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
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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
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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
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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
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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
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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
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AUTONOMOUSHYBRID PETRI NETS
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Influence of the Discrete Parton the Continuous Part
P1
P3
P2 P4
T1T2
T4
T32
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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
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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
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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
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TIMEDHYBRID PETRI NETS
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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
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APPLICATIONEXAMPLES
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Electronic components assembly-test workshop (diodes and transistors)(Motorola in Toulouse – France)
Performance evaluation of a production System
Wafer400-5000 Chips
Chip
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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
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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
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Water Supply System
Tank
Pumps
Lmax
Lmin
Supply from thenatural source
Supply from theground water
Waterconsumption
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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
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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
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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
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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
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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
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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
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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
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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
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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.