UTA

ARRI

Petri Nets for simulation/control of Discrete Event Systems:

introduction & matrix representation.

By

Jose Mireles Jr.

ARRI

Discrete Event SystemsUTA

-The behavior is characterized by a sequence of finite or infinite states delimited by asynchronous events.

states

time

-Almost all are man made:- computer systems (operating, communication, data processing systems)- discrete production systems (manufacturing systems)

-Some continuous systems can dealt as DES if it is considered as events

- thresholds of continuous variables- the beginning and ending of the subsystem operation

Examples of Industrial Use of CP-nets http://www.daimi.au.dk/CPnets/intro/example_indu.htmlThe following projects document large-scale practical use of CP-nets and their tools. Many of the projects have been carried out in an industrial environment. For all projects one or more papers are available.

Protocols and Networks Intelligent Networks at Deutsche TelekomIEEE 802.6 Configuration Control at Telstra Research Laboratories Allocation Policies in the Fieldbus Protocol in Japan ISDN Services at Telstra Research Laboratories Protocol for an Audio/Video System at Bang & Olufsen A/S TCP Protocols at Hewlett-Packard Local Area Network at University of Las Palmas UPC Algorithms in ATM Networks at University of AarhusBRI Protocol in ISDN Networks Modelling and Analysis of Wireless Application Protocol (WAP) Internet Multimedia Communication Standards Internet Open Trading Protocol Distributed Dynamic Channel Allocation Algorithm for Mobile Computing

Software Execution Architecture of Mobile Phone Software at Nokia Feature Interaction in Mobile Phones at Nokia Bank Transactions and Interconnect Fabric at Hewlett-Packard Mutual Exclusion Algorithm at University of AarhusInternet Object Cache at the Hungarian Academy of Science Electronic Funds Transfer in the US Document Storage System at Bull AG ADA Program at Draper Laboratories

Hardware Superscalar Processor Architectures at University of Newcastle VLSI Chip in the US Arbiter Cascade at Meta Software Corp.

Control of Systems Security and Access Control Systems at Dalcotech A/S Mechatronic Systems in Cars at Peugeot-Citroën in France European Train Control System in Germany Flowmeter System at DanfossTraffic Signals in Brazil Chemical Production in Germany Model Train System at University of Kiel Educational Project with Control System for Model Railway

Military Systems Military Communications Gateway in Australia Influence Nets for the US Air Force Missile Simulator in Australia Naval Command and Control System in Canada

Other Systems Bank Courier Network at Shawmut National Coop. Nuclear Waste Management Programme in the US Actor-based Systems at University of CalabriaComputer Simulation of Biochemical Processes

ARRI

Basic concepts of PN

UTA

Token evolution A two part transition rule

- Enabled transition. The input places have at least as many tokens as the weight of the input arcs.

- Transition firing. a) Remove tokens of the input places (indicated by the

weight of the input arcs)b) Add tokens to the output places (indicated by the

weight of the output arcs)

Example:

2 2

ARRI

Formally a Petri net can be defined as follows:

UTA

PN = (P,T,I,O,Mo); where

1. P = { p1, p2,…, pm} is a finite set of places2. T = { t1, t2,…, tn} is a finite set of transitions3. I : ( P x T ) N is an input function that defines

directed arcs from places to transitions, where Nis a set of non-negative integers.

4. O : ( P x T ) N is an output function that defines directed arcs from transitions to places.

5. Mo : P N is the initial marking

For simplicity, let us work with functions Pre, a previous incidence function (explained later)Post, a posterior incidence function (explained later)

Marking vector M = [m1, m2, m3, m4,… mm]T

where mi is the number of tokens in pi.

Usually, Mo denotes the initial marking.

ARRI

UTA

∆, The transition rule:

- Enabling condition: a transition tj ∈ T is enabled iff∀pi ∈ P, M(pi) ≥ Pre(pi, tj)(current marking ≥ Pre function vector of tj).

-Firing of tj:

M(pi)k+1 = M(pi)k – Pre(pi,tj) + Post (pi,tj) ∀pi ∈ P

this expression may be rewritten as:

Mk = Mk+1 + Cvk ( PN state equation )

Where C is the incidence matrix:C = [cij] cij = Post (pi,tj) - Pre (pi,tj)vk is the firing transition vector at instant tk.

Example:

Example, for the following figure, the marking vector is M = [ 1 1 0 1 0 1 ]T

p1

p2

p3

p4

p5

p6

ARRI

UTA

p1 p2 p3t2 t3

p5 p6t4 t5

t1

p4

Marking Mo = [ 2 0 0 1 0 0]T

Pre(t1) = [ 1 0 0 1 0 0]T (only relates precedent places)

Post(t1) = [ 0 1 0 0 1 0]T (only relates posterior places)

Pre = 1 0 0 0 00 1 0 0 00 0 1 0 01 0 0 0 00 0 0 1 00 0 0 0 1

0 0 1 0 01 0 0 0 00 1 0 0 00 0 0 0 11 0 0 0 00 0 0 1 0

Post =

t1 t2 t3 t4 t5 t1 t2 t3 t4 t5

ARRI

UTA

t1 is enabled, since

200100

Mo

100100

Pre(t1)

≥

Firing sequence of t1

110010

M1

200100

Mo

= - +

100100

Pre(t1)

010010

Post(t1)

p1 p2 p3t2 t3

p5 p6t4 t5

t1

p4

Notation: M0 M1

t1

1

ARRI

Introductory examples

Two synchronized processes:

UTA

Sequence:-The wagons start moving (authorized by switch M) simultaneously to the right.-When end position is reached the wagons inverse motion.-When the initial position is detected, every wagon stops.-A new cycle starts when both wagon are stopped.

a b

L1 R1

2d

L2 R2

c

M

abcdM

L1R1

L2R2

p1 p2 p3t2 t3

p5 p6t4 t5

t1

p4

M

d

b

c

aR1

R2

L1

L2

1

ARRI

Synchronization of three processes:

UTA

Sequence:- Wagons move at different speed.- All wagons start moving at same time.- Wagons 1 and 2: The first one arriving to the right position waits for the other one to start return together.- Wagon 3: independent return.

a b

L1 R1

2d

L2 R2

c

M

L2

L1

dM

3f

L3 R3

e

D2

L3D3

D1

f

b

d

f

b

ARRI

Matrix Formulation of Discrete Event Systems.Basic concepts:

UTA

Job

Trans Fv Relates v j

xiRelates xi

Trans.Fr

Resource rj

Relates x j vi

Transition

Jobs Sv Relates

ri

xj

Transition

Releaseresource Sr

)(][)()()1( txFStmxMtmtm TT −+=+=+PN marking transition equation:

m(t) = [ ]TTc

TTT tutrtvtu )(,)(,)(,)(

],,,[ ucT

ucrT

rvT

vuT

uT FSFSFSFSFSM −−−−=−=

mFx ⊗=

)(][][)()( turvuFFFFtmFtx cucrvu ⊕=⊕=

Firing transition equation:

Next examples !!!

UTA

ARRI

pinA p1t1 t2

p3t4 t5

p2 t3

p4 t6pinB

poutA

poutB

r1

r3

r2

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

100001000000001000010000

vF

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

000010000100000000100001

TvS

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

000010000001

uF

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

000010100000010001

rF

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

010100000010001000

TrS

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

100000010000

TyS

p1 p2 p3 p4 r1 r2 r3

p1 p2 p3 p4 r1 r2 r3

pinA pinB

poutA poutB

UTA

ARRI

pinA p1t1 t2

p3t4 t5

p2 t3

p4 t6pinB

poutA

poutB

r1

r3

r2

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

100001000000001000010000

x

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010000001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010100000010001

Fv

Locating firing transitions from current marking:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1110

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

001

⎥⎦

⎤⎢⎣

⎡00

Fr Fu

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

110100

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000000

= , x =

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

110101

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

001010

v r u

UTA

ARRI

pinA p1t1 t2

p3t4 t5

p2 t3

p4 t6pinB

poutA

poutB

r1

r3

r2

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

100001000000001000010000

x

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010000001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010100000010001

Fv

Conflict resolution:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1010

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

101

⎥⎦

⎤⎢⎣

⎡00

Fr Fu

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

100100

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

001001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000000

= , x =

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

101101

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

010010

v r u

Which one?

UTA

ARRI

pinA p1t1 t2

p3t4 t5

p2 t3

p4 t6pinB

poutA

poutB

r1

r3

r2

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

100001000000001000010000

x

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010000001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010100000010001

Fv

Conflict resolution, add extra matrix Fuc:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1010

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

101

⎥⎦

⎤⎢⎣

⎡00

Fr Fu

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

100100

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

001001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000000

= , x =

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

101101

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

010010

v r u

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

001000000100

Fuc r2

⎥⎦

⎤⎢⎣

⎡00

markingnegative!

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000000

UTA

ARRI

pinA p1t1 t2

p3t4 t5

p2 t3

p4 t6pinB

poutA

poutB

r1

r3

r2

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

100001000000001000010000

x

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010000001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010100000010001

Fv

Conflict resolution, add extra matrix Fuc:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1010

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

101

⎥⎦

⎤⎢⎣

⎡00

Fr Fu

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

100100

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

001001

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000000

= , x =

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

111101

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

000010

v r u

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

001000000100

Fuc r2

⎥⎦

⎤⎢⎣

⎡10

Preferenceon t2

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

010000

Preventive / Corrective Maintenance

Deadlock Problem