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2010 | Buch

Bases of Petri Nets Kapitel lesen Erstes Kapitel lesen

Discrete, Continuous, and Hybrid Petri Nets

verfasst von: René David, Hassane Alla

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Petri Nets were introduced and still successfully used to analyze and model discrete event systems especially in engineering and computer sciences such as in automatic control.

Recently this discrete Petri Nets formalism was successfully extended to continuous and hybrid systems. This monograph presents a well written and clearly organized introduction in the standard methods of Petri Nets with the aim to reach an accurate understanding of continuous and hybrid Petri Nets, while preserving the consistency of basic concepts throughout the book. The book is a monograph as well as a didactic tool which is easy to understand due to many simple solved examples and detailed figures. In its second completely reworked edition various sections, concepts and recently developed algorithms are added as well as additional examples/exercises.

Inhaltsverzeichnis

Frontmatter
Bases of Petri Nets
Abstract
The main users of Petri nets are computer and automatic control scientists. However, this tool is sufficiently general to model phenomena of extremely varying types.
Petri nets present two interesting characteristics. Firstly, they make it possible to model and visualize behaviors with parallelism, concurrency, synchronization and resource sharing. Secondly, the theoretical results concerning them are plentiful; the properties of these nets have been and still are extensively studied.
In this chapter, the bases of Petri nets, various classes of these nets and their modeling power are presented in a fairly intuitive manner.
René David, Hassane Alla
Properties of Petri Nets
Abstract
The main properties possessed by certain PNs are presented in Section 2.1. These properties are linked to the concepts of boundedness, liveness and deadlocks, conservative components and repetitive components.
We must now look into how we can find out which PNs possess which properties. Seeking the properties is the goal of Section 2.2. The two basic methods are drawing up the graph of markings, on the one hand, and linear algebra on the other.
René David, Hassane Alla
Non-Autonomous Petri Nets
Abstract
Autonomous PNs which allow a qualitative approach were studied in Chapters 1 and 2. In this chapter, we shall present extensions of Petri nets which enable us to describe not only what “happens” but also when “it happens”. These Petri nets will enable systems to be modeled whose firings are synchronized on external events, and/or whose evolutions are time dependent.
After an introduction in Section 3.1, Section 3.2 is devoted to synchronized PNs. Up to now, it was considered that, in a synchronized PN, a transition could be fired only once at a given time. In [DaAl 89, 2nd edition, & 92], a model called “extended synchronized PN” was briefly presented. In this chapter, this model is developed and rechristened “synchronized PN”. This model, more general than the original synchronized PN, is based on the fact that a transition which is q-enabled may be fired q times at the same instant.
Then, interpreted PNs (for control of discrete event systems) and timed PNs (for performance evaluation) are presented in Sections 3.3 and 3.4. These models appear to be special cases of synchronized PNs. As a consequence, all the properties shown for synchronized PNs are relevant to these models.
This last observation is particularly important. In point of fact, the behaviors of timed continuous and hybrid PNs, presented in Chapters 5 and 6, are derived from the behavior of timed discrete PNs. Thus, it follows that the behaviors of all non-autonomous models (i.e. interpreted PNs, Grafcet, timed PNs – constant or stochastic –, timed and/or synchronized continuous and hybrid PNs) are based on the theoretical behavior of synchronized PNs. It follows that all these models inherit the properties of synchronized PNs.
René David, Hassane Alla
Autonomous Continuous and Hybrid Petri Nets
Abstract
The marking of a place in a PN may correspond to the state of a device, e.g. a machine is or is not available. This marking can be compared to a Boolean variable. A marking can also be associated with an integer, e.g. the number of parts in the input buffer of a machine. In this second case, the number of tokens may be a large number. This may result in such a large number of reachable markings that a limit is formed for use of PNs. A number of authors studying production systems have modeled a number of parts by a real number, an approximation which generally proves very satisfactory. Why not then in a PN?
The continuous Petri net is a model in which the number of marks in the places are real numbers instead of integers. The motivation is explained in Section 4.1 and the model is presented in Section 4.2. Then, hybrid PNs containing a “discrete part” and a “continuous part” are defined in Section 4.3. Properties of continuous and hybrid PNs are presented in Section 4.4. Finally, Section 4.5 is devoted to a model called extended hybrid PN.
All the models in this chapter are autonomous, i.e., not dependent on time or on the environment.
René David, Hassane Alla
Timed Continuous Petri Nets
Abstract
The basic model is an autonomous continuous PN, as defined in Chapter 4, plus a maximal speed associated with every transition. Firing of a transition is continuous, like a flow limited by the maximal speed. For the models considered in this chapter, the maximal speeds do not depend on the marking.
For the basic model, the maximal speeds are constant. In Section 5.1, the behavior of such a PN is presented as a limit case of timed discrete PN behavior. Various basic behaviors are then analyzed, in order to understand clearly the semantics of the model.
After a presentation of the conflicts in a timed continuous PN and their resolutions in Section 5.2, an algorithm used to calculate the behavior for such a PN is proposed in Section 5.3. Some applications and properties are presented in Section 5.4.
The basic model is then generalized in Section 5.5. The maximal speeds may depend on time but remain independent from the marking (models in which the speeds may depend on the marking are presented in Chapter 7).
In this chapter, the expression “continuous PN” means “timed continuous PN” (with constant maximal speeds up to the end of Section 5.4). Whenever an autonomous PN is concerned, this is specified.
René David, Hassane Alla
Timed Hybrid Petri Nets
Abstract
The basic model is an autonomous hybrid PN, as defined in Chapter 4, plus a timing associated with every discrete transition and a maximal speed associated with every continuous transition (or more formally a flow rate, as will be specified in Section 6.1.1).
For the basic model 1, the timings and the flow rates are constant. The behavior and a formal definition of this model are presented in Section 6.1. An algorithm used to calculate the behavior for such a PN is proposed in Section 6.2.
Other models, in which the maximal speeds or timings are not constant, are presented in Section 6.3. Finally, timed extended hybrid Petri nets (based on the autonomous model introduced in Section 4.5) are presented with various application examples in Section 6.4.
In this chapter, the expression “hybrid PN” means “timed hybrid PN” (with constant timings and maximal speeds up to the end of Section 6.2). Whenever an autonomous PN is concerned, this is specified.
René David, Hassane Alla
Hybrid Petri Nets with Speeds Depending on the C-Marking
Abstract
In the basic timed hybrid PN (called CHPN), as well as in the variants, presented in Chapter 6, maximal speeds are independent from the C-marking. In this chapter, various models in which speeds depend on the C-marking are explained.
A model called VHPN, in which the firing speed of a C-transition depends on the minimal marking of its input places, is presented in Section 7.1. This model usually provides an acceptable approximation for a discrete system (exactly modeled by a discrete T-timed PN). The instantaneous speeds are no longer piecewise constant. The explicit model of hybrid PN (Definition 6.4 in Section 6.1.5.2) is required.
An approximation of the previous model, called AHPN, is given in Section 7.2. It provides a satisfactory approximation with piecewise constant instantaneous speeds.
Various other models, adapted for modeling special systems, are presented in Section 7.3.
All these models have a common basis: the autonomous hybrid Petri nets defined in Chapter 4.
René David, Hassane Alla
Backmatter
Metadaten
Titel
Discrete, Continuous, and Hybrid Petri Nets
verfasst von
René David
Hassane Alla
Copyright-Jahr
2010
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-10669-9
Print ISBN
978-3-642-10668-2
DOI
https://doi.org/10.1007/978-3-642-10669-9

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