CS代考程序代写 Exercise 1.1

Exercise 1.1
a) A possible Petri net model for the dental clinic is shown below.
b)
t2 p2
t1p1 p3
t4 p4
p5
0 1 0 1 0 1
t3
t5
0 0 0 0
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 2018/2019
Exercise 1 — Solution
0 0 1 0 0 0 A−=01010, A+=00101
1 0 0 0 
0 0 0 0 1 0 0 0 1 0 00010 00001
1 −1
0 1 ∴ A=A+−A−=0 −1 0 0 0 0
0 −1 0 −1 0 0 1 −1 1 0 1 −1
0 −1 1
The assumption is that three patients have arrived (i.e., t1 has fired three times) and two of them are having their teeth cleaned (t2 has fired twice, but t3 has not fired yet). A total of five transition firings have therefore been observed, so the current state (marking) is x(5). The question whether it is possible for a medical examination to start translates to whether it is possible to fire t4; we know (see lecture notes, page 16) that t4 can fire at x(5) if and only if x(5) ≥ A−u4 = [1 0 1 0 1]′. In order to obtain x(5), we simply apply x(k + 1) = x(k) + Auj for k = 0,1,…,4 according to the observed firings, starting with x(0) = x0 = [0 0 2 0 1]′. We can assume, without loss of
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generality, that the three firings of t1 have occurred in a row in the beginning, followed by the two firings of t2. We then have
x(1) = x(0) + Au1
x(2)=x(1)+Au1 =x(0)+2Au1
x(3)=x(2)+Au1 =x(0)+3Au1
x(4)=x(3)+Au2 =x(0)+3Au1 +Au2 x(5)=x(4)+Au2 =x(0)+3Au1 +2Au2 =[1 2 0 0 1]′
One can easily see that x(5) 􏰀 A−u4, so the medical examination cannot start. c) The modifed Petri net model is shown below.
t2 p2 t3
t1p1 p3
p6
d) The new Petri net model is shown below.
t4 p4 t5
p5
t1p1
t2
t4
t6
p2
p3
p4
t3
t5
t7
p6
2
2
p5
p7
2

Exercise 1.2
a) A possible Petri net model for the workshop is shown below.
t1 p1 t2 p2 t3 p3 t4
b)
1
0
0 A= −1  0
0
−1 0 0
1 −1 0
0 1 −1 1 0 0
−1 1 0 0 −1 1
c) The reachability graph is shown below.
0
2
0 t3
0 0 0 0
1 0 0 0 1 t3 2 t4 1 t4 0 5 5 5 5
p4 p5 p6
5 
1 t2 0 1 2 2 2
1 0 0 1 0 t2 0
2 0
0
3 
1
0 t2 t4 t3
t2
2 1 t3 1
32123
4
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0 1
t4 
Exercise 1.3
4 2
4 2
5 1
233
a) A possible Petri net model for the traffic light system is shown below.
t1 p1 t2 p2 t3
p5
t6 p4 t5 p3 t4
3

b)
10−1 1 c) The reachability graph is shown below.
0 0 0 0 −1 0 0 0 0 1 −1 0 0 0 1 −1
1 −1
0 1 A=0 0 0 0 0 −1
1 t2  010
0 2
0 0 1 t3 0
2 0
0 t5 0
2 0 
1 1 
t6
0 0
1 
0 t3 
1 0 
0
0 t5 0
1 1
2 0 
t5 1
0 1
1 1 1 0
t6 0 0
0 0 
0 t3 110
2 0
2 0

0 1
0
0t3 1
t2
 t61t2  0001
 t5 1 110
t5 2
1 t3 0 1
1 1
1 0
t62t2  
0 0
0 0
0 0
1 t2  t6 0
1 0001
0 1 0 1
t3
t5
2 0
2 1 0 t2 1
t6 
0 1
0 0
0 0
010
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d) The modifed Petri net model for the traffic light system is shown below.
t1 p1 t2 p2 t3
p5 p6
t6 p4 t5 p3 t4
e) The Petri net from item d) is an Event Graph (see Definition 2.3 on page 18 of the Lecture Notes).
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