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Petri Nets

as Discrete Event Models

for Supervisory Control

by Alessandro Giua

A Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy. Major Subject: Computer and Systems Engineering.

Rensselaer Polytechnic Institute (Troy, New York)

July 1992

Contents

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Acknowledgement

Frank DiCesare has guided me in a spirit of friendship through my Master’s and Ph.D. thesis, providing learning, encouragement, and support all at once.

Manuel Silva has given me the wonderful opportunity of visiting the University of Zaragoza and of working with his group. Part of this research has been inspired by him.

Alan A. Desrochers, David R. Musser, and Badrinath Roysam have provided additional guidance as members of my doctoral committee.

The Petri net group meetings at Rensselaer have been a fruitful learning experience. I acknowledge the many discussions with Padma Akella, Tiehua Cao, Xin Chen, Mu-Der Jeng, Jagdish Joshi, Hauke Jungnitz, Jongwook Kim, Inseon Koh, Doo Yong Lee, Jim Watson, Mengchu Zhou.

I am grateful to the Computer Integrated Manufacturing project of the Center for Manufacturing Productivity and Technology Transfer which has supported the research of this thesis during the last two years.

The years of my graduate studies have been a period of freedom and carelessness, marking the transition between youth and maturity. Never will love be more poignant or friends more dear; the memories will last forever. For this I have to thank Philippe Jeanne, Thanos Tsolkas, Ermanno Astorino, Antonio De Dominicis, Gloria-Deo Agbasi, Hauke Jungnitz, Giampiero Beroggi and Penny Spring, Agostino Abbate and Josefina Quiles, Didier Laporte, Elisabetta Bruzzone, Nora Si-Ahmed, Amitava Maulik, Persefone Vouyoukas, Doo Yong Lee, Josep Tornero, Markus Vincze, Michael Eppinger, Rolf Münger, Keith Hanna, Alejandro Beascoechea, Markus Hoerler, Carlos Garcia, Leonardo Lanari, Scott Bieber, Brady Richter, Cristina Gesto, Andreas Glatzl, Jose Neira, Maria Angeles Salas, Antonio Ramírez, Enrique Teruel, Joaquin Ezpeleta, Bruno Gaujal, and Rogerio Rodriguez.

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Abstract

Discrete event systems represents a new field of control theory of increasing importance in the domains of manufacturing, communications, and robotics. Supervisory Control theory, based on formal languages, is a well established framework for the study of discrete event systems.

The thesis discusses the use of Petri nets in Supervisory Control. Place/Transition nets have been used by several authors to represent discrete event systems. In our approach, the supervisor, i.e., the control agent that restricts the behavior of a system within a legal behavior, is represented as a Place/Transition net as well. The advantage of such a supervisor, as opposed to a supervisor given as a feedback function, is that a closed loop model of the controlled system may be constructed and analyzed using the techniques pertaining to Petri net models.

We show that a Petri net supervisor may not exist if the system’s behavior or the legal behavior are nonregular Petri net languages. By defining a new class of Petri net languages, called deterministic P-closed, it is possible to derive necessary and sufficient conditions for the existence of supervisors as nets.

The thesis presents an algorithm, based on the concurrent composition operator, for the design of Petri net supervisors and discusses how a composed net may be validated. In a first approach, based on incidence matrix analysis, important properties of the net, such as controllability or such as the absence of blocking states, may be studied by Integer Programming techniques. In a second approach, we consider a class of specifications, called generalized mutual exclusion constraints, and discuss several possible structures for the supervisor capable of enforcing them.

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Chapter 1

INTRODUCTION

1.1 Introduction

The object of the study of traditional control theory have been systems of continuous and synchronous discrete variables, modeled by differential or difference equations. However, as the scope of control theory is being extended into the domains of manufacturing, robotics, computer and communication networks, and so on, there is an increasing need for different models, capable of describing systems that evolve in accordance with the abrupt occurrence, at possibly unknown irregular intervals, of physical events. Such systems, whose states have logical or symbolic, rather than numerical, values that change in response to events which may also be described in nonnumerical terms, are called discrete event systems (DES) [Ramadge 89b] and the corresponding models are called discrete event models (DEM) [Inan 88].

These systems require control and coordination to ensure the orderly flow of events. As controlled (or potentially controllable) dynamic systems, DES qualify as a proper subject for control theory. Hence a twofold issue presents itself: we need a powerful class of DEM, capable of capturing the essential features of discrete, asynchronous and possibly nondeterministic systems, and a unifying theory for their control.

The goal of the present research is that of applying a Petri net model within the framework of Supervisory Control, a control theory for DES introduced recently by Ramadge and Wonham [Ramadge 83, Ramadge 87]. We study how Petri net models may be used to solve supervisory control problems, i.e, the design of a supervisor capable of restricting the behavior of a system within a legal behavior.

1.2 Background and Motivation

In this section we rather informally discuss the motivation of this research. We want to show why Petri nets are a good discrete event model and a powerful tool for Supervisory Control.

1.2.1 Discrete Event Systems and Models

A DES is a dynamic system with a discrete state space and piecewise constant state trajectories (see Figure 1.1); the time instant at which state transitions occur, as well as the actual transitions, will in general be unpredictable.

The state transitions of a DES are called events and may be labeled with the elements of some alphabet Σ. These labels usually indicate the physical phenomenon that caused the change in state. For example, in a manufacturing environment typical event labels are “machine 1 starts working on part A” , “machine 1 breaks down” , etc.

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