Distributed Diagnosis of Discrete-Event Systems Using ...€¦ · Introduction • Why fault...

Distributed Diagnosis of Discrete-EventSystems Using Petri Nets

Sahika Genc and Stephane Lafortune

Department of Electrical Engineering and Computer Science,University of Michigan,

{sgenc,stephane}@eecs.umich.edu; www.eecs.umich.edu/umdes

June 25, ATPN 2003, Eindhoven, Netherlands

Outline

• Introduction

• Centralized Diagnosis

• Distributed Diagnosis with Communication

• Main Result

• Summary

Sahika Genc and Stephane Lafortune, University of Michigan / June 25, 2003 1

Introduction

• Why fault diagnosis?

? Limited sensor information: Faults are unobservable events.

• Problem:

? Detect and isolate faults during the operation of the system.

• Model-based approach: Normal and failed behaviour.

? Discrete-Event System(DES) models are adequate for large class offaults.


Introduction: Previous Work

• DES Modelling Formalism: Automata (languages)

? “Failure Diagnosis Using Discrete Event Models” by M. Sampath, R.Sengupta, S. Lafortune, K. Sinnamohideen, and D. Teneketzis IEEETransactions on Control Systems Technology Vol. 4, No. 2, March1996, pp. 105-124

? “Diagnosability of Discrete Event Systems” by M. Sampath, R.Sengupta, S. Lafortune, K. Sinnamohideen, and D. Teneketzis IEEETransactions on Automatic Control Vol. 40, No. 9, September 1995,pp. 1555-1575

• Previous theory successfully applied to ...


Introduction: Areas of Application

• HEATING, VENTILATION AND AIR CONDITIONING SYSTEMS

Sinnamohideen, Sampath, et al., Johnson’s Control Inc.



• DOCUMENT PROCESSING SYSTEMSSampath, et al., Xerox Corp.

Document Centre 265 DC/LP/ST



• AUTOMATED HIGHWAY SYSTEMS(AHS)

Sengupta, et al., PATH, UC-Berkeley


Introduction: Diagnoser Approach

• Previous work: Solution methodology based on diagnoser automata.

Theory of diagnosability Which faults can be diagnosed?

Online diagnosis How to diagnose?


Introduction: Diagnoser Approach

• Previous work: Solution methodology based on diagnoser automata.

Theory of diagnosability Which faults can be diagnosed?

Online diagnosis How to diagnose?

• Objective: Develop an analogous methodology based on Petri netmodels and deal with distributed systems.

• Why Petri nets?

? A good mathematical tool to model concurrent, asynchronous anddistributed systems.

• Online diagnosis.


Outline

• Introduction



• Main Result

• Summary


Centralized Diagnosis: Notation

• A Petri net graph: N = 〈P, T,A, w〉.

• A labeled Petri net: (N ,Σ, l, x0, f).

• The labeling function: l : T → Σ.

• The labeling function is extended to strings of transitions: l : T ∗ → Σ∗

l(t) = a, l(t′) = a′ ⇒ l(tt′) = l(t)l(t′) = aa′.

• The set of events: Σ = Σo ∪ Σuo.


Centralized Diagnosis

N Nd

System

LabeledPetri Net

LabeledPetri Net

Diagnoser

• The system to be diagnosed is modelled by a labeled Petri net.

• The diagnoser is a labeled Petri net.



N Nd

System DiagnoserObservable

EventFaultType

LabeledPetri Net

LabeledPetri Net

• The system to be diagnosed is modelled by a labeled Petri net.

• The diagnoser is a labeled Petri net.

• The Petri net diagnoser observes the system online and outputs whichfault types have occurred.



The diagnoser for the labeled Petri net (N ,Σ, l, x0, f) is

Nd = (N ,Σ, l, xd0,∆f , fd)

where

• xd0 is the initial diagnoser state,

• ∆f = {F1, . . . , Fk}: Finite set of fault types,

• fd: Diagnoser state transition function.


Centralized Diagnosis: Diagnoser States

Diagnoser state =States F1 · · ·Fk[

|||

]

• A diagnoser state has multiple states(markings).



Diagnoser state =States F1 · · ·Fk[

|||

]

• A diagnoser state has multiple states(markings).

• Each state in the diagnoser state has a fault label. The fault labelshows which type of faults have occurred.

? If a fault of type i has occurred, then the ith entry in the fault labelis 1, otherwise 0.

• The fault label of the initial state, x0, is lx0f = [0 . . . 0].



Given (N ,Σ, l, x0, f) and Nd = (N ,Σ, l, xd0,∆f , fd),

• Unobservable reach of a state x, UR(x), is found by firing thetransitions labeled with unobservable events.

• The initial diagnoser state is the unobservable reach of the initial stateof the system:

xd0 = UR(x0lx0f ).


Centralized Diagnosis: Example

uo f1 a

uo a a e

p1

t1

t4

t2

t5

t3

t6 t7

p4

p7

p2

p5

p8

p3

p6

p9 p10

N

uo f1 a

uo a a e

p1

t1

t4

t2

t5

t3

t6 t7

p4

p7

p2

p5

p8

p3

p6

p9 p10

Nd**

**

**

xd0 =

[1 1 1 0 0 0 0 0 0 0 | 0 00 1 1 1 0 0 0 0 0 0 | 0 00 0 1 0 1 0 0 0 0 0 | 1 00 1 1 0 0 0 1 0 0 0 | 0 0

] •�4∗



Given (N ,Σ, l, x0, f) and Nd = (N ,Σ, l, xd0,∆f , fd)

• If a ∈ Σo is feasible from the diagnoser state x,

? S(x, a) is the set of states reached from the states of x by firingtransitions labeled with a,

? The next diagnoser state x′ = fd(x, a) is found as

x′ = ∪s∈S(x,a)UR(s).


Centralized Diagnosis: Example

uouo f1f1 aa

uouo aa aa ee

p1p1

t1t1

t4t4

t2t2

t5t5

t3t3

t6t6 t7t7

p4p4

p7p7

p2p2

p5p5

p8p8

p3p3

p6p6

p9p9p10p10

N : xd d0 N : xd d1

**

**

****

**

a

xd1 = fd(xd0, a) =

1 0 0 0 0 1 0 0 0 0 | 0 00 0 0 1 0 1 0 0 0 0 | 0 00 1 1 0 0 0 0 0 1 0 | 1 00 0 0 0 0 1 1 0 0 0 | 0 00 1 1 0 0 0 0 1 0 0 | 0 0

•�4∗�



Certain or Uncertain?

Diagnoser state =F1 F2 F3[

| 1 0 1| 1 0 1| 1 0 0

]

• Certain?

? Fault of type 1 (F1) has occurred.? Fault of type 2 (F2) has not occurred.

• Uncertain?

? Fault of type 3 (F3) may or may not have occurred.


Outline

• Introduction



• Main Result

• Summary


Distributed Diagnosis with Communication

• Objective : Achieve same performance of centralized diagnosis withdistributed diagnosis.

• Why distributed diagnosis? System to be diagnosed is

? too large to perform centralized diagnosis

� large automated manufacturing systems, etc.? truly distributed

� networked systems, etc.


Distributed Diagnosis with Communication: Centralizedvs. Distributed


N Nd


EventFaultType


Nd,1Nd,1

Nd,2Nd,2

FiFi

FjFj

System

Diagnoser

Diagnoser

Observable Event ofFirst Diagnoser

Observable Event ofSecond Diagnoser

FaultType

FaultType

Communication

N1N1

NN

N2N2

CommonPlaces



• Based on design considerations, the labeled Petri net (N ,Σ, l, x0, f) ispartitioned into two labeled Petri nets (N1,Σ1, l1, x0,1, f1) and(N2,Σ2, l2, x0,2, f2) as follows

? Σ = Σ1∪Σ2,

? ∀t ∈ T if l(t) ∈ Σ1, then t ∈ T1; ∀t ∈ T if l(t) ∈ Σ2, then t ∈ T2,

? P1 = ∪t∈T1 (I(t) ∪O(t)), P2 = ∪t∈T2 (I(t) ∪O(t)).Result: Common places; disjoint sets of events, transitions and arcs.

N1N1

NN

N2N2

CommonPlaces



• The partitions must satisfy the following assumptions

1. ∀t ∈ T if (I(t) ∪O(t)) ∩ (P1 ∩ P2) 6= ∅ , then l(t) ∈ Σo.2. ∀t1 ∈ T1 and ∀t2 ∈ T2, if l(t1) ∈ ΣFi and l(t2) ∈ ΣFj, then i 6= j.

N1N1NN

N2N2

CommonPlaces

so

s’os’o

s’o

so

so

N1N1NN

N2N2

Fi

Fj


Centralized vs. Distributed


N Nd


EventFaultType


Nd,1Nd,1

Nd,2Nd,2

FiFi

FjFj

System

Diagnoser

Diagnoser

Observable Event ofFirst Diagnoser

Observable Event ofSecond Diagnoser

FaultType

FaultType

Communication

N1N1

NN

N2N2

CommonPlaces


Distributed Diagnosis with Communication: Messages

• Given Pc the set of common places, define the weighting vector

WPc(t) = [w(t, p1)− w(p1, t), . . . , w(t, p|Pc|)− w(p|Pc|, t)].

• x; label lxmt→ x′ = f(x, t); lx

′m

lx′

m = [ lxm, WPc(t) ],

i.e., message label records how many tokens are put into or removedfrom the common places.

• The message label of the initial state is the empty matrix, lx0m = [ ].



• The message label of a diagnoser state is the listing of the messagelabels of each state(row) in the diagnoser state.

• Message sent is the message label of the diagnoser state.

x′d = fd(xd, σo) ⇒

x′d =

States Fault T. lxdm WPc[

| || || |

]︸︷︷︸

MESSAGE



• s ∈ Σ∗o and xd = fd(xd0, s) is defined, then the length of the message

label is|lxd

m | ≤ |s||Pc|.

• Consider the message label lm = [A, B].

? If B is the zero matrix, then

lm = [A].

? If all the rows of A are same, then

Truncate(lm) = [B].



Algorithm DDC: Given that the sequence s = σo0σo1 . . . σon is observedwhere |s| = n + 1, initialize the algorithm i := 0.

Upon observation of σoi do



Algorithm DDC: Given that the sequence s = σo0σo1 . . . σon is observedwhere |s| = n + 1, initialize the algorithm i := 0.

Upon observation of σoi do { If σoi ∈ Σ1, then go to 1, else go to 2 }

1. {Master is Nd,1 }1.1 Find the next diagnoser state.

1.2 If no message is created, then go to step (1.4).1.3 Send message to Nd,2. Nd,2 “updates” its diagnoser state upon

reception of this message.

1.4 If possible, truncate message labels.

1.5 Increment i.

2. {Master is Nd,2 } Same as 1, but exchange 1 and 2 in every expression.


Distributed Diagnosis with Communication: Example

uo f1 a

uo

a

a a

p1

t1

t4

t8

t2

t5

t3

t6

p4

p7

p11

p2

p5

p8

p3

p6

p9

N1

e

h

e

f2

h

e

g

e

f2

g

t12

t16

t9

t13

t17

t10

t14

t7

t11

t15

p11

p15

p8

p12

p16

p3

p6p9

p13 p10

p14

N2



uo f1 a

uo

a

a a

p1

t1

t4

t8

t2

t5

t3

t6

p4

p7

p11

p2

p5

p8

p3

p6

p9

N1

****

**

1 2 3 4 5 6 7 8 9 11

x01 =

[1 1 1 0 0 0 0 0 0 0 |00 1 1 1 0 0 0 0 0 0 |00 0 1 0 1 0 0 0 0 0 |10 1 1 0 0 0 1 0 0 0 |1

] •�4∗



e

h

e

f2

h

e

g

e

f2

g

t12

t16

t9

t13

t17

t10

t14

t7

t11

t15

p11

p15

p8

p12

p16

p3

p6p9

p13 p10

p14

N2

3 6 8 9 10 11 12 13 14 15 16

x02 = [ 1 0 0 0 0 0 0 0 0 0 0 |0 ]



Upon observation of a ∈ Σ1,

• fd,1(x01, a) = x1

1

1 2 3 4 5 6 7 8 9 11 3 6 8 9 11

x11 =

1 0 0 0 0 1 0 0 0 0 |0| −1 1 0 0 00 0 0 1 0 1 0 0 0 0 |0| −1 1 0 0 00 1 1 0 0 0 0 1 0 0 |0| 0 0 1 0 00 1 1 0 0 0 0 0 1 0 |1| 0 0 0 1 00 0 0 0 0 1 1 0 0 0 |0| −1 1 0 0 00 1 1 0 0 0 0 0 0 1 |0| 0 0 0 0 1

Message =[ −1 1 0 0 0

0 0 1 0 00 0 0 1 00 0 0 0 1

]



• Message

(=

[ −1 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

])is received by Nd,2 and the

diagnoser state is updated from

3 6 8 9 10 11 12 13 14 15 16

x02 = [ 1 0 0 0 0 0 0 0 0 0 0 |0 ]

to

3 6 8 9 10 11 12 13 14 15 16 3 6 8 9 11

x12 =

[0 1 0 0 0 0 0 0 0 0 0 |0| −1 1 0 0 01 0 1 0 0 0 0 0 0 0 0 |0| 0 0 1 0 01 0 0 1 0 0 0 0 0 0 0 |0| 0 0 0 1 01 0 0 0 0 1 0 0 0 0 0 |0| 0 0 0 0 1

].


Outline

• Introduction



• Main Result

• Summary


Main Result: Merge

• Let x and x be the diagnoser states of Nd,1 and Nd,2, respectively, atthe end of an iteration of Algorithm DDC.

• Merge operation is defined as follows

x =

di lxif lm

x =

dj lxj

f lm

Merge(x, x) =

di dj,P2−Pc lxif l

xj

f


Main Result: Theorem

Centralized Diagnosis Distributed Diagnosiswith Communication

Merge

xd,1

xd

xd

xd,2

EQUAL


Main Result: Example

After observation of the string aeh, the diagnoser states of the centralizedand distributed system are as followsx3 = fd(x

0, aeh) =[0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 |0 00 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 |0 1

]x3

1 =1 0 0 0 0 0 0 0 0 0 |0| −1 1 0 0 0 0 −1 0 0 00 0 0 1 0 0 0 0 0 0 |0| −1 1 0 0 0 0 −1 0 0 00 0 0 0 0 0 1 0 0 0 |1| −1 1 0 0 0 0 −1 0 0 00 1 1 0 0 0 0 0 0 0 |0| 0 0 1 0 0 0 0 −1 0 00 1 1 0 0 0 0 0 0 0 |0| 0 0 0 1 0 0 0 0 −1 00 1 1 0 0 0 0 0 0 0 |0| 0 0 0 0 1 0 0 0 0 −1

x3

2 =[1 0 0 0 0 0 0 0 0 1 0 |0| 0 0 0 0 1 0 0 0 0 −11 0 0 0 0 0 0 0 0 0 1 |1| 0 0 1 0 0 0 0 −1 0 0

]


Summary

• Define Petri net diagnosers to detect and isolate faults in systemsmodelled with Petri nets.

• Petri net diagnosers do not require structural changes of system model.

• Two types of implementation are proposed: Centralized and distributed.

• Algorithm DDC recovers centralized diagnoser information after mergeoperation.


Improving Performance

• Avoiding the growth of message labels:

? Truncation,

? Lossless data compression,

? Reset the diagnoser states after merge operation.

• Skip communication?

THANKS!


Distributed Diagnosis of Discrete-Event Systems Using ...€¦ · Introduction • Why fault...

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