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Distributed Fault Detection for untimed and for timed Petri nets

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### Distributed Fault Detectionfor untimed and for timed Petri nets

René Boel,

SYSTeMS Group, Ghent University

with thanks to:

G. Jiroveanu, G. Stremersch, B. Bordbar

Outline

- problem formulation and models
- centralised diagnosers via backward search
- distributed diagnosis for interacting Petri nets
- fault detection for timed Petri nets
- probabilistic fault diagnosis

adaptive 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

feedback control paradigm

observed

output

plant

state

estimator

and

fault detector

controlled

input

feedback law

distributed feedback control paradigm

plant

plant

plant

plant

fault

detector

fault

detector

fault

detector

fault

detector

feedback law

feedback law

feedback law

feedback law

distributed 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)

distributed 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?

distributed 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

distributed 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

distributed 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

distributed 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

distributed 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

Petri 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

interaction of Petri net components

- described by token passing via common places
- token entering boundary place enables events to happen only once in neighboring components

interaction of Petri net components

described by token passing via common places

token entering boundary place enables events to happen only once in neighboring components

interaction 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

backward 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

Petri net: example for explaining backward recursion

places P

transitions Τ

Pre arc

Post arc

t13

token

place with 2 tokens

decompose 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

observer design for one single component p5, p9

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

Petri net: compositional modelling p5, p9

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

Petri net: compositional modelling p5, p9

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

Petri net: compositional modelling p5, p9

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!

Petri net: compositional modelling p5, p9

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

computational complexity p5, p9

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!

Outline p5, p9

- 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

example p5, p9

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

prefix p5, p9

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

example p5, p9

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

unfolding of Petri net p5, p9

- 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"

unfolding of Petri net p5, p9

- 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

unfolding of Petri net p5, p9

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)

forward analysis via unfolding p5, p9

- 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

forward analysis via unfolding p5, p9

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

Outline p5, p9

problem formulation and models

why distributed on-line state estimation?

centralised diagnosers

distributed diagnosis

extensions to timed DES models

open questions and conclusions

backward search p5, p9

- 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

example: backward search p5, p9

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

example: backward search p5, p9

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

example: backward search p5, p9

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

example: backward search p5, p9

in order to determine whether explanations (t9 t10) is also possible FD agent2 must ask niehgboring FD agent1if 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

example: backward search p5, p9

- 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

Algiers, 5/5/07

VECOS'07

example: backward search p5, p9

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

Algiers, 5/5/07

VECOS'07

example: backward search p5, p9

Note that a central

observer

knowing both models

and

seeing all observations

also would not be able

to draw an

unambiguous

conclusion

t13

Algiers, 5/5/07

VECOS'07

example: backward search p5, p9

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

Algiers, 5/5/07

VECOS'07

example: backward search p5, p9

- 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

example: backward search p5, p9

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

example: backward search p5, p9

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

distributed fault diagnosis via backward search p5, p9

- 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)

local explanation p5, p9

= 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

fault diagnosis p5, p9

- Relaxed diagnosis goal!
- Enumerate only the minimal traces containing sequences of events that must have happened for OBSito 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

Minimal explanations p5, p9

- Set ℇi,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

Construction of minimal explanations p5, p9

- 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)

Construction of minimal explanations p5, p9

- 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

faults should not be predictable p5, p9

- 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

Some particularities of model p5, p9

- 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

distributed fault detection p5, p9

- 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

Outline p5, p9

- 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

timed discrete event models p5, p9

- 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

time Petri net model p5, p9

- 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)

time Petri net model p5, p9

- consider choice place p with t1, t2 p●
if L(t2) > 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

diagnosis 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)]

diagnosis 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

diagnosis 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

diagnosis 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

diagnosis 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

diagnosis 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

diagnosis for timed Petri nets model

- variables in linear (in)equalities: execution times tof all events t minimal explanation ℇMin(OBS ) of observed set OBS
- must satisfy equations:

Outline model

- 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

probabilistic diagnosis model

- 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●

probabilistic diagnosis model

- 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 k=1,...,n pk

probabilistic diagnosis model

- the probability of trace (t1 t2...tnO) in the of possible explanations is obtained by normalizing these weights k=1,...,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

probabilistic diagnosis model

- 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

probabilitistic diagnosis of timed Petri nets model

- 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

probabilitistic diagnosis of timed Petri nets model

- 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

other case studies model

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!

main ideas - open problems model

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

conclusions model

- 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

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