Table Of ContentSUPERVISION OF PETRI NETS
THE KLUWER INTERNATIONAL SERIES
ON DISCRETE EVENT DYNAMIC SYSTEMS
Series Editor
Yu-ChiHo
Harvard University
ANALYSIS OF MANUFACTURING ENTERPRISES: An Approach to
Leveraging Value Delivery Processes for Competitive Advantage
N. Viswanadham
ISBN: 0-7923-8671-X
INTRODUCTION TO DISCRETE EVENT SYSTEMS
Christos G. Cassandras, Stephane Lafortune
ISBN: 0-7923-8609-4
OBJECT-ORIENTED COMPUTER SIMULATION OF DISCRETE-EVENT
SYSTEMS
Jerzy Tyszer
ISBN: 0-7923-8506-3
TIMED PETRI NETS: Theory and Application
Jiacun Wang
ISBN: 0-7923-8270-6
SUPERVISORY CONTROL OF DISCRETE EVENT SYSTEMS USING PETRI
NETS
John O. Moody and Panos J. Antsaklis
ISBN: 0-7923-8199-8
GRADIENT ESTIMATION VIA PERTURBATION ANALYSIS
P. G1asserman
ISBN: 0-7923-9095-4
PERTURBATION ANALYSIS OF DISCRETE EVENT DYNAMIC SYSTEMS
Yu-Chi Ho and Xi-Ren Cao
ISBN: 0-7923-9174-8
PETRI NET SYNTHESIS FOR DISCRETE EVENT CONTROL OF
MANUFACTURING SYSTEMS
MengChu Zhou and Frank DiCesare
ISBN: 0-7923-9289-2
MODELING AND CONTROL OF LOGICAL DISCRETE EVENT SYSTEMS
Ratnesh Kumar and Vijay K. Garg
ISBN: 0-7923-9538-7
UNIFORM RANDOM NUMBERS: THEORY AND PRACTICE
Shu Tezuka
ISBN: 0-7923-9572-7
OPTIMIZATION OF STOCHASTIC MODELS: THE INTERFACE
BETWEEN SIMULATION AND OPTIMIZATION
Georg Ch. Pflug
ISBN: 0-7923-9780-0
CONDITIONAL MONTE CARLO: GRADIENT ESTIMATION AND
OPTIMIZATION APPLICATIONS
Michael FU and lian-Qiang HU
SUPERVISION OF PETRI NETS
by
Geert Stremersch
University ofG hent, Belgium
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library ofCongress Cataloging-in-Publication Data
Strernersch, G. (Geert), 1973-
Supervision of Petri nets I Geert Strernersch.
p. crn. - (The Kluwer international series on discrete event dynamic systerns)
Includes bibliographical references and index.
ISBN 978-1-4613-5603-5 ISBN 978-1-4615-1537-1 (eBook)
DOI 10.1007/978-1-4615-1537-1
l. Petri nets. 2. Control theory. 1. Title. II. Series
QA267.S7652001
511.3 --dc21 2001046195
Copyright © 2001 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint ofthe hard.cover Ist ed.ition 2001
AII rights reserved. No part ofthis publication rnay be reproduced, stored in a retrieval systern
or transmitted in any form or by any rneans, rnechanical, photo-copying, recording, or otherwise,
without the prior written permission ofthe publisher, Springer Science+Business Media, LLC.
Printed on acid-free paper.
To HEIDI
Contents
Preface xi
1. THE PETRI NET MODEL 1
1 Discrete event systems 1
2 Notation 4
3 Order theoretical preliminaries 5
3.1 Partially ordered sets 6
3.2 Lattices 6
3.3 Up- and down-sets 7
4 Petri net definition 8
5 Petri nets as discrete event system models 10
6 Reachable sets 12
7 Graphical representation 14
8 Reachability via subsets of transitions 18
9 Other concurrency assumptions 20
9.1 The no concurrency assumption 21
9.2 The concurrency assumption 23
10 A general Petri net definition 24
11 Notes and references 26
2. SUPERVISORY CONTROL 27
1 Control goal and architecture 27
2 Formal definition 29
3 Reachable sets under supervision 31
4 Maximally permissive control laws 32
4.1 Introduction 32
4.2 Definition 33
4.3 Construction 37
5 Specific control sets 38
5.1 Fine control 38
Vlll SUPERVISION OF PETRI NETS
5.2 On/ off control 42
6 Linear inequalities as a legal set 43
6.1 One linear inequality 43
6.2 Conjunctions of linear inequalities 47
7 Control design under the no concurrency assumption 50
8 Notes and references 52
3. UNCONTROLLABLE EVENTS AND TRANSITIONS 53
1 Introduction 53
2 Supervisory control laws 57
3 Specific concurrency and control assumptions 60
4 Maximally permissive control laws 61
4.1 Permissive control laws 61
4.2 Optimality 63
5 Control design 63
6 The supremal controllable subset 64
7 Notes and references 68
4. REDUCTION THEOREMS 69
1 Intuition for A* 69
2 Invariance properties of the legal set 72
3 Sets of places and transitions 75
4 Reduction result for A * 82
4.1 Reduction theorem 82
4.2 Consequences and discussion 87
4.3 Further extensions 90
5 Reduction of the control design 91
5.1 An intuitive example 92
5.2 Sets of controllable transitions 93
5.3 Reduction theorem for control design 94
6 Structural and invariance properties of the legal set 98
6.1 Up- and down-sets in (~,~) 98
6.2 One linear inequality 100
6.3 Unions of linear inequalities 102
7 Notes and references 108
5. ACYCLIC PETRI NETS 109
1 Partitioning of the sets of places and transitions 110
2 Structure of the incidence matrices 112
3 Reachability in acyclic Petri nets 114
3.1 Transition bag assumption 114
3.2 Other concurrency assumptions 118
4 A reachability algorithm 119
Contents IX
5 Acyclic Petri nets free of choice places 121
6 Construction of the supremal controllable subset 124
7 Notes and references 127
6. DECOMPOSITION OF THE CONTROL DESIGN 129
1 Introduction 129
2 Unions of legal sets 131
3 A uxiliary results 133
4 Proof of Theorem 6.1 135
5 Discussion 141
6 Control design 143
7 Notes and references 147
7. CONTINUOUS VERSUS DISCRETE EVENTS 149
1 Continuous Petri nets 149
1.1 Definition 149
1.2 Reachability 150
1.3 Acyclic continuous Petri nets 151
2 A subset of the supremal controllable subset 152
2.1 The approach 152
2.2 Construction of A* 153
2.3 Presence of source transitions 158
3 Construction of the supremal controllable subset 159
3.1 Auxiliary lemmas 159
3.2 Discussion 160
N'A
4 No synchronising transitions in c 164
N'A
5 No choice places in c 169
6 A third class 171
7 Structure of A* 173
8 Notes and references 174
8. STRUCTURAL LINEAR ALGEBRAIC CONTROL DESIGN 175
1 Unobservable events 175
2 Overview of the approach 176
3 Intersection of a linear halfspace with the first orthant 178
4 Candidate sets AQ 180
4.1 The case A ~ 0 181
4.2 The case A ::; 0 184
5 Maximal sets AQ 185
6 Reduction of controllers with disjunctions 186
7 A subset of the supremal controllable subset 189
8 Notes and references 190
x SUPERVISION OF PETRI NETS
References 193
Index 197
Preface
The goal of this book is to present a unified, insightful and mathe
matically sound supervisory control theory for Petri nets. Petri nets are
used to model discrete event systems, dynamic systems whose evolution
is completely determined by the occurrence of discrete events. Control
laws which guarantee that the system meets a set of specifications, are
studied and constructed.
The requirement that the system behaviour always satisfies a number
of conditions is expressed by a legal set, a subset of the state space. These
conditions can express safety, avoidance of deadlock, the language is a
subset of a given language, ... Because the controller can only disable
events, and thus merely limit the set of all possible future paths of the
system without being able to 'force' events, one speaks of supervisory
control. The supervisory control goal is that the state of the Petri net
always belongs to the legal set. Moreover, this needs to be done in an
optimal way, namely by keeping the set of possible future evolutions of
the Petri net as large as possible.
In practice, the presence of uncontrollable and unobservable events
complicates control design. These events cannot be influenced or ob
served, respectively. Consequently, the controller should anticipate the
worst-case uncontrollable behaviour of the system while its state may
not be completely known. When all events are observable, and some are
uncontrollable, the controller should keep the state within the supremal
controllable subset of the given legal set.
Control design for such systems has been studied extensively over the
last ten to fifteen years. Research is mainly motivated by automated
manufacturing systems. Several theoretical problems remain open. The
purpose of this book is to tackle the most fundamental of these:
