Formal methods in manufacturing

xv PART I Modelling and Simulations of Manufacturing Systems Chapter 1 Modelling Manufacturing Systems with Place/Transition Nets and Timed Petri Nets.. 29 Thomas Brunsch, Laurent Hardou

Formal Methods in Manufacturing

Electrical Engineering

Illustrated with real-life manufacturing examples, Formal Methods in Manufacturing provides state-of-the-art solutions to common problems in manufacturing sys- tems Assuming some knowledge of discrete event sys- tems theory, the book first delivers a detailed introduction

to the most important formalisms used for the modeling, analysis, and control of manufacturing systems (including Petri nets, automata, and max-plus algebra), explaining the advantages of each formal method It then employs the different formalisms to solve specific problems taken from today’s industrial world, such as modeling and simu- lation, supervisory control (including deadlock prevention)

in a distributed and/or decentralized environment, mance evaluation (including scheduling and optimization), fault diagnosis and diagnosability analysis, and reconfigu- ration.

perfor-Containing chapters written by leading experts in their

helps researchers and application engineers handle damental principles and deal with typical quality goals in the design and operation of manufacturing systems.

Formal Methods

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Manufacturing

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Preface ixEditors xiiiContributors xv

PART I Modelling and Simulations of Manufacturing Systems

Chapter 1 Modelling Manufacturing Systems with Place/Transition Nets

and Timed Petri Nets 3

Maria Paola Cabasino, Mariagrazia Dotoli and Carla Seatzu

Chapter 2 Modelling Manufacturing Systems in a Dioid Framework 29

Thomas Brunsch, Laurent Hardouin and Jörg Raisch

Chapter 3 Modelling Manufacturing Systems and Inventory Control Systems with Hybrid

Petri Nets 75

Maria Paola Cabasino, Alessandro Giua and Carla Seatzu

Chapter 4 Hybrid Models for the Control and Optimization of Manufacturing Systems 105

Christos G Cassandras and Chen Yao

Chapter 5 Freight Transportation in Distributed Logistic Chains 135

Angela Di Febbraro and Nicola Sacco

PART II Supervisory Control of Manufacturing Systems

Chapter 6 Deadlock Avoidance Policies for Automated Manufacturing Systems Using

Finite State Automata 169

Spyros Reveliotis and Ahmed Nazeem

Chapter 7 Structural Deadlock Prevention Policies for Flexible Manufacturing Systems:

A Petri Net Outlook 197

Juan-Pablo López-Grao, José-Manuel Colom and Fernando Tricas

Chapter 8 Deadlock Avoidance Policies in Production Systems by a Digraph Approach 229

Maria Pia Fanti, Bruno Maione and Biagio Turchiano

v

Chapter 9 Supervisory Control of Manufacturing Systems Using Petri Nets 259

Carla Seatzu and Xiaolan Xie

Chapter 10 Supervisory Control of Manufacturing Systems Using Extended

Finite Automata 295

Martin Fabian, Zhennan Fei, Sajed Miremadi,

Bengt Lennartson and Knut Åkesson

Chapter 11 Inference-Based and Modular Decentralized Control of Manufacturing

Systems with Event-Driven Dynamics 315

Shigemasa Takai and Ratnesh Kumar

Chapter 12 Model Predictive Control of Manufacturing Systems with Max-Plus

Algebra 343

Ton J.J van den Boom and Bart De Schutter

PART III Performance Evaluation of Manufacturing Systems

and Supply Chains

Chapter 13 Performance Evaluation of Flexible Manufacturing Systems by Coloured

Timed Petri Nets and Timed State Space Generation 381

Chapter 14 Performance Evaluation and Control of Manufacturing Systems:

A Continuous Petri Nets View 409

Manuel Silva, Estíbaliz Fraca and Liewei Wang

Chapter 15 Performance Evaluation of Flexible Manufacturing Systems with Timed

Process Algebra 453

María Carmen Ruiz, Diego Cazorla, Fernando Cuartero and Hermenegilda Macia

Chapter 16 Lean Buffer Design in Production Systems 477

Jingshan Li, Semyon M Meerkov and Xiang Zhong

Chapter 17 Inventory Allocation and Cycle Time Estimation in Manufacturing and

Supply Systems 503

Liming Liu and Yang Sun

Chapter 18 Minimizing Total Place Capacity under Throughput Constraint for a Weighted

Timed Event Graph 527

Alix Munier Kordon

Chapter 19 Scheduling of Semiconductor Manufacturing Systems Using Petri Nets 553

Fei Qiao and MengChu Zhou

Chapter 20 Model Synthesis, Planning, Scheduling and Simulation of Health-Care

Delivery Systems Using Petri Nets 571

Vincent Augusto and Xiaolan Xie

PART IV Fault Diagnosis of Manufacturing Systems

Chapter 21 Fault Diagnosis of Manufacturing Systems Using Finite State Automata 601

Stéphane Lafortune, Richard Hill and Andrea Paoli

Chapter 22 Fault Diagnosis in Petri Nets 627

Elvia Ruiz-Beltrán, Antonio Ramirez-Treviño and J.L Orozco-Mora

Chapter 23 Online Control Reconfiguration for Manufacturing Systems 653

Yannick Nke and Jan Lunze

Index 683

Design and operation of manufacturing systems and their supply chains is a domain of significantresearch worldwide The complexity of this domain stems from the large dimension of such systemsthat are highly parallel and distributed, from significant sources of uncertainties and from the degrees

of flexibility Formal methods are mathematical techniques, often supported by tools, for developingman-made systems Formal methods and mathematical rigor enable manufacturing engineers tohandle fundamental design principles, such as abstraction or modular and hierarchical development,and to deal with typical engineering problems and quality goals, like reliability, flexibility, andmaintainability Formal methods can provide both a deep understanding of the system, thus helping

to cover holes in the specification, and an improved system reliability, through verification andvalidation of the desired properties Automata, statecharts, queueing networks, Petri nets, min-maxalgebras, process algebras, and temporal logic–based models are becoming more and more used for

an integrated view of design specification, validation, performance evaluation, planning, schedulingand control of manufacturing systems and their supply chains

This book presents some of the most significant works representing the state of the art in thearea of formal methodologies for manufacturing systems, combining fundamentals and advancedissues It is divided into four main parts, each devoted to a specific issue: modelling and simulation,supervisory control (including deadlock prevention), performance evaluation (including schedulingand optimization), and fault diagnosis and reconfiguration Several formalisms are considered,including finite state automata, Petri nets (discrete, timed, continuous, and hybrid), process algebraand max-plus algebra, exemplifying the advantages of each of them in the solution of a specificproblem Within each part, more detailed problems are considered, and the most significant solutionsare discussed and illustrated with a series of interesting and significant examples in the manufacturingarea Individual chapters are written by leading experts in the field Each topic is illustrated in detail,its significance in manufacturing systems is underlined, and the most important contributions in thatspecific area are surveyed

The book is intended for researchers, postgraduate students and engineers interested in problemsoccurring in manufacturing systems In particular, it provides a comprehensive overview of the mostimportant formal model-based solutions to a series of major problems in manufacturing systems andtheir supply chains, which are based on formal and rigorous modelling of the underlying system.Each chapter in the book aims at providing a balance mixture of (1) fundamental theory, givingthe reader a clear introduction to the most important formalisms used for the modelling, analysisand control of manufacturing systems; (2) tutorial value, providing the state of the art on a series

of problems that occur in manufacturing systems, such as deadlock prevention, supervisory control,performance evaluation and fault diagnosis; and (3) applicability, presenting a series of case studiesand applications taken from the industrial world, that make it particularly appealing to practitioners

A brief description of the book is as follows Part I (Chapters 1 through 5) concerns modellingand simulation of manufacturing systems Chapter 1 focuses on Petri nets (untimed and timed) andshows how these can be effectively used to represent manufacturing systems in a bottom-up andmodular fashion Chapter 2 deals with a particular class of manufacturing systems for which alinear representation in an algebraic structure called dioids can be given Chapters 3 and 4 dealwith hybrid models of manufacturing systems In particular, Chapter 3 focuses on hybrid Petri nets(HPNs), a formalism that combines fluid and discrete event dynamics Particular attention is devoted

to first-order hybrid Petri nets (FOHPNs) whose continuous dynamics are piecewise constant It isshown how FOHPNs can be effectively used to model both manufacturing systems and inventorycontrol systems Chapter 4 focuses on stochastic flow models (SFMs) that preserve the essentialfeatures needed to design effective controllers and potentially optimize performance without any need

to estimate the corresponding optimal performance value with accuracy An overview of recently

ix

developed general frameworks for infinitesimal perturbation analysis (IPA) is also presented, throughwhich unbiased performance sensitivity estimates of typical manufacturing performance measurescan be obtained in such SFMs with respect to various controllable parameters of interest Finally,Chapter 5 deals with a problem occurring in many real manufacturing systems, namely, freighttransportation It reviews the established transportation system modelling, including theory andapplications of transportation supply models, trip demand models and dynamic traffic assignmentmethods, both for passenger and freight transportation, and also points out the characteristics offreight transportation that influence the logistic chain performance.

Part II (Chapters 6 through 12) is devoted to the supervisory control of manufacturing systems

In particular, Chapters 6 through 8 introduce deadlock avoidance/prevention policies; Chapters 9through 12 deal with supervisory control problems Chapter 6 addresses the problem of deadlockavoidance in flexibly automated manufacturing systems through the modelling abstraction of the(sequential) resource allocation system (RAS) The pursued analysis uses concepts and results fromthe formal modelling framework of finite state automata (FSA) Chapter 7 investigates the problem ofdeadlock prevention in the Petri net framework After an overview of the classical approaches based

on the addition of monitors that prevent siphons to become empty, a novel methodology is presented.Chapter 8 also deals with Petri nets Here the digraph theory is used to effectively derive controllaws that avoid deadlocks in single unit RAS, that is, systems where each part requires a single unit

of a single resource for each operation In Chapter 9, Petri nets are used to solve supervisory controlproblems of manufacturing systems Different problem statements, depending on the consideredspecifications, and different solutions are considered, in particular based on the theory of monitorplaces and on the theory of regions Extended finite automata (EFA), that is, automata augmentedwith bounded discrete variables, and updates of these variables on the transitions, are introduced

in Chapter 10 and effectively used to automatically synthesize a supervisor Decentralized controland modular control problems are discussed in Chapter 11 Finally, Chapter 12 discusses howmanufacturing systems can often be modeled as max-plus-linear (MPL) systems and controlled viamodel predictive control (MPC)

Part III (Chapters 13 through 20) addresses the issue of performance evaluation of manufacturingsystems and supply chains Chapter 13 discusses approaches based on coloured Petri nets andstate space analysis Performance evaluation and control of manufacturing systems using fluid (i.e.,continuous) Petri nets are discussed in Chapter 14 Chapter 15, discusses how timed process algebracalled bounded true concurrency (BTC) can be effectively used for performance evaluation of flexiblemanufacturing systems The problem of designing the lean, that is, the smallest buffers necessaryand sufficient to achieve the desired line performance, is addressed in Chapter 16 Chapter 17 dealswith the issue of inventory allocation and cycle time improvement in manufacturing systems andsupply chains Chapter 18 covers timed weighted event graphs, a subclass of Petri nets whosetransitions are associated with workshops or specific treatments and whose places represent storagesbetween the transitions It deals with the minimization of the overall capacities of places, underthroughput constraints Chapter 19 focuses on the application of Petri nets to the scheduling ofsemiconductor manufacturing systems To this aim, a hierarchical coloured timed Petri net (HCTPN)

is proposed and genetic algorithms are extended and then embedded into the constructed HCTPN tofind optimal/suboptimal schedules Finally, Chapter 20 focuses on organization problems of health-care systems, presenting a Petri net–based software for health-care service modelling and simulationcalled MedPRO Resource planning and scheduling are also in the scope of the tool

Finally, Part IV (Chapters 21 through 23) illustrates fault diagnosis approaches for discrete eventsystems that can be successfully applied to manufacturing systems In particular, in Chapters 21 and

22 finite state automata and interpreted Petri nets, respectively, are used as reference formalisms

In both chapters, diagnosability analysis is also performed Chapter 21 also discusses the problem

of sensor selection for diagnosability and the problem of cooperative diagnosis for systems withdecentralized information Chapter 23 addresses a problem strictly related to fault diagnosis occurring

in automated manufacturing systems, namely, that of online control reconfiguration

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as the International Conference on Quantitative Evaluation of Systems, the European PerformanceEngineering Workshop, the International Workshop on Software and Performance, the IEEE Confer-ence on Automation Science and Engineering, and the IEEE International Conference on EmergingTechnologies and Factory Automation.

Dr Campos is a founding member of the Aragn Institute for Engineering Research, a member ofthe Aragonese Informatics Engineering Association, and president of the Spanish Concurrent andDistributed Computing Society He has been a member of the IEEE IES Technical Committee onFactory Automation, cochair of the IEEE IES Technical Sub-Committee on Industrial AutomatedSystems and Controls, guest editor of the special section on formal methods in manufacturing of

IEEE Transactions on Industrial Informatics, and associate editor of IEEE Transactions on Industrial Informatics.

Carla Seatzureceived her laurea degree in electrical engineering and her PhD in electronic neering and computer science from the University of Cagliari, Italy, in 1996 and 2000, respectively

engi-In 2002, she joined the Department of Electrical and Electronic Engineering at the University ofCagliari as an assistant professor of automatic control She currently serves as an associate professor

of automatic control in the same department

Prof Seatzu’s research interests include discrete-event systems, hybrid systems, Petri nets, facturing systems, networked control systems, and control of mechanical systems She has publishedover 150 papers and 1 textbook on these topics

manu-Prof Seatzu has served as general cochair of the 18th IEEE International Conference on ing Technology and Factory Automation (ETFA2013) She was chair of the National OrganizingCommittee of the 2nd IFAC Conference on the Analysis and Design of Hybrid Systems (ADHS’06)and member of the international program committees of around 50 international conferences She is

Emerg-associate editor of the IEEE Transactions on Automatic Control and of Nonlinear Analysis: Hybrid Systems She is chair of the IEEE IES Technical Committee on Factory Automation — Subcommittee

on Industrial Automated Systems and Controls

Xiaolan Xiereceived his PhD from the University of Nancy I, Nancy, France, in 1989, and theHabilitation à Diriger des Recherches degree from the University of Metz, France, in 1995

He is a professor of industrial engineering and head of the Healthcare Engineering Department

at the Ecole Nationale Supérieure des Mines de Saint-Etienne (ENSM.SE), France He is leader ofthe ROGI team of the CNRS laboratory LIMOS UMR 6158 on operations research and industrial

xiii

engineering, a team that received the top A+ ranking in the 2010 national AERES research evaluation.

He is also a chair professor at Shanghai Jiao Tong University, China Before joining ENSM.SE, he was

a research director at the Institut National de Recherche en Informatique et en Automatique (INRIA)from 2002 to 2005, a full professor at Ecole Nationale d’Ingénieurs de Metz from 1999 to 2002,and a senior research scientist at INRIA from 1990 to 1999 His research interests include design,planning and scheduling, supply chain optimization, performance evaluation, and maintenance ofmanufacturing and health-care systems He is the author or coauthor of over 200 publications,including over 80 journal articles and 5 books He has rich industrial application experience withEuropean industries He was co-principal investigator (PI) for various national or EU-funded projects,including ANR-HOST on hospital tension avoidance strategies, Labex IMoBS3 on home health-careservices, NSF-China project on health-care resource planning and optimization, FP6-IST6 IWARD

on swarm robots for health services, FP6-NoE I*PROMS on intelligent machines and productionsystems, the FP5-GROWTH-ONE project for the strategic design of supply chain networks, and theFP5- GROWTH thematic network TNEE on extended enterprises

He has been an associate editor on the conference editorial board of the IEEE Robotics and Automation Society, IEEE Transactions on Automation Science and Engineering, IEEE Transactions

on Automatic Control and IEEE Transactions on Robotics and Automation He has also been guest editor of various special issues on health-care engineering in IEEE Transactions Systems, Man and Cybernetics, Annals of Operations Research (AOR, 2010), Healthcare Management Science (HCMS, 2009), and four other special issues on discrete event systems and manufacturing systems for IJCIM (2005), IJPR (2004, 2001), and IEEE Transactions Robotics and Automation (2001) He served as general chair of ORAHS’2007 and IPC chair of the IEEE Workshop on Health Care Management WHCM’2010 and was IPC member for many other conferences.

Knut Åkesson

Automation

Department of Signals and Systems

Chalmers University of Technology

Gothenburg, Sweden

Vincent Augusto

Department Healthcare Engineering

École Nationale Supérieure des Mines de

Saint-Étienne

Saint-Étienne, France

Ton J.J van den Boom

Delft Center for Systems and Control

Delft University of Technology

Delft, the Netherlands

Thomas Brunsch

Control Systems Group

Technical University of Berlin

Maria Paola Cabasino

Department of Electrical and Electronic

Computing Systems Department

University of Castilla-La Mancha

Albacete, Spain

José-Manuel Colom

University of ZaragozaZaragoza, Spain

Fernando Cuartero

Computing Systems DepartmentUniversity of Castilla-La ManchaAlbacete, Spain

Bart De Schutter

Delft Center for Systems and ControlDelft University of TechnologyDelft, the Netherlands

Angela Di Febbraro

Department of Mechanical, Energy andTransportation Engineering

University of GenoaGenoa, Italy

Mariagrazia Dotoli

Department of Electrical and ElectronicEngineering

Polytechnic of BariBari, Italy

Martin Fabian

AutomationDepartment of Signals and SystemsChalmers University of TechnologyGothenburg, Sweden

Maria Pia Fanti

Department of Electrical and ElectronicEngineering

Polytechnique of BariBari, Italy

Zhennan Fei

AutomationDepartment of Signals and SystemsChalmers University of TechnologyGothenburg, Sweden

xv

Department of Mechanical Engineering

University of Detroit Mercy

Department of Signals and Systems

Chalmers University of Technology

Faculty of BusinessLingnan UniversityHong Kong, China

Juan-Pablo López-Grao

University of ZaragozaZaragoza, Spain

Bruno Maione

Department of Electrical and ElectronicEngineering

Polytechnique of BariBari, Italy

Semyon M Meerkov

Department of Electrical Engineering andComputer Science

University of MichiganAnn Arbor, Michigan

Sajed Miremadi

AutomationDepartment of Signals and SystemsChalmers University of TechnologyGothenburg, Sweden

Alix Munier Kordon

Laboratoire d’Informatique de Paris 6Université Pierre et Marie CurieParis, France

Gasper Mu ˇsiˇc

Faculty of Electrical EngineeringUniversity of Ljubljana

Ljubljana, Slovenia

Department of Electrical and Electronics

Instituto Tecnológico de Aguascalientes

Aguascalientes, México

Andrea Paoli

The University of Bologna

Bologna, Italy

Miquel Àngel Piera

Department of Telecommunication and

Control Systems Group

Technical University of Berlin

Berlin, Germany

and

Systems and Control Theory Group

Max-Planck-Institute for Dynamics of Complex

Technical Systems

Magdeburg, Germany

Antonio Ramirez-Trevi ˜no

Department of Control and Automation

Centro de Investigación y de Estudios

Avanzados del Instituto Politécnico Nacional

Guadalajara, Jalisco, México

Spyros Reveliotis

School of Industrial and Systems EngineeringGeorgia Institute of Technology

Atlanta, Georgia

María Carmen Ruiz

Computing Systems DepartmentUniversity of Castilla-La ManchaAlbacete, Spain

Elvia Ruiz-Beltrán

Department of Electrical and ElectronicsInstituto Tecnológico de AguascalientesAguascalientes, México

Nicola Sacco

Department of Mechanical, Energy andTransportation Engineering

University of GenoaGenoa, Italy

Carla Seatzu

Department of Electrical and ElectronicEngineering

University of CagliariCagliari, Italy

Fernando Tricas

University of ZaragozaZaragoza, Spain

Biagio Turchiano

Department of Electrical and ElectronicEngineering

Polytechnique of BariBari, Italy

Department of Healthcare Engineering

Ecole Nationale Supérieure des Mines de

MengChu Zhou

The Key Laboratory of Embedded System andService Computing

Ministry of EducationTongji UniversityShanghai, Chinaand

Department of Electrical and ComputerEngineering

Institute of TechnologyNewark, New Jersey

Part I

Modelling and Simulations of Manufacturing Systems

1 Modelling Manufacturing

Systems with

Place/Transition Nets and

Timed Petri Nets

Maria Paola Cabasino, Mariagrazia Dotoli and

Carla Seatzu

CONTENTS

1.1 Introduction 31.2 Background on Place/Transition Nets 41.2.1 Petri Net Structure 41.2.2 Marking and State Equation 51.2.3 Language and Reachability Set 71.2.4 Some Behavioural Properties: Reversibility, Liveness and Deadlock 91.2.4.1 Reversibility 91.2.4.2 Liveness and Deadlock 91.3 Place/Transition Nets in Manufacturing 101.3.1 Basic Models in Manufacturing 101.3.2 More Complex Example of a Manufacturing System 131.3.3 Related Literature 151.4 Timed Petri Nets 161.4.1 Deterministic Timed Petri Nets 191.4.2 Timed Marked Graphs 221.4.3 Stochastic Timed Petri Nets 231.4.4 Related Literature 231.5 High-Level Petri Net Models in Manufacturing 241.6 Conclusions 25References 25

1.1 INTRODUCTION

Manufacturing systems are man-made systems that are increasingly used in factory automation sincethey can manufacture products with a high degree of automation and numerous different specifica-tions Manufacturing systems may be defined as discrete production systems in which the handledmaterials are discrete entities, for example, parts that are machined or assembled [35] As a result,discrete event formalisms are particularly suitable for modelling manufacturing systems using thebasic concepts of events and activities Hence, manufacturing systems can be effectively represented

as discrete event systems (DESs), whose dynamics depends on the interaction of asynchronous

dis-crete events, such as the arrival or departure of parts or products in a buffer, the start of an operation,the completion of a task and the failure of a machine Among the numerous DES frameworks,

3

Petri Nets (PNs) are a family of formalisms that allow to effectively model manufacturing systems.

Indeed, PNs are a graphical and mathematical tool that provides a uniform environment for thedesign, modelling, formal analysis and performance evaluation of DES Typically, manufacturingsystems are modelled in the PN formalism such that resources (machines, automated guided vehicles(AGVs, buffers, etc.) are represented by places The corresponding marking represents the capacity

of the resource, while the absence of tokens indicates that the resource is unavailable Therefore,places are useful to capture the decentralized nature of the system and the distributed state of theinformation in a complex manufacturing system More generally, places represent conditions in theoperation of the system For instance, a marked (unmarked) place may represent the fact that amachine is operative (down) Moreover, transitions typically represent the start or the termination of

an event Hence, places and transitions represent conditions and precedence relations for the rence of events driving the manufacturing system dynamics In the related literature, PNs contribute

occur-in a major way to the modelloccur-ing of manufacturoccur-ing systems because they occur-include numerous differentformalisms sharing basic principles in a consistent way, each best suited for the desired specificpurpose or degree of detail

This chapter addresses PN in manufacturing systems modelling, and its remainder is organized

as follows In Section 1.2, we present the basics of place/transition nets (P/T nets), which is thesimplest logic framework among the various PN modelling methodologies In Section 1.3, we showhow such a formalism can be employed to effectively represent manufacturing systems in a bottom-

up and modular fashion Several elementary PN models in the manufacturing application field arerecalled, and a more complex example is described in detail to show how these PN sub-modelscan be connected to obtain a PN representing a significant manufacturing system Further, wediscuss the main application fields of P/T nets for modelling manufacturing systems Subsequently,

in Section 1.4, we summarize the basics of the timed extension of P/T nets, showing how such

an enhanced framework may effectively be employed in manufacturing To this aim, we present adetailed example and survey the related literature Then, in Section 1.5, we survey the foremost high-level extensions of P/T nets models in the literature, with particular reference to their application tothe context of manufacturing system models Conclusions are finally drawn in Section 1.6

1.2 BACKGROUND ON PLACE/TRANSITION NETS

In this section, we provide some background on P/T nets For more details on this topic, we addressthe reader to [10,29,31]

1.2.1 PETRINETSTRUCTURE

A place/transition net, or simply a P/T net, is a bipartite weighted directed graph Vertices may be

of two different types: places, represented by circles, and transitions, represented by bars.

Definition 1.1 The structure of a P/T net is fully described by the quadruple N = (P, T, Pre, Post)

where the following applies:

• P = {p1, p2, p m} is the set of m places.

• T = {t1, t2, t n} is the set of n transitions.

• Pre:P × T −→ N is the pre-incidence function that specifies the number of arcs directed

• Post:P × T −→ N is the post-incidence function that specifies the number of arcs directed



Consider the P/T net in Figure 1.1 In such a case, it holds P = {p1, p2, p3, p4}, T = {t1, t2, t3, t4},

Pre = [1 0 0 0; 0 3 0 3; 0 1 0 0; 0 0 1 0] and Post = [0 0 1 0; 3 0 0 3; 1 0 0 0; 0 1 0 0].

The element Post[p2, t1] = 3 denotes that there are three arcs from transition t1 to place

p2 This is represented in the figure by means of a single barred arc with weight (or

Obviously the incidence matrix does not contain, in general, sufficient information to reconstruct

the net structure For instance, in the net in Figure 1.1, there exist both a pre and a post arc with

directed cycle in the graph of the net only involving one place and one transition In such a case, it

holds C[p2, t4] = 0, hiding the existence of arcs between these two vertices in the incidence matrix.

A net without self-loops is called pure A pure net can be represented by its incidence matrix C.

A net having all arcs with unitary multiplicity is called ordinary A state machine is an ordinary net whose transitions have exactly one input and one output arc, while a marked graph is an ordinary

net whose places have exactly one input and one output transition

p•= {t ∈ T | Pre[p, t] > 0}.

1.2.2 MARKING ANDSTATEEQUATION

The state of a P/T net is called marking.

Definition 1.2 A marking is a function m : P→ N that assigns to each place a nonnegative integer

Graphically, tokens are represented as black bullets inside places See, for example, the token in

Definition 1.3 A PN N with an initial marking m0 is called marked PN or PN system and is denoted

A marked net is a DES with a dynamical behaviour that is governed by the firing of transitions Atransition should satisfy a given condition to fire, and its firing leads to a marking update according

to the following rules

Definition 1.4 A transition t is enabled at a marking m if

In the marked net in Figure 1.1, the only enabled transition is t1 To be enabled, transition t2

A transition with no input arcs is called a source transition A source transition t is always enabled,

since, being in such a case Pre[·, t] = 0, the condition in Equation 1.2 is satisfied for all markings m.

Definition 1.5 A transition t enabled at a marking m can fire The firing of t removes Pre[p, t] tokens

If t1 fires, m = [0 3 1 0]T

is reached

marking m.

Definition 1.6 A firing sequence at a marking m0 is a string of transitions σ= tj1t j2· · · tj r ∈ T∗,

m0[tj 1m1[tj2m2 · · · [tj r mr,

that is, for all k ∈ {1, , r} transition tj k is enabled at mk−1and its firing leads to mk = mk−1+C[·, tj k]

To denote that the sequence σ is enabled at m, we write m[σ To denote that the firing of σ at m

∗The Kleene closure of a generic set X is an infinite set that contains all sequences that can be obtained combining elements

in X.

The empty sequence ε (i.e., the sequence of zero length) is enabled at all markings m and is such

is

consider a sequence σ= tj1t j2· · · tj r ∈ T∗as in Definition 1.6 that leads from m0 to mr, it holds

Equation 1.4 is called the state equation of the net.

Let us now introduce the notion of conflict

Definition 1.7 Two transitions t and tare in structural conflict if•t∩•t

a place p with a pre arc to both t and t

allow the firing of only one of the two transitions On the contrary, the conflict is not behavioural at

1.2.3 LANGUAGE ANDREACHABILITYSET

We now introduce two sets that characterize the possible evolutions of a net in terms of firingsequences and reachable markings, respectively

Definition 1.8 The language of a marked net N, m0 is the set of firing sequences enabled at theinitial marking, that is, the set



Definition 1.9 A marking m is reachable in N, m0 if there exists a firing sequence σ such that

the initial marking, that is, the set

Note that in the previous definition, the empty sequence, which contains no transition, is also

A marked PN with a finite reachability set is said to be bounded; otherwise, it is called unbounded.

As an example, the net in Figure 1.1 is bounded On the contrary, its language is infinite since

enabled at the initial marking However, it is easy to show that from the initial marking, only threemarkings may be reached

The simplest example of an unbounded net is given by a net consisting of a transition with anoutput and no input place (source transition) In such a case the transition may fire an arbitrarynumber of times, always leading to an increasing marking Finally, an example of a net with a finite

state (see the following Section 1.2.4).

main steps for its construction are summarized in the following algorithm

Algorithm 1.1 (Reachability graph)

2 Consider an unlabelled node m of the graph.

b Label node m ‘old’.

The reachability graph of the PN system in Figure 1.1 is given in Figure 1.2

Important conclusions can be drawn by looking at the reachability graph of a marked PN Forinstance, the following two implications hold

m t j1mt j2m· · ·

If the reachability set is not finite, a finite coverability graph can still be constructed using the notion

of ω-marking This obviously leads to a larger approximation of the reachability set and of the net

language However, important properties can still be proven For the sake of brevity, they are notreported here but can be found in [29,31]

[ 1 0 0 0 ] [ 0 3 1 0 ]

1.2.4 SOMEBEHAVIOURALPROPERTIES: REVERSIBILITY, LIVENESS ANDDEADLOCK

Several properties can be defined in the PN framework Some of them are behavioural properties, that

is, they depend both on the net structure and on the initial marking; other properties are structural,

that is, they only depend on the net structure For the sake of brevity in this chapter, we only focus onsome behavioural properties that are extensively used in the following chapters, namely, reversibility,liveness and deadlock

1.2.4.1 Reversibility

Reversibility implies that a system can always be reinitialized to its initial state This is a desirable

feature in many man-made systems

Definition 1.10 A marked netN, m0  is reversible if for all reachable markings m ∈ R(N, m0 ), it

The marked net in Figure 1.1 is reversible

The reachability graph provides necessary and sufficient conditions for reversibility In particular,the following implication holds:

path from each node to every other node of the graph

1.2.4.2 Liveness and Deadlock

Liveness of a transition implies the possibility that it can always eventually fire, regardless of the

current state of the net

Definition 1.11 Given a marked netN, m0, we say that a transition t is

Definition 1.12 A marked netN, m0 is



The marked net in Figure 1.3a is not quasi-live because it contains both dead and quasi-live transitions.The marked net in Figure 1.1 is live

Let us now introduce the notion of deadlock, strictly related to liveness.

Definition 1.13 Given a marked netN, m0, let m ∈ R(N, m0 ) be a reachable marking We say that

m is a dead marking if no transition is enabled at m, that is, if N, m is dead A marked net N, m0

Once again the reachability graph provides necessary and sufficient conditions for the verification

of liveness and deadlock as summarized in the following items Consider a bounded marked net

N, m0 and its reachability graph The following implications hold:

1.3 PLACE/TRANSITION NETS IN MANUFACTURING

In this section, we focus on modelling manufacturing systems with P/T nets In particular, in thefirst subsection, we present a series of elementary structures that occur in most production systems

In Section 1.3.2, we present an application example that shows how the presented modules can beeasily combined to describe more complex behaviours Finally, in Section 1.3.3, we provide a survey

of the literature on P/T nets in manufacturing

1.3.1 BASICMODELS INMANUFACTURING

Different physical meanings may be associated with tokens They may represent either the enablingcondition of a given operation or the availability of a physical resource

In the first case, elementary structures may easily be defined to model constraints on the order inwhich certain operations should be performed

As a first example, consider a robot that performs several operations, for example, cleaningpainting and polishing Assume that such operations should be executed exactly according to the

∗ A strongly connected component of a graph is said to be ergodic if there is no output arc from any node of the component

to any node outside the component If the graph is the reachability graph of a marked PN, this means that, once an ergodic component is reached, the only markings that can be reached are those in the component and the only transitions that may fire are those corresponding to the labels in the arcs of the component itself.

FIGURE 1.4 Basic P/T net structures: (a) sequentiality, (b) parallelism, (c) synchronization and (d) choice.

occur before cleaning has finished and polishing cannot be performed before painting is executed.Such a structure, which can obviously be generalized to an arbitrary number of transitions, is called

sequentiality.

they are completely independent

On the contrary, it may occur that several independent operations should be completed before a

given operation may be executed, that is, such operations need synchronization An example in this

executed independently Assume that processed parts should be identical and such a comparison is

are both marked

Moreover, the P/T net in Figure 1.4d models a choice Consider two robots whose activity is

taken from a common buffer (p1) In particular, they execute the same operation on parts, so each

part is processed either by the first robot or by the second one If the buffer only contains one part,

Other elementary P/T net structures can be introduced assuming that tokens represent available

resources These three are the main ones: assembly, disassembly and mutual exclusion.

inverse operation, namely, disassembly, is represented in Figure 1.5b showing that by disassembling

a table, we get four legs and one shelf In Figure 1.5c, an example of mutual exclusion is presented

Here, a token in p1 represents the availability of a machine to perform a given operation, that is,

painting Such a machine should paint parts of different types, A and B, that are taken from infinite

capacity buffers, or equivalently, they are available in an arbitrary number In particular, the firing of

the only condition that should be satisfied is that the machine is available, that is, it has not been

In such a case, the machine is again available when the operation on a part of type B is complete, for

has been completed Obviously, the previous model can be generalized to an arbitrary number ofentities sharing a common resource

Note that the example in Figure 1.1 uses some of the previous basic P/T net structures and may

By disassembling a part of type A, three parts of type B and one part of type C are obtained, which

simultaneously processed by a machine modelled by transition t4 Such an operation may be executed

output of a part from such a buffer and the simultaneous input of a new part to be processed in the

buffer modelled by p1 Note that this structure is quite common when modelling cyclic operations,

particularly when a timing structure is associated with transitions as discussed in Section 1.4 andthe timed model is used for performance analysis

We conclude this subsection with some remarks and generalizations on the model of a bufferthat is typically present in most manufacturing systems In the example in Figure 1.1, buffers havebeen modelled as places with one input and one output transition When the marking of the place

modelling a buffer may grow indefinitely, we talk about infinite capacity buffer See, for example, the net in Figure 1.6a On the contrary, a finite capacity buffer is a buffer whose content cannot exceed

a given threshold An example in this respect is shown in Figure 1.6b where C denotes the capacity

place that represents the number of parts that can still enter the buffer When a new part enters, the

the buffer Finally, Figure 1.6c is a generalization of the finite capacity buffer that contains parts of

type B Moreover, assume that parts of type B are larger than parts of type A In particular, the space occupied by one part of type B is equal to k times the space occupied by one part of type A This, in terms of residual space, means that k tokens should be removed from the complementary place p3 when one part of type B enters the buffer, while only one token should be removed when one part of

k

C p2

FIGURE 1.6 P/T net models of buffers: (a) infinite capacity buffer, (b) finite capacity buffer and (c) finite

capacity multi-class buffer

type A enters The sum of the number of tokens in p1 and p2denotes the total number of parts in the

buffer, of both types A and B.

1.3.2 MORECOMPLEXEXAMPLE OF AMANUFACTURINGSYSTEM

We now present a quite classical example that, with slight variations, has been extensively used inthe literature [20,40]

two different types of products from the two types of raw materials An unlimited source of rawparts is assumed Finally, it is supposed that there are 20 pallets for each type of product

The layout of the system is sketched in Figure 1.7 In more detail, parts of type A are initially

Finally, as already discussed in the case of Figure 1.1, it is assumed that the output of a processedpart corresponds to the simultaneous input of a raw part This allows to model the cyclicity of theprocess

representing the pallets of input parts of type A and B, respectively Here, we can observe some

FIGURE 1.8 PN model of the manufacturing system in Figure 1.7.

only perform one operation at a time This explains why the number of tokens in their places in the

PN model cannot be larger than one Analogous considerations can be repeated to explain the PNmodels of the other robots

Machines are modelled by one place, one input and one output transition When the place ismarked, it means that the machine is ready to perform its operation Consider, for example, machine

operation Similar considerations apply to the other machines and the two AGVs

of parts that can still enter the buffer In Figure 1.8, since the capacity of the buffer is assumed to

number of parts of type A is equal to the number of tokens in p12, while the number of parts of type

1.3.3 RELATEDLITERATURE

Among the numerous PN techniques, P/T nets are the simplest logic framework that can be employed

to model manufacturing systems As a matter of fact, classical P/T nets theory has been extensivelyused to model complex systems such as manufacturing systems, with applications dating back tothe 1970s [35] Indeed, P/T nets are able to model the material, information and control flow withinthe manufacturing system in an event-driven dynamic environment while effectively representingconcurrency or parallelism (two machines working independently), resource sharing and mutualexclusion (a robot handles parts for two machines but cannot serve both at the same time) andsynchronization (a machine is free and a part is ready to be processed by it) Moreover, the graphicalrepresentation leads to a simple and legible model, and the corresponding mathematical notationallows to straightforwardly represent and simulate the manufacturing system in any engineeringcomputation software The reader is referred to [35] for a comprehensive discussion on the motivationfor modelling manufacturing systems by PNs, illustrated by several examples

In the sequel we discuss the main field of application of P/T nets in the context of manufacturingsystems modelling The application of PN to manufacturing systems has been extensively surveyed

in [12] In particular, P/T nets have been successfully applied to the specification, verification andreal-time control of manufacturing systems The first and most important application of the P/T netsformalism to manufacturing systems lies in the specification of such complex systems Since theP/T net model of a manufacturing system typically suffers from the state explosion problem, twoapproaches have been proposed to cope with this problem:

1 Reduction techniques, allowing to simplify the computational effort of analysing large andcomplex P/T nets

2 Synthesis methods, which construct nets systematically such that the desired properties areguaranteed without analysis in the final nets

Reduction methods may roughly be classified into (1) place transformations, (2) fusion of transitionsand (3) addition of nets The reader is referred to [24] and [22] for a critical review on reductiontechniques

P/T net synthesis techniques address the computational complexity problem arising from the statespace explosion by a modular approach, in which the manufacturing system parts or subsystemsare specified separately The numerous systematic synthesis techniques for modelling by P/T netsDES and particularly manufacturing systems may be divided into three classes [6,22,41]: (1) top-down decomposition or refinement methods, using subnet abstractions to reduce the PN whilepreserving the logical property of interest; (2) bottom-up or composition techniques, which aresystematic methods for the synthesis of models starting from partial views of the manufacturingsystem; (3) hybrid methods The first class of methodologies have the advantage to offer a globalview of the system, whereas the second type of techniques are useful for specifying systems whiledescribing resource sharing, although these techniques do not always guarantee that the synthesizednet preserves important properties of the initial subnets Hybrid methods are combinations of thetwo The reader is referred to [12,22,41] for a review on the three subclasses of P/T net synthesistechniques

A second significant application of the P/T net framework to manufacturing systems consists

in their capacity of being analysed to check if the constructed model verifies some specificationproperties [35] Among these, synchronic properties and activity properties can be studied: the first

are, in temporal logic terminology, safety properties, while the second group are liveness properties.

More specifically, P/T nets have been widely applied for the verification of the following

prop-erties of manufacturing systems: reachability, boundedness, safeness, conservativeness, liveness, reversibility and invariance Numerous analysis techniques are provided in the related literature, and

the reader is referred to [12] and [41] for some comprehensive overviews on such methods

Third, a significant application area of P/T nets in the field of manufacturing systems is time control, that is, (1) sequencing the operations in the system and (2) preventing the systemfrom being in some undesired states The exploitation of the structural properties of PN models inorder to obtain computationally efficient algorithms for computing controls is nowadays one of themain goals of PN research The reader is referred to [21] for an overview of the literature on theapplication of untimed PN models and methods to problems in the logical control of manufacturingsystems Moreover, the supervised control approach is a general theory for controlling DES given

real-a specificreal-ation describing its real-allowed real-and desired behreal-aviour The resulting controller (supervisor)restricts the behaviour of the manufacturing system so as to fulfill the specifications NumerousPN-based approaches to supervisory control design have been proposed (see, for instance, [28] and[19] and the references therein) In addition, a comprehensive review of deadlock prevention policiesfor manufacturing systems is available in [26] while an overview of deadlock avoidance strategiesfor manufacturing systems including PN-based strategies may be found in [18]

We complete this section by referring the reader to a number of basic references where somesignificant examples on manufacturing systems may be found: Desrochers and Al-Jaar [12] report anumber of simple manufacturing systems modelled by PNs, and in such a framework, they provideseveral performance analysis with deadlock detection policies Numerous more complex examplesmay be found in [41] with detailed analysis In addition, further examples developed in detail may

be found in [30,33–35]

1.4 TIMED PETRI NETS

The P/T net framework described in the previous sections can be used to analyse logical properties;however, it cannot be used to analyse the performance of a system, such as the execution time of agiven process, the identification of bottlenecks and the optimization of the use of resources In fact,such a model does not consider the duration associated with the activities occurring in the system

In this section, PNs with time are introduced When defining PNs with time, three main elements

should be specified: the topological structure (which is the same of a P/T net), the timing structure and the transition firing rules.

Several timing structures have been proposed in the literature to extend P/T nets, and differentfiring rules are also based on them

m0[tj1, θ1 m1[tj2, θ2 m2 [tj3, θ3 m3 m k−1[tj k, θk mk .

where

θk= τk− τk−1(for k = 2, 3, 4 ) denote the delay between the firing of transition tj k−1and tj k

A timing structure specifies the value that these delays may assume.

Note that although in a timed evolution the delays denote the time elapsed between the firing oftwo transitions, from a structural point of view, a delay can be associated with different elements

of a net, such as places, transitions or arcs In the rest of this chapter, we assume that the timing

In the literature, two main classes of PNs with time have been defined: timed Petri nets (TdPNs) and time Petri nets (TPNs) In TdPNs a delay is represented by a single value θ As an example,

consider a net with delays associated with transitions: if a transition becomes enabled at time τ and

remains enabled henceforth, it must fire at time τ+ θ In TPNs a delay is represented by a time

T-TdPNs and T-TPNs indicate Petri nets where delays are associated with transitions, while P-TdPNs and P-TPNs indicate PNs where delays are associated with places In the following, we use TdPNs to indicate T-TdPNs, and we focus on TdPNs since they are more used in the literature

with respect to TPNs For an introduction to TPNs, we refer to [31] where a series of interestingreferences are also contained

The timing structure of a net can be deterministic, when the delays are known a priori, or stochastic, when the delays are random variables.

priori, or it can also be marking dependent.

If the delay θi has an exponential distribution fi (τ) = λ i e−λiτ(with λi > 0), transition

different from the exponential one, the transition is called generalized stochastic Finally,

if the parameters of the distribution depend on current marking of the net, the transition is

called stochastic marking dependent.

In this chapter, only immediate, deterministic constant and exponential stochastic transitions areconsidered Therefore, in the following, the last two types of transitions are briefly called deter-

For a detailed comparison of the various timing mechanisms, we refer to [3]

In a TdPN, the notion of server semantics is fundamental Different choices can be made:

an infinite number of operation units that work in parallel; as an example, this is the case

units can process all tokens simultaneously

FIGURE 1.9 Examples of TdPNs (a) A stochastic TdPN, (b) and (c) deterministic TdPNs.

time instants θ1, ., αθ1(with α≥ 1) since the single operation unit can only process onetoken at a time.

number k of operation units; as an example, if we consider the net in Figure 1.9c, assuming

three operation units can process only three tokens at a time In particular, at time instants

In the rest of this chapter, we always assume infinite server semantics In fact, starting from thisnotion, we can always represent the other choices limiting the maximum number of times a transitionmay fire by simply adding a self-loop to the considered transition whose place contains as many

tokens as the number of servers we want to consider for such a transition See, for example, place p3

Another essential notion is the memory associated with transitions Two different kinds of memory

can be specified:

been disabled by the firing of some other transition In the case of total memory policy,

2 Enabling memory: In such a case, each transition has only memory of its current enabling

and can only fire after being consecutively enabled for a time interval equal to its firingdelay

To better explain the notion of memory, consider the PN in Figure 1.10 that can model a system

previously been enabled for one time unit

We consider as basic notion the enabling memory policy

1.4.1 DETERMINISTICTIMEDPETRINETS

In this subsection we introduce the so-called deterministic timed transitions Petri nets (DTdPNs) or

deterministic T-TdPNs where the delays associated with transitions are deterministic We prefer to

focus on this kind of model instead of stochastic TdPNs (STdPNs), because manufacturing systemsare very well modelled by DTdPNs since the timed transitions typically represent the duration ofthe activities of the considered system Also in such a case, the delay can be associated with places

or arcs, but we focus on deterministic T-TdPNs that are more used in literature.

Definition 1.14 A deterministic TdPN is characterized by the algebraic structure Nd = (N, )

where the following applies:



Even for TdPNs it is possible to define a marked PN In general, the marking vector at the time

Definition 1.15 A deterministic TdPN Nd with a marking m0at the initial time instant τ0 is said to

The state of a DTdPN is determined not only by the marking, as for P/T nets, but also by theclocks associated with transitions

α is infinite is a degenerate case Then the marked net in Figure 1.9c has enabling degree equal to

At each time instant, the number of clocks associated with a transition is equal to its currentenabling degree Thus, each time the net evolves from one marking to another, the number of clocksassociated with each transition may change

The net evolution occurs in an asynchronous way on the basis of the events occurrence that

is regulated by the clocks associated with the events according to the algorithm of evolution herereported [31]

Algorithm 1.2 (Temporal evolution of a DTdPN) Consider a DTdPN at the time instant τj, and let

associated with the transitions The marking evolution of the DTdPN can be obtained by repeatingthe following steps:

with tihave to be discarded: clocks that are discarded from set{oi,1, , o i,α i ( j)} are those

having the higher values



The algorithm is based on the assumption of enabling memory and infinite server semantics However, if total memory is used, step 2 of the algorithm should be appropriately modified.

Example 1.2

Let us consider the net in Figure 1.11a whose temporal evolution is shown in Figure 1.11b

Transition t1has a delay time θ1= 2 and an enabling degree α1(0) equal to 1 at the initial time

instant τ0= 0; it has only one active clock o1,1 After two time instants, that is, at τ1= τ0+ θ1,

the clock is deleted, transitions t1fire once and the clock is again set to θ1= 2 At the reached

marking m1 = [1 3]T , transition t1still has an enabling degree equal to 1, and the clock o1,1isagain active The net continues to evolve following Algorithm 1.2 

m(p2 )

m(p1 ) τ 1

2 4 6 8

p2

θ1= 2 (a) (b)

FIGURE 1.11 (a) Deterministic TdPN system and (b) its evolution.