[SOLVED] CS Exercise 3.1

Exercise 3.1
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 3
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The Petri net in Figure 1 represents a manufacturing system, where one robot and two machines (1 and 2) process workpieces of types A and B, respectively. Finished workpieces are taken to a depot. The displayed transitions have the following meaning:
t1 . . . robot takes an unprocessed workpiece of type A,
t2 . . . robot loads an unprocessed workpiece of type A into machine 1 and machine 1 starts processing
the respective workpiece,
t3 . . . machine 1 finishes processing workpiece A and puts it in the depot,
t4 . . . robot takes an unprocessed workpiece of type B,
t5 . . . robot loads an unprocessed workpiece of type B into machine 2 and machine 2 starts processing the respective workpiece,
t6 . . . machine 2 finishes processing workpiece B and puts it in the depot,
t7 . . . one workpiece of type A and one of type B leave the system together.
t1 p1 t2 p2 t3 p3
t4 p4 t5 p5 t6 p6 Figure 1: Petri net for Exercise 3.1.
t7
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Fachgebiet Regelungssysteme
The only task performed by the robot is to take unprocessed workpieces and load them into the machines. Calculate the least restrictive control to achieve the following specifications:
The capacity of each machine is 1.
The robot can only hold one workpiece at a time.
You can assume that all transitions are observable and preventable (controllable).
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Exercise 3.2
t1 p1 t2 p2
t5 p5 t6
t3 p3 t4 p4
Figure 2: Petri net for Exercise 3.2.
Consider the Petri net shown in Figure 2. It may model, for example, part of a manufacturing system, in which transitions t1 and t3 represent the arrival of unprocessed workpieces, and each place, a part of the production process. In such a scenario, the firing of transition t6 represents the finishing of a product. Now, consider the following specification:
x3(k) + x4(k) 2 x1(k) + x2(k) x4(k) 1 x1(k) + x2(k) + x3(k) + x4(k) 4
Assuming that all transitions are controllable and observable, consider a controller with incidence matrix Ac and initial state x0c given as follows:
0 0 1 0 1 0 2 Ac=1 0 0 1 0 0 x0c=1
1 0 1 0 0 0 4
a) Does the given controller enforce1 the specification? Is it minimally restrictive?
b) How many products can be completely produced if the given controller is being used?
c) Without changing the number of controller places or the initial marking x0c, modify the given controller so that the specification is enforced without limiting the number of products that can be produced. Provide the modified incidence matrix of the controller. Is your new controller minimally restrictive?
1 In this context, we say that a controller enforces a specification if, under its action, the inequalities represeting the specifica- tion can never be violated.
2

Exercise 3.3
A plant to be controlled is modeled by the Petri net shown in Figure 3. The firing of transitions t1 and t2 represent, respectively, the arrival of unprocessed workpieces of types A and B, which are then deposited in input buffers represented by places p1 and p2. A machine then receives one workpiece of each type (transition t3) and processes them (place p3); the resulting processed product is then picked up by a robot (transition t4) and transported out of the system (transition t5).
An unprocessed workpiece of type A weighs 5kg, whereas one of type B weighs 2kg. As represented by the initial marking in Figure 3, initially there is one workpiece of type A in the corresponding buffer; the other input buffer, the machine, and the robot are initially empty. The following specifications are given:
both input buffers combined cannot support more than 20kg of weight;
the robot has capacity for only one processed product at a time;
in the input buffers, there can never be more workpieces of type B than of type A.
t1 p1
t2 p2
t3 p3 t4 p4 t5
Figure 3: Plant model for Exercise 3.3.
a) Write the specifications in the form x(k) b and compute the corresponding least restrictive (ideal) controller. Provide the controllers incidence matrix, Ac, and initial marking, x0c.
b) Assume, now, that transitions t3 and t5 are observable but not controllable, and that t4 is uncon- trollable and unobservable. Are the given specifications ideally enforceable? Justify your answer.
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