Modeling with ordinary Petri Nets

Events: Actions that take place in the system

The occurrence of these events is controlled by the state of the system. The state of the system is described as a set of conditions.

A condition: a predicate or logical description of the state of the system.

Events may occur. Occurrence of an event may require some conditions to hold (preconditions).

Occurrence of events may cause some preconditions to ceaseand may cause other conditions (postconditions) to become true.

Modeling - machine shop example (1)

• Conditions

a The machine shop is

waiting.

b An order is arrived and is

waiting.

c The machine shop is

working on the order.

d The order is complete.

• Events

1 An order arrives.

2 The machine shop starts on

the order.

3 The machine shop finishes

the order.

4 The order is sent for

delivery.

Modeling - machine shop example (2)

E ven t P recon d ition s P ostcon d ition s

1234

N o n ea ,bcd

bc

d ,aN o n e

A n o rd e ra rriv es

A n o rd e r is w a itin g

P ro cessin gsta rts

T h e o rd e r isb e in g p ro cessed

P ro cessin g is co m p le te

T h e o rd e ris co m p le te

T h e o rd e r is sen t fo r d e liv e ry

T h e m ach in e sh o pis id e l, w a itin g fo r w o rk

1 2

3 4

a

cb d

Feedback control - Assembly cell (1)

T 1

T 2

T 3

T 4

T 5

T 6

T 7

T 8

P1

P2

P3P4

P5

P6

P7

W o rk a rea c lea r, en g in e b lo ck an d c ran k read y to b e p ro cessed

S -3 8 0 ro b o t a lig n s th e c ran k sh a ft

S -3 8 0 ro b o t p ick s u p n ewp is to n ro d an d p o s itio n s itin th e w o rk sp ace

M -1 ro b o t p ick s u p th e p is to n p u llin g to o l

M -1 ro b o t p u lls th e en g in ero d in to th e en g in e b lo ck , re tu rn s p u llin g to o l.

M -1 ro b o t p ick s u p th e cap an d p o sitio n s it o n p is to n ro d

M -1 ro b o t read ie s tw o b o ltsan d u ses th em to fa s ten cap to p is to n ro d

Feedback control - Assembly cell (2)

Constraints:

Each robot shall perform a task at a time: m2 + m3 1 m4 + m5 + m6 + m7 1

M-1 robot does not interrupt S-380 robot:m1 + m2 + m3 + m5 + m6 + m7 1

<<<Resources to be taken care of>>>:

A piston rod shall be ready at P3 : m3 1A pulling tool is required in P4 and P5 : m4 + m5 1A cap is required in P6: m6 1

Two nuts are required in P7: m7 1

Feedback control - Assembly cell (3)

Uncontrollability conditions:

Operation of M-1 robot not be interrupted from the point that it pullsthe piston rod into the engine block until it has completed fastening the cap on the piston rod. T6, T7, T8 uncontrollable.

1- Check L.Wuc

2- Find R1 and R2 so that R1. Wuc + R2L. Wuc 0L´= R1 + R2L.

R2 = I, find R1 by row operation.

Feedback control - Assembly cell (4)

T 1

T 2

T 3

T 4

T 5

T 6

T 7

T 8

P1

P2

P3P4

P5

P6

P7

c1

c4c3

c7c6

c2

c5

Uncontrollabilityconditions

Feedback control - Assembly cell (5)

Vision system used in M-1 robot has been obscured:Starting and completing the task can be observed, buttracking the robot’s performance in between is not possible T5, T6, T7 unobservable.

1- Check L´.Wuc

2- Find R1 and R2 so that R1. Wuo + R2L”. Wuo 0L”= R1 + R2L’.

R2 = I, find R1 by row operation.

Feedback control - Assembly cell (6)

T 1

T 2

T 3

T 4

T 5

T 6

T 7

T 8

P1

P2

P3P4

P5

P6

P7

c1

c4c3

c7c6

c2

c5

Uncontrollableand

unobservableconditions

Concepts in Petri Net (1)

Autonomous PN: Neither time nor external synchronization are involved

in the model.

Boundedness: For every reachable marking, the number of tokens in every place is bounded.

Safeness: The marking of every place is either 0 or 1 (Boolean marking).

Liveness: Regardless of the evolution, no transition will becomeunfireable on a permanent basis.

Invariants: P-invariants and T-invariants.

Concurrency: Firing of transitions are causally independent (I.e. concurrent,they may occur in any order).

Synchronization: ….

Concepts in Petri Net (2)

Source transition: without input place.

Sink transition: without output place.

Deadlock: no transition is enabled.

Conflict: between transitions.

Conservation: A PN is conservative if it does not lose orgain tokens but merely moves them around.

Petri net classes (1)P

etri

Net

s

Abb

revi

atio

ns

Ext

ensi

ons

P a rticu lar s tru c tu re

S y n ch ro n ized P N

Tim ed P N

In te rp re ted P N

S to ch as tic P N

In h ib ito r a rcs P N

C o n tin u o u s an d h y b rid P N

N o n -au to n o m o u s P N

G en era lized P N

F in ite cap ac ity P N

C o lo red P N

E v en t g rap h s

S ta te g rap h s

C o n tin u o u s P N

H y b rid P N

Ord

inar

y P

etri

net

s

Petri net classes (2)

Three main classes:

• Ordinary PNs: All arcs have weight 1, one kind of tokens, infinite capacityfor places, no time involved.

• Abbreviations: Simplified representations (useful graphical representations),can be mapped to an ordinary PN.

• Extensions: Some functioning rules are added, to enrich the initial model.

Applications

Modeling:- Communication protocols in computer systems; (concurrency, synchronization, and resource sharing).

- Manufacturing systems; • concurrency (two machines working independently)• synchronization (a machine is free and a part is ready to be processed by it).• Resource sharing (a robot is assigned to handle parts in two machines but not at the same time).

- Hybrid Systems (continuous + discrete parts); ex. batch production processes in biotechnological industry, manufacturing systems.

Analysis

PNs exhibit how a system work.

Analysis of a PN consists of seeking properties of the constructed model such as liveness, boundedness, deadlock, ….(they show whether the specifications are fulfilled).

Main categories of methods for seeking these properties:

• Drawing the graph of marking and coverability tree.• Using linear algebra.• Reducing the PNs.