# CS代考程序代写 Exercise 7.1

Exercise 7.1

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 7

CoSntrGol

Consider the production line shown in Figure 1. It consists of three machines; machines 1 and 3 are able to process only one workpiece at a time, whereas machine 2 has capacity for two workpieces. The first machine has unlimited storage capacity for unprocessed workpieces, which it receives from outside the line. Machine 2 also has unlimited storage capacity for workpieces coming from machine 1. The third machine, however, has no storage facility, i. e., a workpiece can only be passed from machine 2 to machine 3 when the latter is free. The processing times of the machines are v1 = 3, v2 = 5, v3 = 2. Suppose that, at the beginning, the first machine contains one workpiece, the second contains two workpieces, and the storage between machines 1 and 2 contains three workpieces. The input storage as well as machine 3 are initially empty.

96

9696

Sys

tem

s

Fachgebiet Regelungssysteme

E EM1 E EM2 EM3 E

87

8787

Figure 1: Production line for Exercise 7.1.

a) Model the system as a timed Petri net with holding times.

b) Considering the arrival of unprocessed workpieces to the storage of machine 1 as an input for the system and the finishing of processing by machine 3 as an output, write max-plus state equations for the model of the production line.

Exercise 7.2

The firing times of the transitions of a given system are described by the max-plus equations x(k + 1) = A0x(k+1)⊕A1x(k)⊕B0u(k+1) and y(k) = C0x(k), where B0 = [e ε ε]′, C0 = [ε 1 ε], and A0, A1 are matrices whose corresponding precedence graphs are shown in Figure 2. Obtain the state equations — of the form x ̃(k + 1) = Ax ̃(k) ⊕ Bu(k + 1) and y(k) = Cx ̃(k) — and draw a timed event graph that models the system.

25

12 12

G(A0): e 4 G(A1): 33

Figure 2: Precedence graphs associated with matrices A0 and A1.

1

Exercise 7.3

The firing times of the transitions of a given system are described by the following recursive equations, in which ui(k) represents the earliest possible firing times for input transitions tui , with i ∈ {1, 2, 3}, and y(k) represents the earliest possible firing times for the output transition ty.

x1(k + 1) = maxu1(k) , x2(k)

x2(k + 1) = u2(k + 1)

x3(k + 1) = maxx1(k + 1) + 2 , x2(k − 1) + 7 , u3(k − 1)

y(k) = maxx1(k) + 12 , x3(k) + 8

a) Draw a timed event graph with holding times whose transitions’ earliest possible firing times can

be described by the given equations.

b) Determine,forthesystemofequationsabove,astate-spacerepresentationinmax-plus,oftheform

x(k + 1) = Ax(k) ⊕ Bu(k + 1) y(k) = Cx(k) .

Exercise 7.4

Consider the timed event graph provided in Figure 3, in which tu is an input transition and ty is an output transition.

3

tu t12 ty

1

t2

t3

1

a) Let the variables u, xl, and y represent the firing times of transitions tu, tl, and ty, respectively, and let x = [x1 x2 x3]′. Consider the following recursive equations for the firing times of the transitions:

31

x(k+1)=Aix(k+1−i) ⊕ Bju(k+1−j) i=0 j=0

y(k) = C0x(k) .

Provide matrices Ai, Bj, and C0, for all i ∈ {0,…,3} and all j ∈ {0,1}.

Figure 3: Timed event graph for Exercise 7.4.

2

b) We now want to represent the above recursive equations in the state-space form:

x(k + 1) = Ax(k) ⊕ Bu(k + 1) y(k) = Cx(k) .

How would you define x(k)? Provide also the resulting matrices A, B, and C in terms of matrices Ai, Bj, and C0. (You do not have to explicitly compute each element of matrices A, B, and C.)

3