Discrete, Continuous, and Hybrid Petri Nets (eBook)

Foreword 4
Preface 7
Contents 11
Notation 17
Bases of Petri Nets 21
BASIC CONCEPTS 21
Places, Transitions and Arcs 21
Marking 22
Firing of a Transition 23
Autonomous and Non-autonomous Petri Nets 24
The Essential Characteristics 25
SPECIAL PETRI NETS 25
Particular Structures 26
Abbreviations and Extensions 29
MODELING OF SOME CONCEPTS 37
Properties of Petri Nets 41
PRESENTATION OF THE MAIN PROPERTIES 41
Notations and Definitions 41
Bounded Petri Net, Safe Petri Net 44
Liveness and Deadlock 45
Conflicts 50
Invariants 54
SEEKING THE PROPERTIES OF PETRI NETS 57
Graph of Markings and Coverability Root Tree 58
Linear Algebra 62
Reduction Methods Preserving Some Properties 72
Other Results 73
Concluding Remarks 77
Non-Autonomous Petri Nets 81
INTRODUCTION 81
SYNCHRONIZED PETRI NETS 83
Principle 84
Iterated Firing On Occurrence of an External Event 90
Properties of the Synchronized PNs 99
INTERPRETED PETRI NETS 108
Definition of a Control Interpreted Petri Net 109
Interpretation Algorithm of a Control Interpreted PN 113
Interpreted PN Without Outputs: Generalization of the Concept of Synchronized PN 116
TIMED PETRI NETS 118
General Information 118
Constant Timing 121
Stochastic Petri Nets 129
Autonomous Continuous and Hybrid Petri Nets 137
MOTIVATION 137
AUTONOMOUS CONTINUOUS PETRI NETS 139
From Discrete Petri Net To Continuous Petri Net 139
Definition 141
Reachability and Conflicts 143
Comments about units 149
AUTONOMOUS HYBRID PETRI NETS 150
Intuitive presentation 151
Definition 154
Reachability and conflicts 157
PROPERTIES OF AUTONOMOUS CONTINUOUS AND HYBRID PETRI NETS 163
Definitions and Properties Similar for Discrete and Continuous Petri Nets 163
Reachability and Limit Reachability for a Continuous Petri Net 165
e-Liveness for a Continuous Petri Net 168
Lim-Liveness for a Continuous Petri Net 169
Properties for a Hybrid Petri Net 171
EXTENDED HYBRID PETRI NETS 173
Threshold Test 173
Zero Test and Arc Weight 0+ 174
Marking 0+ 176
Definition 177
Timed Continuous Petri Nets 179
DEFINITION OF THE MODEL 179
Limit Case of a Discrete Timed Petri Net 180
Analysis of Some Basic Behaviors 181
Definitions 194
CONFLICTS 200
Existence of an Actual Conflict 200
Conflict Resolution 202
SPEED CALCULATION ALGORITHMS 204
There is No Structural Conflict 205
Resolution By Priorities 207
Resolution By Sharings And Priorities 220
Complete Algorithm For All IB-states 233
PROPERTIES 236
Illustratory Examples 236
General Properties 240
Modeling Power 243
MAXIMAL SPEEDS FUNCTIONS OF TIME 245
Timed Hybrid Petri Nets 250
DEFINITION OF THE MODEL 250
Flow Rate and Maximal Firing Speed 251
Illustrative Example 254
Events To Be Considered 256
Conflict Resolutions 257
Formal Definitions 260
ALGORITHM 267
Resolution for a Case 4 Conflict 268
Consequences of Various Events 273
Timed Hybrid PNs Automatically Treated in Algorithm 6.1 275
Algorithm for Building the Evolution Graph 281
Resolution of a Case Not Treated by Algorithm 6.1 287
VARIANTS OF THE MODEL 288
Synchronized D-Transitions 289
Stochastic Timings for D-Transitions 292
Hybrid Petri Nets with Time Transitions 293
C-Transitions with Flow Rates Functions of Time 294
Colored Hybrid Petri Net 296
EXTENDED TIMED HYBRID PETRI NETS 297
Modeling of Zero Buffers 298
Arc Weight 0+ for Testing if a C-Place is Empty 301
Pure Delay of a Continuous Flow 304
Conclusion on Timed Extended Hybrid Petri Nets 311
Hybrid Petri Nets with Speeds Depending on the C-Marking 314
APPROXIMATION OF TIMED DISCRETE SYSTEMS BY VHPNs 314
Weakness of Basic Timed Hybrid PNs for Small Numbers 315
Simple Cases of Variable Speed Hybrid PN 316
General Case of VHPN 320
Application Examples 329
ASYMPTOTIC HYBRID PETRI NETS (AHPNs) 335
A C-Transition Has a Single Input C-Place 335
Several Input C-Places 340
Generalization 342
Differences Between VHPN and AHPN Behaviors 345
OTHER MODELS 349
Liquid Flow 349
Differential Hybrid Petri Nets 350
Transfer Line with Operation-Dependent Failures 354
Postface 357
Appendix 363
Regular Expressions and Languages 363
Conflict Resolution 365
Elements of Graph Theory 369
Algebra of Events 371
About Grafcet 375
Modeling Power of Synchronized PNs 381
Timed Petri Nets Are Special Cases of Synchronized Petri Nets 383
Time Petri Nets 389
Linearity of the Fundamental Equation for Continuous Petri Nets 393
Notation 0+ and Non-Standard Analysis 397
Sharing Between Two Transitions 399
Graph of Relations Among Conflicts 405
Piecewise Constant Maximal Speeds 409
From Hybrid PNs to Hybrid Automata 417
P& T-Timed Petri Nets and Modeling Power
Exercises 429
Solutions to Exercises 470
References 542
Index 558