Exercise 2.1
Prof. Dr.-Ing. Jo ̈rg Raisch
Germano Schafaschek
Soraia Moradi
Behrang Nejad
Fachgebiet Regelungssysteme
Fakulta ̈t IV Elektrotechnik und Informatik Technische Universita ̈t Berlin Lehrveranstaltung ”Ereignisdiskrete Systeme“ Wintersemester 2020/2021
Exercise sheet 2
CoSntrGol
Consider the Petri nets given in Figure 1. Analyze each Petri net regarding its boundedness, liveness, and persistence properties.
Another potential feature of a Petri net is reversibility. A Petri net is called reversible if the initial marking x0 is reachable from all possible x ∈ R(N, x0). Are the Petri nets in Figure 1 reversible?
Sys
tem
s
Fachgebiet Regelungssysteme
a)
t1
t2
t3
t4
b)
t1
t2
t3
t4
c)
t1
t2
t3
t4
t5
d)
t1
t2
t3
t4
e)
t2
t1
t4
t3
f)
t2
t1
t4
t3
Figure 1: Petri nets for Exercise 2.1.
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Exercise 2.2
The Petri net in Figure 2 is given.
p1
p2
t2
t1
p3
2
p4
a) State the incidence matrix of the Petri net. Does the incidence matrix provide all information on the Petri net graph? Explain your answer.
b) Construct the coverability tree for the Petri net.
c) Is the Petri net • bounded?
• reversible?
• persistent?
What can be said about the liveness of each transition?
d) A Petri net is called blocking if a state x ∈ R(N, x0) exists in which no transition can fire. Is the Petri net blocking? Justify.
e) Is the Petri net conservative with respect to the vector γ1 = [1 1 0 1]′? How about γ2 = [1 1 0 0]′?
Exercise 2.3
The Petri net in Figure 3 is given.
t3
Figure 2: Petri net of Exercise 2.2.
p1 t4
p5 p2
t1t2 t5
p3 t3 p4 Figure 3: Petri net of Exercise 2.3.
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a) Construct the coverability tree for this Petri net.
b) Are the following states reachable? Check possible sequences of transitions in the Petri net, if no statement can be made based on the coverability tree.
0 0
ξ1 =1 0
2 0
ξ2 =0 0
1 1
ξ3 =0 0
1 1
ξ4 =0 1
0000
c) Determine – if possible – a sequence of transitions that can be generated by the coverability tree, but not by the Petri net.
d) Check if the Petri net is conservative with respect to the following vectors: γ1 = [2 0 2 0 1]′, γ2 =[1 0 1 0 0]′,γ3 =[0 1 0 0 −1]′.Isitpossibletodecidebasedonthecoverabilitytree?
Exercise 2.4
Consider the Petri net shown in Figure 4.
t3
t1 p2 t2
t4
t5
p4
Figure 4: Petri net of Exercise 2.4.
a) Evaluate the liveness of each transition.
b) Construct the coverability tree for the Petri net.
c) Is the Petri net bounded? Justify your answer based on the coverability tree.
d) Is the Petri net conservative with respect to the following vectors?
p3
p1
t6
p5
1 0
γ1 = 0 1
1 1
γ2 = 1 0
0 0
γ3 = 1 1
100
e) Repeat items a) to d) for the initial marking x0 = [0 1 1 1 0]′.
f) Find an initial marking for which transition t6 is L3-live but not live. For this initial marking, is the Petri net blocking?
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