Petri Net

1
Lecture 9

• Petri Net
• Reaction tank an application example

2
Petri Net

• 9.1 Petri Net - introduction
• 9.2 Petri Net - history
• 9.3 Petri Net and state diagram
• 9.4 Petri Net and PLC in Industrial Control
• 9.5 Petri Net - terminology
• 9.6 Petri Net - formal definition
• 9.7 Petri Net - types
• 9.8 Petri Net - primitive structures
• 9.9 Petri Net - properties
• 9.10 Petri Net - interpretation for Industrial
  Control

3
Reaction tank -scheme
Raw material
Water for cooling
Steam for heating
Product
4
Reaction tank an application example

• Reaction tank problem
• Reaction tank - scheme
• Technological conditions
• Variables list
• Petri net for reaction tank control places and
  transitions
• Petri net for reaction tank control

5
Reaction tank control - Petri net
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
6
Reaction tank control - Petri net

• P1,.P5 places
  (circle)
• T1,.T6 transitions
  (rectangle)
• oriented connection - edge (line,arc)
• token, mark
  (filled point)
• Static character structure of the net
• Dynamic charecter movement of the mark through
  the net

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9.1 Petri Net - introduction

• Mathematical tool suitable for analysis and
  modelling of discrete events systems
• Grafical tool (comprehensive)
• Very helpful and favourite tool both for
• theoretical research
• and for practical design of control systems

8
Petri Net - introduction - 2

• PN Petri Net
• SIPN Steuerung Interpretierte Petri
• Netz (Germany)
• It is possible to simply express and model
• Parallel behaviour
• Resource sharing
• Real time relationships

9
Petri Net introduction - 3

• This introduction deals with the graphical aspect
  of Petri Net for system description, not the
  algebra of Petri Net.
• While the mathematical properties of Petri Net
  are interesting and useful, the beginner will
  find that a good approach is to learn to model
  systems by constructing them graphically, aided
  in construction and analysis by computer software
  for simulation and analysis of Petri Nets.

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10.2 Petri Net - history

• Author Carl Adam Petri Ph.D. dissertation
  thesis in 1962 on the topic Formal methods in
  dicrete events modelling (communication betweeen
  sequential machines - automatons)
• See Ref. 1
• Further development at MIT Massachusets
  Institute of Technology (seventies of the
  previous century)

11
Petri Net - history - 2

• Since then, Petri Nets and their concepts have
  been extended and developed, and applied in a
  variety of areas
• Office automation, work-flows, flexible
  manufacturing, programming languages, protocols
  and networks, hardware structures, real-time
  systems, performance evaluation, operations
  research, embedded systems, defence systems,
  telecommunications, Internet, e-commerce and
  trading, railway networks, biological systems !

12
9.3 Petri Net and state diagram

• State diagram suitable for typical sequential
  systems (one event following another event in
  clear order, only a few alternative pathes, no
  paralell pathes)
• Petri nets suitable for parallel processes and
  communications
• SW for Petri net applications makes possible to
  create and run model in modular way

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9.4 Petri Net and PLC in Industrial Control

• There are very efficient graphical tools
  (languages) at disposal today for PLC programming
  which are very similar to the Petri Nets
  formalism (they are based on it)
• Two terms directly onnected with Petri Nets
• GRAFCET (Graphe Fonctionenel de Connexion Etapes
  Transitions - France, origin 1975-1977) since
  1987 International standard (IEC 848)
• SFC (Sequential Function Chart)
• according to the international standard
• IEC 1131-3 (standard for PLC programming)

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9.5 Petri Net - terminology

• A Petri Net is a collection of directed arcs
  connecting places and transitions. Places may
  hold tokens.
• The state or marking of a net is its assignment
  of tokens to places.
• Here is a simple net containing all components of
  a Petri Net

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Petri Net - terminology - 2
Simple net
weight
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Petri Net - terminology - 3

• Arcs have capacity (weight) 1 by default if
  other than 1, the capacity is marked on the arc.
• Places have infinite capacity by default.
• Transitions have no capacity, and cannot store
  tokens at all.
• With the rule that arcs can only connect places
  to transitions and vice versa, we have all we
  need to begin using Petri Nets.
• A few other features and considerations will be
  added as we need them.

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Petri Net terminology - 4

• Petri net models consist of two parts
• 1. the net structure that represents the static
  part of the system
• 2. a marking that represents the overall state
  on the structure. State can change - this
  represents dynamic behavior of the system.

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Petri Net - terminology - 5

• The token distribution among the places of a
  Petri net is called its marking.
• When one or more tokens reside in a place, the
  place is said to be marked, otherwise it is
  unmarked.
• The number of tokens at a place represents the
  local state of the place.
• The marking of the net represents the overall
  state of the system.

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Petri Net - terminology - 6

• A transition is enabled when the number of tokens
  in each of its input places is at least equal to
  the arc weight going from the place to the
  transition.
• An enabled transition may fire at any time. When
  fired, the tokens in the input places are moved
  to output places, according to arc weights and
  place capacities. This results in a new marking
  of the net, a state description of all places.

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Petri Net - terminology - 7

• Example
• This Petri net has
• 2 places p1, p2
• 1 transition t1
• p1 has one token f(p1) 1
• p2 has 0 tokens f(p2) 0

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Petri Net - terminology - 8

• Firing a Transition
• When a transition tj fires
• Each pi that has an edge from pi to tj removes a
  token from pi
• Each pj that has an edge from tj to pj adds a
  token to pj

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Petri Net - terminology - 9
Petri net after t1 fires
Petri net before t1 fires
23
Petri Net - terminology - 10

• A transition must be enabled before it fires
• There is a token in each pi that has an edge
• to the transition
• An enabled transition may or may not fire
• (it can be another condition asociated with
  the transition)

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Petri Net terminology - 11
Petri net after t1 fires
Petri net before t1 fires
25
Petri Net - terminology - 12

• A place can contain non-negative whole number of
  marks
• At the moment of transition firing (activation,
  skipping) the marks are withdrawn from input
  places and added to the output places of the
  transition
• Oriented edges connect places and transitions

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Petri Net - terminology - 13

• initial marking (marks displacement in the
  places before the first skip) describes the
  initial state of the system
• system development is represented by transfer of
  the marks in the net on the basis of activating
  of the transitions (skip)
• every new marking of the system represents a new
  state of the system

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Petri Net - terminology - 14 Transition enabling
- example

• weight of the P9 - T3 edge is 4
• place P9 contains only 3 3 tokens
• transition T3 is not enabled (for one car we
  need four wheels)
• transition T3 cannot be fired

wheels
motor
P9
car body
P10
P8
4
T3
P11
car
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9.6 Petri Net - formal definition
A Petri Net is a 5 tuple PN (P,T,F,W,M0), where

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9.7 Petri Net - types

• Original Petri Nets
• Only 1 token can be removed/added from a place
  when a transition fires
• (i.e., the weight is always 1)
• Weighted Petri Nets
• Generalized the original Petri net to allow
  multiple tokens to be added/removed when a
  transition fires.
• The edges are labeled with the weight (i.e.,
  number of tokens)
• If there is no label, then the default value is 1

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Petri Net - types - 2
Weighted Petri net after t1 fires
Weighted Petri net before t1 fires
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Petri Net - types - 3

• Other Types of Petri Net
• Petri nets have been extended over the years in
  many directions including time,data, and
  hierarchy.
• Time Extended Petri Nets
• Coloured Petri Nets
• Hierarchical Petri Nets

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Time Extended Petri Net

• First developed in the mid 1970s
• For real systems it is often important to
  describe the temporal behavior of the system,
  i.e., we need to model durations and delays.
• There are 3 basic ways to introduce time into the
  Petri net. Time can be associated with
• tokens
• places
• transitions
• The first introduction of time in Petri nets is
  in the Timed Petri net model
• C. Ramchandani, Analysis of Asynchronous
  concurrent systems by timed petri nets, MIT
  MAC-TR-120, 1974. In this model, a time duration
  is associated with each transition.

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Time Extended Petri Net - 2

• The firing rules in this model are that the
  transition must fire as soon as it is enabled,
  and firing a transition takes a fixed, finite
  amount of time.
• The notion of instantaneous firing of transitions
  is not preserved in the Timed Petri net model.
• When a transition becomes enabled, the tokens are
  immediately removed from its input places.
• After the time delay, tokens are deposited in
  the output places.
• The result is that the state of the system is not
  always clearly represented during the process.
• // Note. There are many other extensions to Petri
  Nets that consider time.

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Coloured Petri Net

• Developed in the Late 1970s ( K. Jensen,
  Colored Petri nets and the invariant method,
  Theoretical Computer Science, volume 14, 1981,
  pp. 317-336.)
• Tokens often represent objects (e.g. resources,
  goods, humans) in the modeled system
• To represent attributes of these objects, the
  Petri net model is extended with coloured or
  typed tokens.
• each token has a value often referred to as
  colour'.
• Transitions use the values of the consumed tokens
  to determine the values of the produced tokens
• a transition describes the relation between the
  values of the input tokens and the values of
  the output tokens'
• It is also possible to specify preconditions'
  which take the colours of tokens to be consumed
  into account.

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Hierarchical Petri Nets -1

• Developed in the Late 1980s
• Specifications for real systems have a tendency
  to become large and complex
• An abstraction mechanism, hierarchical
  structuring, is used to make constructing,
  reviewing, and modifying the model easier
• The hierarchy construct is called a subnet

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Hierarchical Petri Net - 2

• A subnet is an aggregate of a number of places,
  transitions, and subsystems
• Such a construct can be used to structure large
  processes
• At one level we want to give a simple
  description of the process (without having to
  consider all the details).
• At another level we want to specify a more
  detailed behavior.
• Each subnet is represented with a rectangular box
  that encapsulates part of the Petri Net model

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Hierarchical Petri Net - 3

• Each subnet is represented with a rectangular
  box that encapsulates part of the Petri Net model

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9.8 Petri Net - Primitive structures
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Primitive structures - 2

• Sequence is obvious - several things happen in
  order.
• Conflict is not so obvious. The token in P4
  enables three transitions but when one of them
  fires, the token is removed, leaving the
  remaining two disabled. Unless we can control the
  timing of firing, we don't know how this net is
  resolved.

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Primitive structures - 3

• Concurrency, again, is obvious many systems
  operate with concurrent activities, and this
  models it well.
• Synchronization is also modeled well using Petri
  Nets when the processes leading into P8, P9 and
  P10 are finished, all three are synchronized by
  starting P11.

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Primitive structures - 4

• Confusion is another not so obvious construct. It
  is a combination of conflict and concurrency. P12
  enables both T11 and T12, but if T11 fires, T12
  is no longer enabled.
• Merging is not quite the same as synchronization,
  since there is nothing requiring that the three
  transitions fire at the same time, or that all
  three fire before T17 this simply merges three
  parallel processes.
• The priority/inhibit construct uses the inhibit
  arc to control T19 as long as P16 has a token,
  T19 cannot fire.

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Primitive structures - 5
Example Enabling and activation - effective
conflict between transitions T1a T2
hungry human B
hungry human A
dinner
P2
P1
P3
T1
T2
P4
P5
P6
full human B
full human A
dirty dishes
hladový strávník A
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Primitive structures - 6
Conflict development between transitions T1 and
T2 version A
hungry human B
hungry human A
dinner
P2
P1
P3
T1
T2
P4
P5
P6
filling human B
filling human A
dirty dishes
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Primitive structures - 7
Conflict development between transitions T1 and
T2 version B
hungry human B
hungry human A
dinner
P2
P1
P3
T1
T2
P4
P5
P6
filling human B
filling human A
dirty dishes
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10.9 Petri Net- properties

• We can ask some interesting questions about the
  Petri Nets (i.e., we can ask interesting
  questions about the models of a particular
  system)
• Some properties of interest include
• Terminate
• Does the Petri Net terminate?
• Immediately Reachable
• Is a state reachable when a transition fires?
• Reachable
• Is a state eventually reachable?

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Petri Net- properties -2

• Live
• In all states, is there at least one transition
  that can fire?
• Partial deadlock
• Is there a state in which at least one
  transition that can never fire?
• Deadlock
• Is there a state in which none of the
  transitions can fire?

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Petri Net- properties -3

• Safe
• In all states, does each place contain at most
  one token?
• Bounded
• In all states, is there a limit to the number
  of tokens that can be in one place?
• Conservative
• Is the total number of tokens in the Petri Net
  constant?

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9.10 Petri Net - interpretation for Industrial
Control

• Petri net can expres the desired control system
  behaviour (SIPN)
• Another condition must be fulfilled before
  transition firing, e.g. some event asociated
  with the transition must occure
• Places actions of the control system (outputs
  of PLC)
• Transitions conditions (e.g. from sensors,
  inputs of PLC)

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Reaction tank - problem

• typical partial task from food industry or
  chemical industry
• analogue variables, e.g. temperature, level,
  pressure, are monitored via binary sensors (only
  limit values of this analogue variables - it is
  cheaper
• concurency of activities in technological process
• SIPN application

50
Reaction tank -scheme
Raw material
Water for cooling
Steam for heating
Product
51
Technological conditions for control system
(e.g. PLC)

• Specific reaction is to be run in the reaction
  tank at the temperature hold between ltT1,T2gt.
  Steam is used for heating (double cover of the
  tank). Cold water is used for cooling (radiator
  tube inside of the tank). The liquid is mixed
  with the mixer during the reaction.
• In the initial state all valves are closed,
  both heating, cooling and mixing do not work.

52
Technological conditions - 2

• Start of the process via Start button
• The tank is filled up to the high level sensor
  and the mixer is started
• The mixer is running until the reaction is
  fulfilled
• According to the comming raw material temperature
  steam heating or water coolling is starting up

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Technological conditions - 3

• If the comming raw material temperature matches
  the desired interval ltT1,T2gt, neither heating nor
  cooling is started
• If the desired temperature is reached when
  heating or cooling, the heating or cooling is
  shut down. The reaction is completed.
• The liquid from the tank is drained (low level
  sensor)

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Variables list inputs (of PLC)

• STA.Starting button (the reaction is to start
  má zacít STA1)
• H_LS.. High level of the raw material sensor
  (the level is above the high sensor H_LS1)
• L_TS.. Low temperature of the raw material
  (Tltlow limit T1 L_TS1)
• H_TS.. High temperature of the raw material
  (Tgthigh limit T2 H_TS1)
• L_LS.. Low level of the raw material sensor (the
  level is under the low sensor L_LS1)

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Variables list outpus (of PLC)

• V1_S.. Valve for raw material input
  (V1_S1 open the valve)
• M... Mixer motor
• (M1 start the mixer)
• V2_TP valve for heating steam input (V2_TP1
  open the valve)
• V3_P.. Valve for product output (V3_P1
  open the valve)
• V3_CHV..Valve for cooling water input
  (V3_CHV1 open the valve)

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Limit values identification

• Analogue variables are often monitored via binary
  sensors (cheaper)
• There are two types of limit values in industrial
  processes control operating and critical
• operatingH - High
• L- Low
• critical HH - High High
• LL - LowLow
• These limits are highlighted on the operator
  panel (for better overview )

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Petri Net (PN) for reaction tank control -
places - and output variables values

• P1 (standstill) V1_S0, V2_TP0, V2_TP0,
  V3_P 0, V3_CHV 0, M0
• P2 (filling) V1_S1
• P3 (coolling) V3_CHV1
• P4 (heating) V2_TP1
• P5 (draining) V3_P1
• P6 (mixing) M1

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Petri Net for reaction tank control -
transitions - and input variables values

• STA, (start) T1
• H_LS.NotL_TS , (filled, cool) T2
• L_TS, (low temperature of the raw material)
  T3
• L_LP, (low level of the product )
  T4
• H_LS.H_TS, (filled, warm) T5
• NotH_TS, (not enough high temperature of the raw
  material T6
• H_LS.NotH_TS.L_TS, (filled, temperature between
  ltT1,T2gt T7

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PN for reaction tank control -1
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
60
PN for reaction tank control -2
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
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PN for reaction tank control - 3
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
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PN for reaction tank control - 4
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
63
PN for reaction tank control - 5
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
64
PN for reaction tank control - 6
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
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PN for reaction tank control -7
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
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PN for reaction tank control - 8
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
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PN for reaction tank control - 9
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
68
PN for reaction tank control -10
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
69
PN for reaction tank control - 11
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
70
PN for reaction tank control -12
P1
T1
P2
T5
T2
T7
P3
P4
P6
T3
T6
P5
T4
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Reference

• 1 Petri, C.A., Kommunikation mit Automaten.
  Petri, C.A., Bonn Institut für Instrumentelle
  Mathematik, Schriften des IIM Nr. 2, 1962, Second
  Edition, New York Griffiss Air Force Base,
  Technical Report RADC-TR-65--377, Vol.1, 1966,
  Pages Suppl. 1, English translation

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Reference - 2

• http//robotica.pardini.net/Documentos/Petri01.pdf
  
• www.hait.ac.il/staff/leonidm/1F99L15.htm
• http//www.ite.his.se/ite/utbildning/kurser/auc111
  /pdf/pn_informal.pdf
• http//www.techfak.uni-bielefeld.de/mchen/BioPNML
  /Intro/pnfaq.html
• A Petri Net Tool DNAnet (Free software for
  students)

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