1

MODELING

BY HYBRID PETRI NETS

In collaboration with

René DAVIDDirecteur de recherche émérite CNRS

Hassane ALLAProfesseur à l'UJF

2

Discrete Event Systems

état de la vanne

temps

fermée

en ouverture

ouverte

t1 t2 t3 t4

en fermeture

3

Motivation

M1 B1 M2 B2 M3

C1 C2C2 10 machines

9 buffers capacity 20

8 x 1014 states

Objective : Systems Dynamic Analysis

4

Motivation (cont’d)

t

t

Large numbers: continuous approximation may be convenient

5

Outline

DISCRETE, CONTINUOUS and HYBRID PETRI NETS

TIMING IN DISCRETE & CONTINUOUS PETRI NETS

AUTONOMOUS and TIMED HYBRID PETRI NETS

APPLICATION EXAMPLES

CONCLUSION

6

DISCRETE PETRI NETS

CONTINUOUS PETRI NETS

HYBRID PETRI NETS

7

Discrete PN

P1

P3

P2 P4

T1T2

T4

T3

P1

P3

P2 P4

T1T2

T4

T3

Firing of T1

T1

m2

m10

1

0 1 2

T3

T2

T1

T2

T4T3 T4T3 T4

T2 T2

Mk = Mi + W · S

8

Continuous PN

Mk = Mi + W · S

Firing of T1quantity 0.1

P1

P3

P2 P4

T1T2

T4

T31.9

1.0

0.1

P1

P3

P2 P4

T1

T4

T32.0

1.0 m2

m10

1

0 1 2

T1

T3

T2

T4

9

Hybrid PN

Mk = Mi + W · S

Firing of Tquantity 0.3

P1

P3

P2 P4

T1T2

T4

T32

P1

P3

P2 P4

T1T2

T4

T31.7

0.3

m2

m1

0

1

0 1 2

4

T 1 T 2

T 3 T 4

T 2

10

TIMING INDISCRETE PETRI NETS

&CONTINUOUS PETRI NETS

11

Timing in a Discrete PN

Machine M1

Buffer 2

Machine M2

Buffer 1

P1

P2

T1

T2

P3

P4

12

13

12

Timing in a Continuous PN

Pump

Tank 1

Tank 2

V1 = 3 liter/sec V2 = 2 liter/sec

180P1

P2

T1

T2

V1=3

V2=2

Unit = 1 liter

13

Discrete System : Approximation

Discrete model

P1

P2

T1

T2

P3

P4

21

31

d1=

d2=

Continuous model

P1

P2

T1

T2

75

V1= 2

V2= 3

14

Continuous System : Approximation

75

P1

P2

T1

T2

V1=2

V2=3

m2(t)=75-t

m2

75

m1(t) = (3-2) t = t

t

m1

00 75

75pour 3)(

2)(

22

11

t

Vtv

Vtv

75pour

2)(

2)(

12

11

t

Vtv

Vtv

15

Continuous System : exact model

180P1

P2

T1

T2

V1=3

V2=2

Unit = 1 liter

0 1800

m1 (l)

t

180 m1(t) = 180 - t

16

Maximum Firing SpeedsDepending on Time

T11V1(t)

P2

T21

P1

P3V2(t)

0

0

V1(t) = v1(t)

2

10 20 30

0

0

V2(t)

2

10 20 30

0

0

v2(t)

2

10 20 30

0

0

m2(t)

2

10 20 30

4

6

8

17

Generalization

The basic rules related to an autonomous continuous PN are verifiedThe instantaneous firing speeds may be defined in various ways

First example

v2(t) = k h(t)

Tank

h(t)Surface S

v1(t)

m1(t)

P1

T1

T2

v1(t)

m1

v2(t) = k S

Second example

P2P1

T1 v1(t) = l m1(t) m2(t)

m1 m2

18

AUTONOMOUSHYBRID PETRI NETS

19

Influence of the Discrete Parton the Continuous Part

P1

P3

P2 P4

T1T2

T4

T32

20

Influence of the Continuous Parton the Discrete Part

P1

P3

P2 P4

T1 T4

2

14.8

14.8

8.2

6.5

Production allowed

Production stopped

Input buffer

Production

Outputbuffer

21

Transformation of Continuous Markinginto Discrete Marking

Transition T1

not enabled

P1

P2

T1

20.12

0.75

P1

P2

T1

20.75

0.75

P1

P2

T1

20

0.75

Transition T1

enabled

Firing of T1

22

Transformation of Discrete Markinginto Continuous Marking

P3

P4

T2

20.10

0.75

Transition T2

enabled

Firing of T2 P3

P4

T2

20.85

0.75

23

General Case

P1P3

P2P4

T1

2 1.84.0

2

Firing of T1

P1P3

P2P4

T1

2 1.82.2

2

1.0

P5

P6

P7

T2

2

2.0

4.0

Quantity offiring of T2 0.1

P5

P6

P7

T2

2

2.0

3.9

0.2

24

TIMEDHYBRID PETRI NETS

25

Example of Hybrid PN

60

P3

P4

T3

T4

V3 = 3

V4 = 2

d1 = 90

d2 = 72

120

Valve L open

Pump working

P1

P2d5 = 45

d6 = 58

Pump stopped

Valve L closed

V3 = 3 liter/sec

Pump

V4=2 liter/sec

Tank 1

Tank 2

Valve L

26

APPLICATIONEXAMPLES

27

Electronic components assembly-test workshop (diodes and transistors)(Motorola in Toulouse – France)

Performance evaluation of a production System

Wafer400-5000 Chips

Chip

28

Cutting Molding Furnace Test

Electronic components assembly-test workshop (diodes and transistors)(Motorola in Toulouse – France)

Performance evaluation of a production System (cont’d)

Wafer

29

Production System

P1

P10

P8

P6

T1

T9

T3

V9 = 1

T11

P12

P2

P9

P11

P7

P13

P3

T7

T2

T10

T12

T4

T8

30 000 20 000

500

500

600

600

20 00030 000

d2 = 1 000

V10 = 1

V11=0.5 V12=0.33

d5 = 300 d6 = 360

Upstreambuffer

Processing

Upstreambuffer

Processing

P4 P5

T6T5

30

Water Supply System

Tank

Pumps

Lmax

Lmin

Supply from thenatural source

Supply from theground water

Waterconsumption

31

Water Supply System

t

V2(t)

T1

T4

V1

Supply from theground water

T5Lmax

Lmax

Lmin

T3 V3(t)

T2 V2(t)

Supply from thenatural source

Water in the tank

Water demand

Pumpsworking

Pumpsstopped

32

Transfer Line

S1 (or V1)11

C122

C233

M1 B1 M2 B2 M3

S2 (or V2) S3 (or V3)

V1 V2 V3

C1 C2

11 22 33

Placeavailable

Machinesoperational

Machinesout of order

Placeoccupied

Placeavailable

Placeoccupied

33

Controlled system via Communication Networks

Network Control System NECS Project

Not deformed reception of informations: (delays, flows capacities, losses)

Real time System : Temporal accuracy

Actuators SensorsSystem

Controller

Communication Networks

Use of the communication networks to carry out tasks of control

34

Open loop control

Messages: Continuous flows

Decisions of routing or emission: Discrete events

Networks communications can be represented by hybrid tools of modeling :

Modeling of the networks

Communication networks

Communication networks

Control deviceControl device ProcessProcess

hybrid Petri Nets

Communication networks

Communication networks

Controlled system via Communication Networks -Motivations

35

Modeling ToolsModeling Tools

buffer

Emitter1 Emitter2

Receiver2Receiver1

Communication network

Water supply system:

Natural sources

Consumptions

Tank

valve valve

valve valve

36

An analogy

Source

d’alim entation

Source

d’alimentation

buffer

Consumption 1

Tank

V1

Natural source 1 Natural source 2

T3

T2

V3

V2 T1

T4 V4

Consumption 2

t

V4

t

V3

V1 V2

Flow of messages

V1

Transmittingsources

V2

V4

Receivingsources

V3

t

V4

t

V3

37

Internet = Set of interconnected sub-network

Network IPbus

Network IPring

Network IPbus

Backboneof US Backbone of

Europe

National network

Regional network

Selected reference model: TCP/IP

Communication Network: Network architectures

38

Network capacity?

CongestedCongestednetworknetwork

E1

E2

E3

Rt4 R

Rt3 Rt2

Rt1

Networks architecture

39

CongestionCongestion control ontrol

E R

Tempo

cwnd = 1acq

acqcwnd = 2

Tempo

acqcwnd = 4

Tempo

124

8

16

Number of transmission

cong

esti

on w

indo

w

(Ko)

22

11

Expiration of TempoLosses of data

Thres2

 

Thres1

Slow start

CongestionAvoidance

Resume of slow start

Algorithms of TCP: Slow startCongestion Avoidance

40

Case of transmission line merged in Internet environment

VT3

RtV1V1 V2

V2E1 R

T1T’1 T2 T’2

Cnl 1

0+

Cnl 2

0+

C

T ”1

V1

0 1 2 3 4 5 6 7 8 9 10-5

0

5

10

15

20

25

t

VA transmission line

41

Related Models

BRINKMAN et al. CCPN + timed places(practical)OLSDER CCPN + timed places(theoretical)TRIVEDI et al. Fluid stochastic Petri netsSIFAKIS, HALWACKS et al. Hybrid automataALUR, COURCOUBETIS et al. Hybrid automataPETTERSSON, LENNARTSON Modeling, with Bond graphsCOHEN et al. Relat. Markov decisionWEITING Hybrid High-Level NetsDEMONGODIN, KOUSSOULAS + differential equationsBALDUZZI et al. + stochastic

42

Conclusion

Continuous systems have been studied for a long time

Modeling, analysis and control of DES have undergonemajor developments in recent decades

In recent years a need has emerged to consider systems whichare partially continuous and partially discrete

CONTINUOUS Petri nets and HYBRID Petri netscan be used for modeling these systems

43

Bibliography

SURVEY REFERENCES

BIBLIOGRAPHY ON HYBRID PETRI NETS

www.diee.unica.it/~aldo/bibliohpn.htmlby Alessandro Giua & Aldo Piccaluga, Dip. di Ingegneria Elettrica edElettronica, Università di Cagliari, Italy.

R. David and H. Alla, Discrete, Continuous, and Hybrid Petri Nets, in preparation, to be published by Springer, Heidelberg, 2004.

H. Alla, R. David, Continuous and Hybrid Petri Nets, Journal of Circuits, Systems and Computers, Special Issue on Petri Nets, Vol 8 No 1, 1998 pp. 159-188.

R. David, H. Alla, On Hybrid Petri Nets, Discrete Event Dynamic Systems, Theory and Applications, Kluwer Academic Publishers, 11, 9-40, 2001.