Control and Deadlock Recovery of Timed Petri Nets Using Observers Alessandro Giua DIEE –...

Control and Deadlock Recoveryof Timed Petri Nets

Using Observers

Alessandro GiuaDIEE – Department of Electrical and Electronic Engineering

University of Cagliari, Italy

Joint work with: - Carla Seatzu (U. of Cagliari) - Francesco Basile (U. del Sannio)

A. Giua, Control and Deadlock Recovery of Timed Petri Nets Using Observers, MOSIM’04 - Sept 2, 2004 2

OUTLINE

0) Petri nets

1) Motivation for discrete event observers

2) Relevant literature

3) Main idea

4) Marking estimation

5) Marking estimation with initial macromarking

6) Control using observers

7) Deadlock recovery and estimate after net time out

8) Using timing information to improve the procedure

9) Conclusions and future work


0 – PETRI NETSA place/transition net is a 4-ple : N=(P,T,Pre,Post)

• P={ p1, p2, …, pm} set of places (circles);

• T={ t1, t2, …, tn} set of transitions (bars);

• Pre: matrix denoting # of arcs from places to transitions• Post: matrix denoting # of arcs from transitions to places

p1

p2 p3

p5

t6 t1

p8

t2p4

p5 p6 p7

t3

t4 t5


0, MN

MwMMNLwNMMNR m000 :),(),(

wMTwMNL 0*

0 ),(

• Net system (a net N with initial marking M0):

• Set of firable sequences:

• Set of reachable markings:

Siphon: a set of places S such that if a transition inputs into S then it also outputs from S (Ex: S = {p1, p2} )

p1

p2

p3

t1

t3

t2

An empty siphon will always remain empty all its output transitions are deadlocked

PETRI NETS (cont’d)


OUTLINE

0) Petri nets



3) Main idea








1 - MOTIVATION FOR DISCRETE EVENT OBSERVERS

Two approaches to design of observers for discrete event

Systems

• Computer science approach (CSA): the state is unknown because the system structure is nondeterministic

• Control theory approach (CTA): the system structure is deterministic but the initial state is unknown


• CTA - Supervisory control theory is based on

language specifications (a set of legal words):

plant

w

word of events

controllerKlegalwords

control

language specification event-feedback

MOTIVATION (cont’d)


• When dealing with Petri nets it is natural to use state specifications (a set of legal markings):

plant

M

state / marking

controller

control

L

legal markings

state specification state-feedback



• A mixed structure is often used:

State specification = marking of the net

Output events = transitions firing

plant

w

model

controller

M=Mww

L

M00

legal markings

• When the net structure and the initial marking is known (and the net labeling is deterministic) event observation is sufficient to reconstruct the net marking.



If the initial marking is not completely known:

use “observers” to estimate the marking after the word of events w has been observed

wM0M

w Bw

In our approach the observer determines two parameters

Estimate: Bound:

plant

w

observer

controller

ww Bww

L


legalmarkings


C(w)

Mw

),( ww Bfw C

),( tg wwt

),,( tBhB wwwt

wwB


Unlike other approaches based on automata, the PN

structure allows one to “describe” the set of consistent

markings in terms of these two parameters that

are recursively updated.

wC

Linear constraint set


In this talk we present: Algorithms for computing estimate and consistent set Algorithms for control using observers Algorithms for deadlock recovery Deadlock analysis of the closed loop system

PROBLEM: incomplete information due to the presence of an observer in the control loop may lead to deadlock.


All these problems are solved using the same approach

based on integer programming


OUTLINE

0) Petri nets



3) Main idea








2 - RELEVANT LITERATURE • State-feedback control with partial observability

- Li & Wonham [CDC88] [T-AC93] (state observ.)- Takai, Ushio & Kodama [T-AC95] (state observ.)- Zhang & Holloway [Allerton95] (event observ.)

plant

controller

control

Llegal

markings

Mmarking

mask

equivalenceclass

Derived nec & suff condition for optimality given a mask.


• DES state estimation for FSM / Predicate Transformers- Ramadge [CDC86] (FMS)- Caines, Greiner, Wang [CDC88] [CDC89] (FMS)- Özveren, Willsky [T-AC90] (FMS)- Kumar, Garg & Markus [T-AC93] (PT)

DRAWBACKThese approaches enumerate at each step the set of consistent states (high complexity). No notion of “estimate error”.

LITERATURE (cont’d)


• Diagnosis- Wang, Schwartz [T-net 93] (state estimation)- Ushio, Onishi & Okuda [SMC98] (place observation)

• Petri net observability- Meda, Ramirez [SMC98] (interpreted nets)- Ramirez, Riveda, Lopez [ICRA2000]

• Partial knowledge of the marking- Cardoso, Valette & Dubois [ICATPN90] Concept of macromarking and “membership function”.

LITERATURE (cont’d)


OUTLINE

0) Petri nets



3) Main idea








3 - MAIN IDEA

• Initially observed sequence:

• Initial marking

• Estimate:

• Set of consistent markings:

0w

Tw MM 11100

Tw 0000

00 w

m MMw NC

t 2 p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3


firing is detected

t 2 p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3

1t

MAIN IDEA (cont’d)


After fires

• Observed sequence:

• Actual marking

• Estimate:

• Set of consistent markings: t 2 p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3

Tw 0011

TwM 0121

11 tw 1t


11 w

m MMw NC


firing is detected2t

t 2 p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3



• Observed sequence:

• Actual marking

• Estimate:

• Set of consistent markings: t 2 p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3

Tw 0102

212 ttw TwM 021

2

After fires21tt


22 w

m MMw NC


OUTLINE

0) Petri nets



3) Main idea








4 - MARKING ESTIMATION

• Hypothesis:

- The net structure is known

- The transition firing can be observed

- The initial marking is not known

PostPreTPN ,,,

0M

• Algorithm

1 - Initial estimate: Let

2 - Wait until fires

3 - Update previous estimate:

4 - New estimate:

5 - ; goto 2.

00

w string)(empty 0ww

t

)(Premax ,t,wwt ,t)C(wtwt

wtw


ESTIMATION (cont’d)

• Can define place error:

ww M

)()(),( ppMMe wwwwp

),(),(),( wwpwtwpwtwtp MeMeMe

),(),(),( wwwtwwtwt MeMeMe

wmmm MMMwMMMw NNNC 00 :)(

• Estimate is a lower bound:

• The set of markings consistent with observation w is:

and estimation error:

• Error functions are non-increasing:

wwww MMe 1),(

Properties


Properties

• An observed word is marking complete if w ww M

w L(N,M ) 0

ww

wt

• A net system is:

- Marking Observable (MO) if there exists a complete word

- Strongly Marking Observable (SMO) in k steps if:

a) all with are complete

b) all with that are not complete can be continued in a word

N M, 0



Observer reachability graph

• Each node of the graph is labeled with:

The real marking Mw The estimation error uw = Mw - w

t1

t2

t3

p1 p2

p3

2 0 0 / 2 0 0

1 1 0 / 1 0 0

1 0 1 / 1 0 0

2 0 0 / 1 0 0

t2

t2

t3

t1

0 2 0 / 0 0 0 1 1 0 / 0 0 0

0 1 1 / 0 0 0 2 0 0 / 0 0 0

0 0 2 / 0 0 0 1 0 1 / 0 0 0

t3

t3 t3 t2

t1

t1

t2

t2

t1

t3

t2

t2



Observer coverability graph

• If the net is unbounded, is it possible to construct an observer coverability graph (OCG). The error vector u is now only an upper bound.

t1 t2 p

1 / 1

/ 1

t1

t1, t2

t1, t2

t2 t1

t1

0 / 0

1 / 0

/ 0

t2



Analysis of properties

Theorem 1

A net system is:N M, 0

),(),( 0000MNLMNLMM

• marking observable iff

• marking observable if there exist a node in the

OCG with

• strongly marking observable iff in the OCG for each

dead node and for each node in a cycle



OUTLINE

0) Petri nets



3) Main idea








5 – MARKING ESTIMATION WITH INITIAL MACROMARKING

• Sometimes partial information on the initial marking is available

0M

),(0 MNRM

X

MXMXMMNR TTmN),(

• Example: assume the net starts from marking (known) evolving unobserved until it reaches ; at this point we start observations.

Then we may use the information that

• This characterization in terms of PN reachability is hard to use but we can approximate it using a matrix of invariants :

M


Generalizing, we define an initial macromarking.

bMVMbV Tm NM ,

P rPPPP 10

jP1j jb

jPjv

Tr

r

bbbb

vvvV

21

21

0P

• The set of places is written as:

• For each , the token content of is known to be

Nothing is known about the marking in

• Let be the char vector of and define

• Macromarking:

MACROMARKING (cont’d)


A MANUFACTURING EXAMPLE

p1

p10

p3 p4

p5

p6

p7

p8 p9

p2

t6

t7

t1

t4

t2

t3

t5

p11

p12

M0(p11)+M0 (p12) = 1M0(p1)+M0 (p3)+M0 (p4) = 5M0(p1)+M0 (p5)+M0 (p6) = 5M0(p1)+M0 (p3)+M0 (p6) +M0 (p11) = 6M0(p1)+M0 (p4)+M0 (p5) +M0 (p12) = 5

Initial macromarking: we know the token content in each cycle

M0(p2)+M0 (p8) +M0 (p9) = 6M0(p2)+M0 (p3)+M0 (p4)+M0 (p7)+M0 (p10) = 6M0(p2)+M0 (p5)+M0 (p6)+M0 (p7)+M0 (p10) = 6M0(p2)+M0 (p3)+M0 (p6)+M0 (p7)+M0 (p10)+M0 (p11) = 7M0(p2)+M0 (p4)+M0 (p5)+M0 (p7)+M0 (p10)+M0 (p12) = 6



Algorithm (estimation with macromarking)

1 – Initial estimate with

2 - Initial bound

3 - Let the current observed word be w=w0.

4 - Wait until t fires.

5 - Update the estimate to

6 - New estimate:

7 - New bound:

8 - Goto 4.

0w 00

(p)w

bBw

0

,t)C(wtwt

)(VBB wwtT

wwt


),(Pre),(max)( tppp wwt w


Elementary results

• The estimate is a lower bound: ww M

MwMbVMMMw m [:),( 00

def NC

• The error functions are non-increasing

• The set of markings consistent with the observation w is:


wwwTTm

ww M,BVMVMBw Ndef

,CC

• This set can also be characterized as:


OUTLINE

0) Petri nets



3) Main idea








6 - CONTROL USING OBSERVERS

• GMEC specifications: a set of linear constraints

for j = 1, …, q.

Example:

jT

j kMx

})(|{ jT

jm kMxjM NL

• The set of legal markings is:

1)()(2)()(

43

21pMpMpMpM


• Control with observer

Prevent the firing of t after w has been observed iff there exists a

legal consistent marking M such that the firing of t from M leads to a

forbidden marking i.e., if exists j such that where

enabledcontrolisif1disabledcontrolisif0

, )(tt

tf wCControl pattern:

CONTROL (cont’d)

),('),,1(

)(..

'max

tCMMpikMx

wMts

Mx

iT

i

Tjj

C

jj k*


• EXAMPLE

t2

p1 p2

p3t1 t3

}2)(|{ 1 pMML

3)()()(),( 321 pMpMpMMbVM

The firing of t1 is legal from but

111

0M

),()(102

0 bVwM MLCL

L

003

102

1tt1 is disabled


• EXAMPLE

t2

p1 p2

p3t1 t3

1 0 2 / 0 0 0 / 0

1 1 1 / 0 1 0 / 1

0 2 1 / 0 1 0 / 1

0 1 2 / 0 1 0 / 1

0 0 3 / 0 0 0 / 0

t3

t2

t3

t3

t1

2 1 0 / 0 1 0 / 1

1 2 0 / 0 1 0 / 1

0 3 0 / 0 1 0 / 1

t2

t1

t1

t3

t3

0 2 1 / 0 1 1 / 2

0 1 2 / 0 1 1 / 2

0 0 3 / 0 0 1 / 1

1 0 2 / 0 0 1 / 1

t2

t3

t3

t1

0 1 2 / 0 0 1 / 1

1 1 1 / 0 0 1 / 1

0 2 1 / 0 0 1 / 1

1 2 0 / 0 0 0 / 0

t2

t1

t2

t1

2 1 0 / 1 1 0 / 2

t1

1 1 1 / 0 1 1 / 2

t1

t1

2 0 1 / 0 0 1 / 1

t1

t3

2 1 0 / 0 0 0 / 0 t1

t3

t3

1 0 2 / 1 0 1 / 2 t3

2 0 1 / 1 0 1 / 2

t1

t2

1 1 1 / 1 1 1 / 3

Actual marking Mw Estimation error uw = Mw - w

Bound


• Usually, the control law using observers is not optimal

since it can disable the firing of transitions that do not

yield illegal markings.

• Such a control law may easily cause the controlled plant

to block.

• We want to add to the observer the possibility of

recovering from deadlocks caused by the incomplete

information.

CONTROL (cont’d)


A MANUFACTURING EXAMPLE p1

p10

p3 p4

p5

p6

p7

p8 p9

p2

t6

t7

t1

t4

t2

t3

t5

p11

p12

M(p9) 3

M(p3)+M(p5) 3


( 4 5 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 1 5 5 6 5 6 6 6 7 6 )

( 4 5 0 1 0 1 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 0 4 5 5 4 6 5 6 6 5 )

( 5 5 0 0 0 0 1 1 0 0 1 0 / 1 0 0 0 0 0 1 0 0 0 1 0 / 0 4 4 4 4 6 5 5 5 5 )

( 5 6 0 0 0 0 0 0 0 0 1 0 / 1 1 0 0 0 0 0 0 0 0 1 0 / 0 4 4 4 4 5 5 5 5 5 )

t1

t4

t3

Deadlock

Actual marking Mw Estimate w

Bound w


p1

p10

p3 p4

p5

p6

p7

p8 p9

p2

t6

t7

t1

t4

t2

t3

t5

p11

p12

M(p9) 3

M(p3)+M(p5) 3

Only the green tokens have been detected

t6 and t7 are disabled by the controller

?

?

?


OUTLINE

0) Petri nets



3) Main idea








IDEA:

use the info that the net is deadlocked to improve the estimate (reducing the set of consistent markings)

B,C

bM

Theorem: In an ordinary net a marking

M is dead iff:

• is a siphon

• for all

0)( pMpS

StTt ,

Mw

7 - DEADLOCK RECOVERY AND ESTIMATE UPDATE AFTER NET TIME-OUT

Set of blocking markings


• Given a structurally bounded net N, a marking M is dead iff a vector ({0,1}m such that:

1Pre1

1PostPre

)( 22

1

sMs

KMsKssK

N

T

TT

D

s

DEADLOCK RECOVERY (cont’d)

)(),(:}1,0{|)( NsMsMN mmb DM N

The set of blocking markings of N:

is the characteristic vector of a siphon s

contains only empty places s

contains all empty places s

each transitions has at least apre arc coming from s


DEADLOCK RECOVERY (cont’d) Algorithm (Control pattern updating after net time-out)

Let C =C(,B). Assume f(.,C) has led the net to a time-out.

1. Let i=0 and f0 = f(., C).

2. Let Ti={tT | fi(t)=1} and let Ni the net obtained by N

removing all transitions not in Ti.

3. Update the control pattern to fi+1=f(., C Mb(Ni))

4. If fi+1= fi THEN exit: (the deadlock procedure has failed)

5. Wait until

(a) a transition fires (net has recovered from deadlock)

(b) a new net time-out occurs: let i=i+1 and go to 2.


A unique linear algebraic formalism for:

– state estimation

– control

– deadlock recovery


Main advantages of the approach:

This procedure is denoted NTO procedure (net time-out procedure).


A MANUFACTURING EXAMPLE (cont’d) p1

p10

p3 p4

p5

p6

p7

p8 p9

p2

t6

t7

t1

t4

t2

t3

t5

p11

p12

M0(p11)+M0 (p12) = 1M0(p1)+M0 (p3)+M0 (p4) = 5M0(p1)+M0 (p5)+M0 (p6) = 5M0(p1)+M0 (p3)+M0 (p6) +M0 (p11) = 6M0(p1)+M0 (p4)+M0 (p5) +M0 (p12) = 5

M(p9) 3

Initial macromarking: we know the token content in each cycle

M(p3)+M(p5) 3

M0(p2)+M0 (p8) +M0 (p9) = 6M0(p2)+M0 (p3)+M0 (p4)+M0 (p7)+M0 (p10) = 6M0(p2)+M0 (p5)+M0 (p6)+M0 (p7)+M0 (p10) = 6M0(p2)+M0 (p3)+M0 (p6)+M0 (p7)+M0 (p10)+M0 (p11) = 7M0(p2)+M0 (p4)+M0 (p5)+M0 (p7)+M0 (p10)+M0 (p12) = 6


( 4 5 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 1 5 5 6 5 6 6 6 7 6 )

( 4 5 0 1 0 1 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 0 4 5 5 4 6 5 6 6 5 )

( 5 5 0 0 0 0 1 1 0 0 1 0 / 1 0 0 0 0 0 1 0 0 0 1 0 / 0 4 4 4 4 6 5 5 5 5 )

( 5 6 0 0 0 0 0 0 0 0 1 0 / 1 1 0 0 0 0 0 0 0 0 1 0 / 0 4 4 4 4 5 5 5 5 5 )

t1

t4

t3

NTO


( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

t6

( 4 5 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 1 5 5 6 5 6 6 6 7 6 )

( 4 5 0 1 0 1 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 0 4 5 5 4 6 5 6 6 5 )

( 5 5 0 0 0 0 1 1 0 0 1 0 / 1 0 0 0 0 0 1 0 0 0 1 0 / 0 4 4 4 4 6 5 5 5 5 )

( 5 6 0 0 0 0 0 0 0 0 1 0 / 1 1 0 0 0 0 0 0 0 0 1 0 / 0 4 4 4 4 5 5 5 5 5 )

t1

t4

t3

NTO


NTO

NTO

( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

t6

( 5 5 0 0 0 0 0 0 1 1 1 0 / 5 0 0 0 0 0 0 0 1 1 1 0 / 0 0 0 0 0 5 5 5 5 5 )t7

( 4 5 1 0 1 0 0 0 1 0 1 0 / 4 0 1 0 1 0 0 0 1 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )t2

( 4 5 1 0 0 1 0 0 1 0 0 1 / 4 0 1 0 0 1 0 0 1 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t5

( 4 5 1 0 0 1 0 1 0 0 1 0 / 4 0 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t1

( 4 5 0 1 0 1 0 1 0 0 1 0 / 4 0 0 1 0 1 0 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

( 5 5 0 0 0 0 1 1 0 0 1 0 / 5 0 0 0 0 0 1 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

t4

t3


NTO

NTO

( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

t6

( 5 5 0 0 0 0 0 0 1 1 1 0 / 5 0 0 0 0 0 0 0 1 1 1 0 / 0 0 0 0 0 5 5 5 5 5 )t7

( 4 5 1 0 1 0 0 0 1 0 1 0 / 4 0 1 0 1 0 0 0 1 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )t2

( 4 5 1 0 0 1 0 0 1 0 0 1 / 4 0 1 0 0 1 0 0 1 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t5

( 4 5 1 0 0 1 0 1 0 0 1 0 / 4 0 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 5 5 5 5 5 ) t1

( 4 5 0 1 0 1 0 1 0 0 1 0 / 4 0 0 1 0 1 0 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

( 5 5 0 0 0 0 1 1 0 0 1 0 / 5 0 0 0 0 0 1 1 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 1 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 5 5 5 5 5 )

t4

t3

( 5 6 0 0 0 0 0 0 0 0 1 0 / 5 6 0 0 0 0 0 0 0 0 1 0 / 0 0 0 0 0 0 0 0 0 0 )


Proposition: if the initial macromarking is such that

(i.e., each column of V is a P-invariant) then, for all

observed words w,

bV

,M0CV T

bVww

,0 MCC


If a marking is consistent with the observation w then it

is also consistent with the initial observation


Theorem 1: if the initial macromarking is such that

then the closed loop system will never time out if the following constraint set does not admit feasible solutions

wt, T | ft T 1))(( 00 C

bV

,M0CV T

)( 0N M

bMVb

T

M


where


Definition: the maximal control pattern for a set C is:

where and i

iff

lim),(max C

),(0 C ff ))(N, f()g(ff ibii MC1


When a controlled system times out, if it is deadlocked

eventually a control pattern is reached ),(max Cf


Theorem 2: if the initial macromarking is such that

then the closed loop system will always recover from a time-out if the following constraint set does not admit feasible solutions

bVt, T | ft T 1)),((maxmax

M

bV

,M0CV T

)( maxN M

bMVb

T

M


where


p1

p10

p3 p4

p5

p6

p7

p8 p9

p2

t6

t7

t1

t4

t2

t3

t5

p11

p12

M(p9) 3

M(p3)+M(p5) 3

A MANUFACTURING EXAMPLE (cont’d)

Initial macromarking: the net is a marked graph

each cycle corresponds to a P-invariant

the initial macromarking is such that

bV

,M 0CV T


A MANUFACTURING EXAMPLE (cont’d)

Theorem 1 does not apply:

the following constraint set admits feasible solutions

the net might time out (it actually does)

)( 0N M

bMVb

T

M

Theorem 2 does apply:

the following constraint set does not admit feasible solutions

the closed loop system with net time-out recovery

is deadlock-free

)( maxN M

bMVb

T

M


OUTLINE

0) Petri nets



3) Main idea








8 - USING TIMING INFORMATION TO IMPROVE THE PROCEDURE

We extend the previous approach to exploit available information on the timing structure so as to obtain a better estimate of the set of consistent markings.

A known delay time (t) is associated to each transition.

We say that a transition t has timed-out at time now if it has been control enabled without firing during [now- (t), now].

We can be sure that at time now the actual marking Mw is such that Mw |t.


IDEA:

If Tto is the set of transitions that have timed out at time now, we know for sure that the actual marking is such that

We compute a (possibly) less restrictive control pattern using as set of consistent markings

i.e., for all we compute

USING TIMING INFORMATION (cont’d)

)( tobw N M M

)()( tob N w MC Tt

)()( tob N w MCt,f


USING TIMING INFORMATION (cont’d)

Main Advantages:

• Accelerates the state estimation

• Accelerates the deadlock recovery procedure

• Enables to recover from partial deadlocks

The new approach is denoted TTO procedure (transition time-out procedure).


A MANUFACTURING EXAMPLE (cont’d) p1

p10

p3 p4

p5

p6

p7

p8 p9

p2

t6

t7

t1

t4

t2

t3

t5

p11

p12

M(p9) 3

M(p3)+M(p5) 3

(t1) = 2 (t2) = 5

(t3) = 3 (t4) = 1

(t5) = 2 (t6) = 6

(t7) = 3

Transition time-out

Transition firing

Delays:


( 3 4 1 0 1 0 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 6 5 5 5 5 5 4 4 4 1 )

( 3 4 0 1 1 0 0 1 0 0 1 0 / 0 0 0 1 1 0 0 1 0 0 1 0 / 4 4 4 4 4 3 3 3 3 0 )

( 3 4 1 0 1 0 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0 0 / 6 5 5 5 5 5 4 4 4 1 )

( 3 4 0 1 1 0 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1 0 / 5 4 5 4 5 4 3 4 3 0 )

t1

( 3 4 0 1 1 0 0 1 0 0 1 0 / 0 0 0 1 1 0 0 1 0 0 1 0 / 4 4 4 4 4 3 3 3 3 0 )

( 3 4 0 1 1 0 0 1 0 0 1 0 / 0 0 0 1 1 0 0 1 0 0 1 0 / 4 4 4 4 4 3 3 3 3 0 )

t2

( 3 4 0 1 0 1 0 1 0 0 0 1 / 0 0 0 1 0 1 0 1 0 0 0 1 / 4 4 4 4 4 3 3 3 3 0 ) t4

( 4 4 0 0 0 0 1 1 0 0 0 1 / 1 0 0 0 0 0 1 1 0 0 0 1 / 4 4 4 4 4 3 3 3 3 0 )

now = 1

now = 2

now = 2

now = 3

now = 4

now = 7

now = 8

{ t4 }

{ t5 }

{ t3, t4, t5 }

{ t1, t3, t4, t5 }


t4

( 4 4 0 0 0 0 1 1 0 0 0 1 / 1 0 0 0 0 0 1 1 0 0 0 1 / 4 4 4 4 4 3 3 3 3 0 )

( 4 4 0 0 0 0 1 1 0 0 0 1 / 4 0 0 0 0 0 1 1 0 0 0 1 / 4 4 4 4 4 0 0 0 0 0 )

t3

( 4 5 0 0 0 0 0 0 0 0 0 1 / 4 1 0 0 0 0 0 0 0 0 0 1 / 4 4 4 4 4 0 0 0 0 0 )

( 4 5 0 0 0 0 0 0 0 0 0 1 / 4 1 0 0 0 0 0 0 0 0 0 1 / 4 4 4 4 4 0 0 0 0 0 )

( 4 5 0 0 0 0 0 0 0 0 0 1 / 4 5 0 0 0 0 0 0 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 )

now = 8

now = 9

now = 11

now = 12

now = 14

{ t1, t4, t5 }

{ t1, t2, t4, t5, t7 }

{ t1, t2, t3, t4, t5, t7 }


OUTLINE

0) Petri nets



3) Main idea








• We provided a unique linear algebraic formalism for: state

estimation, control, deadlock recovery.

• We showed how timing information can be used to accelerate the

state estimation and to detect the observer induced deadlock.

• Some sufficient conditions for deadlock recovery have been derived.

9 – CONCLUSIONS


FUTURE WORK

• Language completeness:

A word is language complete if

This may allow to use observers in event feedback.

),(),( ww MNLNL

• Partial event observability:

assume some events are unobservable or

undistinguishable. This may destroy the linear algebraic

formalism in the general case. Look for restricted cases.


• Associate a probabilistic structure to the transition

firing and define , where is the

probability of having a complete word after k firings.

Under which conditions ?

]1,0[: N )(k

1)( kk

FUTURE WORK (cont’d)


• A. Giua, C. Seatzu, “Observability of place/transition nets,” IEEE Trans. on

Automatic Control, Vol. 47, No. 9, pp. 1424-1437, September, 2002.

• F. Basile, A. Giua, C. Seatzu, “Observer based state-feedback control of

timed Petri nets with deadlock recovery,” IEEE Trans. on Automatic Control,

Vol. 49, No. 1, pp. 17-29, Jan 2004.

• F. Basile, A. Giua, C. Seatzu, "Observer-based state-feedback control of

timed Petri nets with deadlock recovery: theory and implementation," Proc

CESA'2003 Multiconference (Lille, France), Jul 2003.

• A. Giua, C. Seatzu, J. Júlvez, "Marking estimation of Petri nets with pairs of

nondeterministic transitions," Asian Journal of Control, June 2004. To appear.

• A. Giua, D. Corona, C. Seatzu, “State estimation and control of

nondeterministic -free labeled Petri nets”, Proc. WODES’04. To appear.

REFERENCES


- Structurally Strongly Marking Observable (sSMO) if

the system is SMO for all

- Structurally Marking Observable (sMO) if the system

is MO for all

)R(N,MM 0

• A net system is:

- Uniformly Marking Observable (uMO) if the system

is Marking Observable for all

N M, 0

MN ,

)R(N,MM 0

- Uniformly Strongly Marking Observable (uSMO) if

the system is SMO for all MN ,

mNMMN ,

MN , mNM



• uniformly MO iff the semi-linear set

is a home-space for all

Theorem 2

A net system is:N M, 0

),(Pre)(),(Pre0)( tppMtMpMMAptp

Pp

• uniformly SMO only if it is bounded

Similar results hold for structural MO and SMO.

This is a finite union of linear sets with the same period and the

home space property is decidable (Johnen & Frutos Escrig; 89)




B,C

bM

Mw

• After a deadlock recovery procedure is invoked we should remember the set of consistent markings is

• The linear characterization of this set is rather complex (it involves also a vector ). We propose to use a simpler approximation.

bB MC ,

s



B,C

bM

Mw

• Compute for all places:

b

iiBμMts

pMzMC

,..

)(min

Tmw zz 1~

)~(~

wwT

ww VBB

wwwTT

ww μM,BμVMVMBμ ~~~~,~ C

• Define the new set:

• Compute:

• Define

ww Bμ ~,~C

Control and Deadlock Recovery of Timed Petri Nets Using Observers Alessandro Giua DIEE –...

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Transcript of Control and Deadlock Recovery of Timed Petri Nets Using Observers Alessandro Giua DIEE –...