable transition
•  

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Diagnosis based on PN problem statement
G. Jiroveanu, R.K. Boel, and B. Bordbar. On-Line
Monitoring of Large Petri Net Models Under
Partial Observation. Journal Discrete Event
Dynamic Systems, 2008 Minimal context, minimal
explanation, minimal marking.
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Diagnosis based on PN problem statement
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Centralized diagnosis of DES based on minimal
explanations
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Probabilistic settings

• The probability of firing a transition should not
  depend on what concurrent transitions do, and the
  order on which concurrent transitions fire should
  not be randomized
• Firing should not necessarily be reduced to one
  transition at a time.
• The probability of firing a given transition
• depends only on its own recourses.

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Probabilistic settings
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Probabilistic settings
The probability function on the set of
configurations is defined as follows
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Probabilistic settings

• A stochastic analysis of faults that either
  occurred in the past or that may occur in the
  future prior to the next observed event
  occurrence (Flochová et al. 2007)
• so that the explanation only includes
  unobservable future events not belonging to the
  minimal explanations.
• A deterministic analysis of faults that must have
  occurred in the past (Jiroveanu, Boel, Berdbar
  2008) and a probabilistic analysis of faults that
  may occur in the future prior to the next
  observed event occurrence.

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Probabilistic settings
Having the set of minimal configurations C(On),
respectively the set of minimal explanations of
the received observations LN (On) is defined
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Probabilistic settings
Having the set of minimal configurations C(On),
respectively the set of minimal explanations of
the received observations LN (On) is defined
The plant diagnosis after observing On based on
the set of minimal explanations - obtained by
projecting the set of minimal explanations onto
the set of fault events
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Probabilistic settings
Having the set of minimal configurations C(On),
respectively the set of minimal explanations of
the received observations LN (On) is defined
The plant diagnosis after observing On based on
the set of minimal explanations - obtained by
projecting the set of minimal explanations onto
the set of fault events
30
Probabilistic settings
Having the set of minimal configurations C(On),
respectively the set of minimal explanations of
the received observations LN (On) is defined
The plant diagnosis after observing On based on
the set of minimal explanations - obtained by
projecting the set of minimal explanations onto
the set of fault events
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Probabilistic settings
All explanations - similar expressions after
removing all underscores.
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Probabilistic settings
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Probabilistic settings
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Probabilistic settings

• Steps needed in order to derive fault
  probabilities
• Compute the set of minimal explanations of the
  most recent observed event. Derive minimal
  explanations of the last observed event t0 and
  minimal explanations of a sequence of observed
  events.
• (2) Compute the unnormalized probability of all
  minimal explanations
• (3) Sort explanations in descending order
  starting from the most probable ones. Shellsort
  can be used, branch and bound like improvements
  can be useful in order to avoid enumerating very
  unlikely explanations.
• (4) Accept top x (0-100 ) of explanations
  according to the input requirements.
• (5) Compute the set of maximal explanations of
  the most recent observed event, if required.

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Probabilistic settings
(6) Compute the unobservable continuations, which
follow after the next observable transitions and
partition the continuations into the following
sets the set of configurations, which contain
at least a faulty event a set of
configurations, which contain at least a faulty
event of the fault of the type i and the set of
configurations, which dont contain any faulty
event. A modification of classical AI depth
search, which evaluates at first the node that
has the most nodes between itself and the last
observed transition, can be used for computing
the set of continuations equipped with
probabilities.
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Probabilistic settings
(7) Compute the unnormalized probabilities of the
faults (faults of the type i) of all
continuations (of unobservable reaches after the
last observation). (8) Compute the unnormalized
probabilities of the faults (faults of the type
i) based on the sets of all explanations. (9)
Normalize the probabilities
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Example
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Example
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Laboratory example- older Fischertechnik-modelold
unreliable sensors and all parts, AB PLC control
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• !!!!Possibly a model, shortly

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• !!!!Possibly a model, shortly

Minimal explanations of the last event
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Conclusions

• Two methods of probabilistic diagnosis were
  presented, both methods use minimal explanations
  and contexts concept, probabilities assigned to
  conflicting transitions and , reverse Petri
  nets.  They both are based on George and you or
  better George, you and Bordbar, and Benveniste
  et al. approaches.
• 1. the method uses the probabilistic analysis of
  the plant evolution before the last observed
  event and the probabilistic estimation of the
  future evolution of the plant after the last
  observed event NYC.
• 2. The second method  (novel approach) is based
  on the deterministic analysis of the plant
  evolution before the last observed event and the
  probabilistic estimation of the possible future
  failure evolution of the plant.

43
Conclusions

• Two methods of probabilistic diagnosis were
  presented, both methods use minimal explanations
  and contexts concept, probabilities assigned to
  conflicting transitions and , reverse Petri
  nets.  They both are based on George and you or
  better George, you and Bordbar, and Benveniste
  et al. approaches.
• 1st method uses the probabilistic analysis of
  the plant evolution before the last observed
  event and the probabilistic estimation of the
  future evolution of the plant after the last
  observed event NYC.
• 2. The second method  (novel approach) is based
  on the deterministic analysis of the plant
  evolution before the last observed event and the
  probabilistic estimation of the possible future
  failure evolution of the plant.

44
Conclusions

• Two methods of probabilistic diagnosis were
  presented, both methods use minimal explanations
  and contexts concept, probabilities assigned to
  conflicting transitions and , reverse Petri
  nets.  They both are based on George and you or
  better George, you and Bordbar, and Benveniste
  et al. approaches.
• 1st method uses the probabilistic analysis of
  the plant evolution before the last observed
  event and the probabilistic estimation of the
  future evolution of the plant after the last
  observed event NYC.
• 2nd method  (a novel approach) is based on the
  deterministic analysis of the plant evolution
  before the last observed event and the
  probabilistic estimation of the possible future
  failure evolution of the plant.

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Advantages of the approach

• The probabilistic setting allows us to
  incorporate statistical knowledge on the
  production of faults some event may be more
  likely than the others depending on reliability
  tests on devices, on the previous experience on
  monitoring the plant or the network (relative
  frequencies of spontaneous faults), on the loss
  of information on faults (e.g. masking of an
  alarm, temporally unavailable links, faults of
  protocols).
• Methods allow  some smoothness of observation,
  i.e. including of misleading observations and not
  observing of a normally observable events in the
  model.
• Randomization of the model also provides a
  convenient way of introducing robustness of the
  model against modeling errors on faults
  propagation.

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Problems and open questions

• The process of randomization has to be done very
  carefully and one has to tackle several problems
  in assigning probabilities. 
• Decentralized diagnosis algorithms and
  distributing setting are needed to allow fault
  detection in large plants
• possible solution
• - several communicating probabilistic Petri nets
  components computing local probability assignment
  for all locally possible traces explaining
  observations.
• components can interact by exchanging tokens via
  boundary places (or boundary synchronizing
  transitions), common normalization for both
  interacting component
• Relaxing the assumption of well formed free
  choice Petri nets following Haar 2003

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• Benveniste, A. et al. Fault detection and
  diagnosis in distributed systems an approach by
  partially stochastic Petri nets. Discrete Event
  Dynamic Systems Theory and Applications, vol. 8,
  pp. 203-231, June 1998.
• A. Benvensite, E. Fabre, and S. Haar. Markov
  nets Probabilistic models for distributed and
  concurrent systems. IEEE Transactions on
  Automatic Control, 48(11)19361950, 2003.
• Benveniste, A. et al. Diagnosis of asynchronous
  discrete event systems, a net unfolding
  approach. IEEE Transactions on Automatic
  Control, 48(5), pp. 714-727, May 2003.
• S. Haar, Probabilistic cluster unfoldings for
  Petri nets,Technical report 1517, IRISA, Rennes,
  France, 2003.
• J. Esparza. S. Romer and W. Vogler. An
  improvement of McMillans unfolding algorithm.
  Lect. Notes in Computer Science 1055, 87106,
  Springer-Verlag, 1996.
• J. Flochova, R. K. Boel, and G. Jiroveanu. On
  Probabilistic Diagnosis for Free-Choice Petri
  Nets. Proceeding of ACC, NYC, US, 56555656,
  2007.
• G. Jiroveanu, R.K. Boel, and B. Bordbar. On-Line
  Monitoring of Large Petri Net Models Under
  Partial Observation. Journal Discrete Event
  Dynamic Systems, 18323354, 2008.
• M. Nielsen, G. Plotkin, and G. Winskel. Petri
  nets, event structures and domains, part I.
  Theoret. Computer Science, 1385108, 1981.

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Thank you for your attention

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