Petri Nets And Industrial Applications A+ Tutorial Pdf
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- Marcello La Rosa · Peter Loos Oscar Pastor (Eds.) Business Process Management
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Marcello La Rosa · Peter Loos Oscar Pastor (Eds.) Business Process Management
This paper presents a mathematical framework for model-ing prognostics at a system level, by combinmodel-ing the prognos-tics principles with the Plausible Petri nets PPNs formal-ism, first developed in M. The main feature of the resulting framework re-sides in its efficiency to jointly consider the dynamics of dis-crete events, like maintenance actions, together with multiple sources of uncertain information about the system state like the probability distribution ofend-of-life, information from sensors, and information coming from expert knowledge.
In addition, the proposed methodology allows us to rigorously model the flow of information through logic operations, thus making it useful for nonlinear control, Bayesian updating, and decision making. A degradation process of an engineer-ing sub-system is analyzed as an example of application us-ing condition-based monitorus-ing from sensors, predicted states from prognostics algorithms, along with information coming from expert knowledge.
The numerical results reveal how the information from sensors and prognostics algorithms can be processed, transferred, stored, and integrated with discrete-event maintenance activities for nonlinear control operations at system level. In prognostics, the integration of the predicted information at a system level encompasses two distinct research challenges.
First is about predicting the change in system performance and remaining life estimation through an adequate combina-tion of the degradacombina-tion rates and states of health of individ-ual components. Second, and perhaps most important, to. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.
Here, system-level nonlinearities are understood as uncertain environmen-tal elements that affect the system operation irrespectively of the component-wise state of health or degradation, like intervention processes e. In the literature, the majority of prognostics research to date has been focused on individual components, and determining theirend-of-life EOL and re-maining useful life RUL , e.
Besides, few attempts can be encountered describing system-level prognostics methodolo-gies. Generally, these attempts provide the EOL of a sys-tem based on its constituent components and how they inter-act, like in Gomez, Rodrigues, Galvo, and Yoneyama , where a system-level approach was developed using fault tree analysis from the RUL of individual components.
Daigle, Bregon, and Roychoudhury provided a distributed ar-chitecture to model system-level prognostics based on the concept of structural model decomposition whereby the so-lution of independent local prognostics subproblems were in-tegrated to obtain prognostics at system-level. Liu, Xu, Xie, and Kuo studied multi-component maintenance mod-els with economic dependence for components that degrade in a continuous manner.
Nonetheless, to the authors best known, holistic methodologies for prognostics with integration of system or sub-system level nonlinearities and uncertainties, still remain missing in the literature. Consequently, the approach has the advantage of being able to integrate by first time informa-tion from prognostics e. Moreover, the uncertainty is intrinsically accounted for in PPNs, since its formulation stems from combining the in-formation theory principles with the PNs technique, as will be shown further below.
To exemplify some of the problems that can be solved by the proposed methodology, several toy examples are formulated through a set of PPN architectures.
Next, the framework is tested using a numerical example to model decision-making over a two-component engineering system under degradation using prognostics information, expert knowledge, and main-tenance actions, which is just some of the challenges faced in system-level prognostics applications using PPNs. The remainder of the paper is organized as follows. Section 2 briefly overviews basic concepts about prognostics and PNs before introducing the fundamentals of PPNs.
Section 5 illustrates our approach through a numerical exam-ple about two-component degrading system. Finally, Section 6 gives concluding remarks. In the PHM community, the aforementionedremaining time.
The potential of prognostics in positively contributing to safety and cost relies in its capac-ity to provide anticipated information about an anomalous or faulty condition. For the estimation problem, the component, sub-system, or system state of health or degradation is typ-ically assumed to be represented using a stochastic variable. The functionsfkandhkare. This model is sequentially evaluated at every time stepk, and produces updated informa-tion about the system statexkas long as new measurements.
Next, prognostics algorithms can be employed to project the state predictions into future in absence of new observations. It can be computed as:. Finally, it is important to remark that in this work, the focus is not on predicting the EOL or RUL of a system as a probabil-ity, but on methodology that integrates EOL and RUL within an asset management framework at a system level, as will be described next. A place represents a particular state of the system or activity being modeled e. Places are temporarily visited bytokens, the abstract moving units of a PN.
The distribution of tokens over the PN at a specific time of execution is referred to as. The transitions are responsible of the dynamic be-havior of the PN, and enable the system to move from one state to another. In graphi-cal representation, places are typigraphi-cally expressed using circles while transitions are drawn as bars or boxes.
Arcs are labeled with their correspondingweights, non-negative integer values indicating the amount of parallel arcs 1 by default. Figure 1 is provided to serve as illustrative example of a PN of three places p1,p2,p3 , and one transition t1. From a mathematical point of view, a PN can be defined as an ordered 6-tupleNas follows Murata, :. As mentioned above, each edge has assigned a weight 1by default within the set of weights. For example in the PN from Fig. Note that by means of PNs and their marking, the behavior of complex engineering systems can be described in terms of discrete system states and their changes over time.
The following rules summarize the algebra of PNs as explained above:. Transitions always consume from all the input arcs at the same time and produce from all out-coming arcs the same amount;.
Plausible Petri nets PPNs are a class of PNs recently de-veloped by the authors, which are based on a combination of discrete and continuous numerical processes whose val-ues may be uncertain plausible. Two interacting subnets form the PPN graph: 1 a symbolic subnet, where the to-kens are objects in the sense of integer moving units, as in classical PNs Petri, , 2 a numerical subnet, where tokens arestates of information 1.
The resulting framework. Super-scriptsnp, n0p represent the number of numerical and. In prac-tical terms, these states of information can be understood as probability density functions PDFs except for a normalizing constant. Thus, the markingMkof a PPN at a certain timek. Note that the arc weights from the symbolic subnet, e.
In graphical representation, numerical nodes are represented us-ing double line, whereas sus-ingle line is used for the rest. A PPN model is shown in Fig. The dashed rectangles shown in Fig. As stated in the last section, the marking at timekof PPNs consists of both types of information given byM k N for the numerical places, andM k S for the case of symbolic places.
The marking evolution of the symbolic subnet corresponds to the state equation of a PN Murata, recall Eq. Figure 3. Illustration of the conjunction left and disjunction right of two arbitrary states of information, namelyfa x and. The rules given above are sufficient to explain the informa-tion flow dynamics of PPNs. Notwithstanding, observe that they are mostly based on conjunction of states of information which requires the evaluation of normalizing constants in-volving an intractable integral.
Also, note that there are situa-tions where the conjunction is conducted using density func-tions which are not completely known, perhaps because they are defined trough samples. In particle methods, a set ofNsamplesx n N. It can be evaluated for the case of.
This argument is important in terms of using PPNs in practical applications, as will be demonstrated in next section. In this section, a set of PPN sample architectures are provided to illustrate how our PPNs can be used for decision mak-ing at a system level in applications where information from prognostics along with other sources of information like ex-pert knowledge, sensors, etc.
The examples have been kept as simple as possible since they are mostly intended to serve as guideline to built more complex PPNs. Moreover, the examples provide represen-tative architectures whereby to conceptualize the proposed PNN methodology in a prognostics and health management context.
A couple of PPN architectures are exemplified here for mod-eling decision making aspects in presence of multiple infor-mation about the EOL from different components of an en-gineering system. This example assumes that numerical placesp 1N andp. Transition t. The resulting information is collected in place p 3N which acts as a buffer of information.
As stated. Figure 5. In this section, an example of PPN is provided which includes information about the PDF of EOL of a component or sub-system, along with information coming from expert knowl-edge. Figure 5a illustrates the PPN consisting of three numer-ical places, two symbolic places, and two transitions. As in the last example, transitiont1is defined based on condition, such that it is fired if the uncertainty2 of the PDFfEOL reaches or exceeds a specific threshold value.
For example, this can be as a consequence of a faulty sensor, or a perturbed prognostic estimation. Fi-nally, observe that the PDFfp3 can be further used in an. As in. In such case,p. The PPN framework presented above is exemplified here us-ing a numerical example to illustrate some of the advantages of using PPNs for integration of prognostics at a system level.
Figure 6 illustrates the idealized system model through a PPN of four numerical places, four symbolic places, and. The termvk represents a measurement error which. In the. In 15c , H denotes the differ-ential entropy of EOLk, that can be obtained by evaluating.
Figure 7. The expectation of the states of information is represented using gray circles. The dashed lines represent the5thth probability band. The latter represents expert knowledge. The differential entropy DE is used in this example as a qual-ity indicator of the information in place p 3N , so that the transitiont4is activated if the DE offkp3 is higher than the. For the numerical evaluation of the PPN in Fig. The algorithm.
Note that the disjunction of states of information. Table 2. Summary of the discrete events taking place when running the PPN shown in Fig. The second and third col-umn show the symbolic marking and firing vector, respec-tively. Figure 8. Note from Fig.
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With the continuing decline of the cost of computer hardware, and processors in particular, there is increasing interest in concurrent processing to achieve greater speed and efficiency. The use of Petri nets, graph models of concurrent processing, is one method of modeling and studying concurrent processing. Any edge e in E is incident on one member of P and one member of T. The set P is called the set of places and the set T is called the set of transitions. Less formally, a Petri net is a directed, bipartite graph where the two classes of vertices are called places and transitions.
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This paper presents a mathematical framework for model-ing prognostics at a system level, by combinmodel-ing the prognos-tics principles with the Plausible Petri nets PPNs formal-ism, first developed in M. The main feature of the resulting framework re-sides in its efficiency to jointly consider the dynamics of dis-crete events, like maintenance actions, together with multiple sources of uncertain information about the system state like the probability distribution ofend-of-life, information from sensors, and information coming from expert knowledge. In addition, the proposed methodology allows us to rigorously model the flow of information through logic operations, thus making it useful for nonlinear control, Bayesian updating, and decision making. A degradation process of an engineer-ing sub-system is analyzed as an example of application us-ing condition-based monitorus-ing from sensors, predicted states from prognostics algorithms, along with information coming from expert knowledge.