Title: Distributed Fault Detection for untimed and for timed Petri nets
1Distributed Fault Detectionfor untimed and for
timed Petri nets
- René Boel,
- SYSTeMS Group, Ghent University
- with thanks to
- G. Jiroveanu, G. Stremersch, B. Bordbar
2Outline
- problem formulation and models
- centralised diagnosers via backward search
- distributed diagnosis for interacting Petri nets
- fault detection for timed Petri nets
- probabilistic fault diagnosis
3adaptive supervisory control?
- improve behaviour of large plants
- consisting of many interacting components
- controlled by disabling events
- in order to guarantee that certain specifications
are always satisfied - requires on-line estimation
of current mode of operation of
plant
4feedback control paradigm
observed output
plant
state estimator and fault detector
controlled input
feedback law
5distributed feedback control paradigm
6distributed fault detection for large discrete
event systems
- locally observable events at fault detection
agenti (possibly small) subset of all events
happening in planti - assume all observable events
are always seen immediately
by local fault detection (FD) agent,
with their exact occurrence time (global clock)
7distributed fault detection for large discrete
event systems
- Can cooperation between local fault detection
agents guarantee same quality of fault detection
as would be achievable by a centralized fault
detection agent that - observes all observable events (planti, i
1,...,I) - knows model of complete plant
- what is minimal information to be exchanged
between FD agents for achieving same performance
as centralized FD?
8distributed state estimation for discrete event
models
- model includes unobservable events that cause
deterioration of plant behavior, i.e. faults - each local fault detection agent implements
local FD algorithm,
in order to adapt supervisor's
reaction to current mode of operation of plant - local agent uses sequence of observable
events generated locally
9distributed state estimation for discrete event
models
- one fault detection (FD) agent per component
- limited communication between local FD agents
allows performance as good as centralized FD
agent that would see all
observable events, would know global model
10distributed observers
- distributed implementation reduces on-line
computational complexity of FD agents - distributed implementation makes fault detection
more robust against communication errors that
centralized FD - ...provided local FD agents provide
overestimate of set of possible faults
11distributed observers
- distributed implementation of FD agents makes
fault detection less sensitive to errors in
knowledge about model - especially knowledge of distant components often
outdated when components change often
12distributed observers
- but design more complicated because
- local FD agents must understand effects of
interaction between components - communication strategy between local FD agents
must be designed
13Petri net model
common places may be unobservable parts of
physical network
interactions via common places - no common
transitions
assume at least one observable event in cycle
covering more than one component
only very few transitions generate
(locally) observable signal
14interaction of Petri net components
- described by token passing
via common places - token entering boundary place enables
events to happen only once in
neighboring components
15interaction of Petri net components
- described by token passing
via common places - token entering boundary place enables
events to happen only once in
neighboring components
16interaction of Petri net components
- tight interaction between components
- local FD agenti may not know initial marking of
places in planti - forward explanations not possible for generation
of allowed trajectories explaining observations
17backward explanation
- generate all trajectories that
- are compatible with observations up to present
time - are compatible with plant model
- starting from most recently observed execution of
transition t and recursively moving upward to its
input places t, its input transitions (t), and
so on until possible previous marking is obtained
18Petri net example for explaining backward
recursion
places ? P
transitions ? ?
Pre arc
Post arc
t13
token
place with 2 tokens
19decompose Petri net in 2 components that interact
via places p5, p9
each component contains one fault transition,
resp. t1, t8
t13
each component contains one observable
transition, resp. t6, t10
Algiers, 5/5/07
VECOS'07
20observer design for one single component
- behaviour Petri net model generates sequence of
events - observation only some of the events are observed
by control agent - model set T of transitions partitioned in
observable transitions t ? To - and unobservable transitions t ? Tu
21Petri net compositional modelling
Large plants can be represented by several
Petri net components, interacting with each
other by unobservably exchanging tokens
via common places
component 1
component 2
22Petri net compositional modelling
- set P of places of Petri net model consists of
- "local places" in each component i
- for component i "input places PIN,i,j
that have input transitions (Pre) in
component j and output transitions (Post) in
component i - for component i "output places POUT,i,j
that have output transitions (Pre) in
component j and input transitions (Post) in
component i - decomposition not constrained by limitations on
sensors
23Petri net compositional modelling
- To avoid unnecessary complications in analysis
assume Petri net bounded, i.e. all
reachable markings have bounded number of tokens
in each place - Problematic assumption boundedness
depends on the global structure of the Petri
net, cannot be verified
locally!
24Petri net compositional modelling
fault detection agenti for componenti only
observes local observable events
agent1 observes each occurrence of t6, at
clock times ?(t6)n agent2 observes each
occurrence of t10 at clock times ?(t6)n
component 1
component 2
25computational complexity
- combining two components of similar size leads to
much more complicated behaviour - number of possible traces of combination of two
components is much larger than twice the size of
the behaviour of each component separately! - exponential explosion of computational
complexity!
26Outline
- problem formulation and models
- why distributed on-line state estimation?
- backward generation of explanations
- distributed diagnosis
- extensions to timed DES models
- open questions and conclusions
27example
only observable event t10 fault event t8
p5
p8
t8
t9
p6
if only p8 is marked initially then only normal
behaviour is trace t12 and empty trace while
possible faulty behaviour contains all prefixes
of t8, t10, t11 including empty trace
p11
t10
p7
t13
t12
t11
p9
28prefix
- set of all prefixes of t8, t10, t11
- ?, t8, t8, t10, t8, t10, t11
- since untimed model does not specify any upper
bound on time delays for events the model can
never guarantee that an enabled event will have
happened
29example
only observable event t10 fault event t8
p5
p8
t8
t9
p6
if only p5 is marked initially then normal
behaviour includes all prefixes of the trace
t9, t13, t10, t11 where moreover t13 can
also occur after t10 and after t11
p11
t10
p7
t13
t12
t11
p9
30unfolding of Petri net
- set of all prefixes and permissible reorderings
of the trace t9, t13, t10, t11 - ?, t9, t9, t13, t9, t10, t9, t13,
t10, - t9, t10, t13, t9, t10, t11, t9, t13, t10,
t11, - t9, t10, t13, t11, t9, t10, t11, t13
- described by unfolding of the net, obtained by
- by "opening all cycles" and
- by "copying all places with more than 1 input
transition"
31unfolding of Petri net
- example does not
- contain cycles
- place p6 and place p9 must be replaced by 2,
resp. 3 copies of the place
p5
p8
t8
t9
p6,1
p6,2
p11
t10,1
t10,2
p7,1
t13
p7,2
t12
t11,1
t11,2
p9,1
p9,4
p9,2
p9,3
32unfolding of Petri net
- after unfolding a Petri net each token is
generated by a uniquely defined sequence of
events - problem how to make an unfolding finite?
- It is possible to obtain a finite unfolding of a
Petri net so that "same behaviour" is generated
(taking into account that repeating a cycle
infinitely often does not generate new states)
33forward analysis via unfolding
- if initial marking is known then one can
enumerate all possible traces that end with an
observable event occurrence - and select as possible explanations of the
observed events only those traces that contain
the observed events in correct order - using unfoldings avoids need for enumerating all
possible orderings of unobservable events
34forward analysis via unfolding
- but requirement that all initial markings are
known (or an upper bound on these markings) is
not acceptable for distributed fault detection - since componenti does not know how many tokens
componentj puts in input places PIN,i,j and when
it puts these tokens there
35Outline
- problem formulation and models
- why distributed on-line state estimation?
- centralised diagnosers
- distributed diagnosis
- extensions to timed DES models
- open questions and conclusions
36backward search
- can avoid this difficulty
- finding minimal explanations via backward search
determines where tokens should be available and
by what time they must be in that place
37example backward search
observe t10 at time ?(t10)
p5
p8
t8
t9
p6
token must have arrived in p6 before ?(t10)
p11
t10
p7
t13
t12
either fault t8 or unobservable event
t9 must have fired before ?(t10)
t11
p9
38example backward search
observe t10 at time ?(t10)
p5
p8
t8
t9
p6
token must be present in p6 before
?(t10) or have been sent to p5 by
neighboring component before ?(t10)
p11
t10
p7
t13
t12
t11
p9
39example backward search
FD agent2 knows token was present in p8 prior to
?(t10) and hence it can determine that fault t8
may have occurred but determining whether
fault t8 occurred for sure requires
information from neighboring component that it
can put token in p5 before ?(t10)
p5
p8
t8
t9
p6
p11
t10
p7
t13
t12
t11
p9
40example backward search
in order to determine whether explanations (t9
t10) is also possible FD agent2 must ask
niehgboring FD agent1 if it is possible that a
token arrived in p5 before ?(t10) note that
FD agent2 does not have to know model of plant1
except for fact that p5 is common place
p5
p8
t8
t9
p6
p11
t10
p7
t13
t12
t11
p9
41example backward search
- possibility of token in p5
- depends on whether
- agent1 knows that 2
- tokens in p0 present
- before ?(t10)
- if so then
- irrespective of how
- many times t6 has
- been observed by
- FD agent1 a token
- could may reach p5
- prior to ?(t10)
t13
VECOS'07
Algiers, 5/5/07
42example backward search
agent1 responds that it is possible that
p5 became marked before time ?(t10) from this
response FD agent2 concludes that the fault t8
may or may not have occurred
t13
VECOS'07
Algiers, 5/5/07
43example backward search
Note that a central observer
knowing both models and seeing all
observations also would not be able
to draw an unambiguous
conclusion
t13
VECOS'07
Algiers, 5/5/07
44example backward search
FD agent1 will return the same response if only
1 token is in p0 but t6 has not been
observed yet, and the conclusion of FD
agent2 will be the same however
FD agent1 will know then that the
token in p0 can no longer be used for
future explanations
t13
VECOS'07
Algiers, 5/5/07
45example backward search
if only 1 token were present in p0 prior to
?(t10) and FD agent1 has observed the occurrence
of t6 prior to ?(t10) then the token in p5 can
only occur via (t0, t3, t6) which requires a
token in p9 prior to ?(t10)
t13
Algiers, 5/5/07
VECOS'07
46example backward search
but then ... agent1 must return the question to
agent2 and ask it is possible that p9 has
received a token prior to ?(t10) FD agent2 will
answer that this is indeed possible, and will
receive a positive answer from FD agent1
t13
Algiers, 5/5/07
VECOS'07
47example backward search
moreover FD agent2 knows that in that case the
only explanation of t10 occurring at ?(t10) is
the sequence of events (t9,t10) then FD agent2
can deduce that the fault t8 has not occurred
t13
Algiers, 5/5/07
VECOS'07
48distributed fault diagnosis via backward search
- method described in simple example can be applied
for all compositions of Petri nets interacting
via common places - using general algorithm for generating minimal
explanations (backward unfolding) - at random times any FD agent can initiate
communication round that is assumed to reach a
conclusion instantaneously (before any other
fault or observable event can occur)
49local explanation
- ordered sequence of local unobservable and
observable events, so that - ordering of observable events corresponds to
observations - uppermost transition of local explanation is
either a place that is known locally to be marked
initially, or a place where a token can enter the
local component from a neighbouring component
50fault diagnosis
- Relaxed diagnosis goal!
- Enumerate only the minimal traces containing
sequences of events that must have happened for
OBSi to be allowable - do not expand minimal traces,
i.e. do not include transitions that do not
lead to satisfaction of constraints necessary for
occurrence of observable event
51Minimal explanations
- Set Ei,Min(OBSi ) of minimal local explanations
using model of component i and local observations
in model i - would allows us
- to decide if a fault happened for sure in
component i if we could detect the tokens
entering via boundary places
52Construction of minimal explanations
- assume OBS t1o, 1st observation at time
?(t1o) - necessary constraint for execution of event t1o
is marking by at least 1 token of each place pink
in Pre(t1o) - this in turn requires that for each place pink
at least one of the input transitions (determined
by Post(., pink) of pink) has fired prior to
?(t1o)
53Construction of minimal explanations
- assumption no unobservable cycles with choice
places sufficient to guarantee search stops in
finite time - (ensures no problems occur due to tokens moving
unobservably through cycles containing several
initially marked places) - all unobservable cycles must be trap circuits
54faults should not be predictable
- theorem about equivalence of distributed and
centralized diagnosis only true if reasonable
assumption is made that faults are not
predictable, - i.e. there does not exist a marking that does
inevitably lead to a fault transition
55Some particularities of model
- Unlike many other distributed anayses (Fabre,
Jard, Su,...) - we assume global clock available
- but each agent only knows local model and
interactions with neighbouring models - justification
- GPS timer sufficiently accurate for applications,
- but many reconfigurations make it difficult for
each agent to know global model - applications "slow" networks
56distributed fault detection
- From time to time local agents should exchange
enough information - so that local diagnosis result in component i
detects all the local faults that global
diagnoser would detect at same time - i.e. after communication between agents
- local diagnosis projection of global diagnosis
57Outline
- problem formulation and models
- why on-line state estimation?
- centralised diagnosers
- distributed diagnosis
- fault diagnosis for time Petri net models
- probabilistic DES, open questions and conclusions
58timed discrete event models
- if minimal and maximal time delays for executing
transitions are specified by model - then not every untimed prefix of a possible trace
is possible for timed model - alternatively stated adding a token may reduce
reachable space - analysis much more difficult, since set of
reachable states not monotonely growing - simplify notation only 1-safe nets
59time Petri net model
- assume t becomes enabled at ?p then
t must be executed at some time ??
?en(t) L(t), ?en(t) U(t) - where ?en(t) maxp??t ?p
- if t has not been executed yet, and no other
enbabled transition has removed a token form one
place in ?t, then t is forced to execute at
?en(t) U(t)
60time Petri net model
- consider choice place p with t1, t2 ? p?
- if L(t2) gt U(t1)
- then t2 always
pre-empted by t1 - drop t2 from model
- ? backward explanation of observations for timed
model should remove such a path, even if it
appears in explanation for
timed model
t1
t2
61diagnosis can be refined by adding timing
information to model
- start diagnosis for time Petri net model by
developing, via backward search, set of minimal
explanations of observations, compatible with
untimed PN model - check whether there exists for each event
occurrence in untimed minimal explanation a
non-empty interval of execution times that
satisfies all the constraints ??
?en(t) L(t), ?en(t) U(t)
62diagnosis can be refined by adding timing
information to model
- diagnosis for timed model starts with diagnosis
for untimed model - and then checks if there exists a legal valuation
of all the event times in the untimed minimal
explanation
63diagnosis can be refined by adding timing
information to model
- valuation of variables in set of linear
inequalities possible execution times of events
in set of possible untimed explanations of
observed events - at time when observation occurs fix this
execution time, and check if each element in set
of explanations is still compatible with timed
model
64diagnosis can be refined by adding timing
information to model
- for timed model also need to expand set of
explanations forward since timed model may force
events to happen by a certain and not observing
such a forced event implies elimination of such
an explanation from set of possible explanations
65diagnosis can be refined by adding timing
information to model
- need to recalculate set of solutions to
conjunctive/disjunctive set of linear
inequalities at each time when - an observed event is executed
- an enabled unobservable transition is forced to
occur - reduce problem on real valued sets
to finite state problem
by using state classes of time
Petri nets
66diagnosis can be refined by adding timing
information to model
- problem becomes further complicated if one takes
into account that concurrently executed traces
may be forced to remove a token from a place and
thus may also eliminate a traces from set of
possible explanations of an observed event
67diagnosis for timed Petri nets
- variables in linear (in)equalities
execution times ?t of all events
t ? minimal explanation EMin(OBS )
of observed set OBS - must satisfy equations
68Outline
- problem formulation and models
- why on-line state estimation?
- centralised diagnosers
- distributed diagnosis
- extensions to timed DES models
- probabilistic fault diagnosis, open questions and
conclusions
69probabilistic diagnosis
- for free choice PNs it is possible to "easily"
derive probability distribution
over set of possible explanations of observed
event sequence - if ?t ? Tu (?t)1 then it suffices to define
for each place p a probability distribution over
set of transitions in set p? -
70probabilistic diagnosis
- if trace (t1 t2...tnO) ? possible explanations
of observation tnO at time ?(tnO) - and if pn probability that token in unique
place pk ?tk moves to tk (i.e. pk
probability that tk is executed if pk becomes
marked) - then the weight of trace (t1 t2...tnO) in the of
possible explanations is ?k1,...,n pk
71probabilistic diagnosis
- the probability of trace (t1 t2...tnO) in the of
possible explanations is obtained by normalizing
these weights ?k1,...,n pk over all elements in
the set of explanations - summing the probabilities of all traces that
contain a fault defines the Bayesian
(conditional) probability that the fault
occurred, given the probabilistic model and given
the observed sequence of events
72probabilistic diagnosis
- for free choice PNs need to assign probability
per set of places in ?tj for set of concurrent
transitions with common ?tj - calculation remains largely identical then as
before - for non-free choice nets the compatible
definition of the probabilities of choices made
by tokens in places becomes very difficult
73probabilitistic diagnosis of timed Petri nets
- timed petri nets where each firing time
distribution is exponentially leads to Markov
process ( stochastic PN) - probababilistic diagnosis in principle easy
(Bayesian recursive algorithm) in that case - but set of explanations of sequence of observed
events for stochastic PN set of explanations of
untimed PN
74probabilitistic diagnosis of timed Petri nets
- combining forced transitions ( probability
distribution over interval with finite upper
bound) leads to very complicated analysis - computational complexity probably same as using
finite state abstraction of such a probabilistic
PN
75other case studies
- traffic modelling via hybrid systems
- ? incident detection via failure diagnosis
- differences
- stochasticity much more important
- really hybrid system use fluid Petri nets or
hybrid automata - initial state belongs to large set of possible
initial states - concept of minimal explanation may be relevant in
this case study too!
76main ideas - open problems
- backward search more efficient/more easily
distributed than forward search - for minimal explanations (faults that must have
occurred - diagnosis versus prognosis) - computational complexity
- stopping criteria/saturated languages
- concurrency expressed via Petri nets
- interaction between Petri net components via
common places
77conclusions
- applying fault diagnosis to realistic plant model
requires computationally efficient algorithms - combine analysis of this talk with computer
science approaches for describing large sets, and
for reachability analysis - abstraction leads to pessimistic fault detection
results but may be inevitable