The diagnosis of bearing and gear faults solely through non-intrusive current signal analysis is an interesting approach. This paper’s primary contribution lies in the extraction of a novel indicator for identifying these faults irrespective of speed and load variations. In fact, comprehensive monitoring systems encompassing variable speed and load regimes alongside combined gearbox faults solely using electrical signals are scarce in the existing literature. To address this gap, we propose a pioneering method. The proposed method uses a regime normalization technique with different current sensors, allowing the grouping of various regimes under the same health state. By minimizing dispersion among class observations and distinguishing between different health states, including variations in speed and load, this method promises heightened diagnostic accuracy. This paper proposes also to improve the diagnosis of faults by using the Naïve Bayes classifier with the introduction of a criterion called the threshold in order to address the uncertainty. Addressing uncertainty is crucial to prevent false alarms during diagnosis. To verify the effectiveness of the proposed method, current data collected from a test bench composed of a gearbox system and operating under variable speed and load conditions is tested and validated. Also, the proposed method is compared with diverse machine learning classifiers to test the effectiveness of introducing the uncertainty to improve the diagnosis. The proposed model is performed based on accuracy of 97.55%, precision of 99.8% and F1-socre of 98.73%.

In Predictive Maintenance (PM) of manufacturing industries, the main objective is to improve production time and increase the quantity of produced parts. However, product quality management is less addressed. Indeed, each non-conform part represents a direct loss, underlining the importance of product conformity assessment. This research provides a new methodology for assessing the impact of systemic parameters, such as the characteristics of raw materials and the state of machines, on the quality of the final product. The methodology begins with the collection and preparation of industrial data which, due to their specific nature, require detailed processing to meet the required quality standards. We then analyze and compare three distinct neural architectures: Long Short-Term Memory (LSTM), Gated Recurrent Units (GRU), and an LSTM-GRU. These models were deployed to compare their capacity to predict the global quality of the part directly, compared to an indirect prediction which requires the determination of the conformity of each dimension of the part. The implementation of this methodology was carried out on a real dataset, provided by the French company SCODER, which reveals promising prospects for the PM sector. The results highlight the effectiveness of using the LSTM model to predict 3D measurements and improve quality management in manufacturing production processes.

Predicting the remaining useful life (RUL) is a critical step before the decision-making process and developing maintenance strategies. As a result, it is frequently impacted by uncertainty in a practical context and may cause issues. This article proposes a new hybrid deep architecture that predicts when an in-service machine will fail to overcome the latter problem, allowing for an improved data analysis and dimensionality reduction capability providing better spatial distributions of features and increasing interpretability. A deep convolutional variational autoencoder with an attention mechanism (ACVAE) has been developed and tested using the aero-engine C-MAPSS dataset. We defined two adapted threshold settings ($\alpha 1, \alpha 2$) by analyzing the spatial distribution and minimizing the overlapping area between the degradation classes. To reduce the conflict zone, we used the soft voting classifier. The performance of our visual explainable deep learning model has reached a higher level of accuracy compared with previous existing models.

Fault detection and diagnostics are important steps in the predictive maintenance of industrial systems, especially faults in the mechanical parts most susceptible to fail, such as bearings and gears in rotating machines. These two components represent more than 50% of causes of the operational downtime. Therefore, the detection of their appearance allows anticipating the total failure of the machine and schedule in advance maintenance actions. However, in the presence of a combined gear and bearing faults, it is difficult to isolate their states. To remedy this situation, this paper proposes a data processing methodology that exploits the three-phase current signals of the rotating machine and build a health indicator (HI) from each current phase that reveals the different health states. This indicator is constructed by extracting features from the collected raw data in frequency and time domains, and then they properly combined with a physical significance. After that, all health indicators (HIs) of the three phase current data are fed to a machine learning model for an online pattern recognition of the bearing and gear states, including the combined faults. The proposed approach is demonstrated through a test bench that studies bearing and gear defects of a gearbox under different operating conditions.

In the framework of rotating machines prognosis considering naturally progressing degradations, this work proposes a new adaptive strategy for the estimation of the Remaining Useful Life (RUL) of bearings. In fact, although they are the indispensable mechanical elements of rotating machines, bearings are the most stressed part and their damage causes unexpected stops. The most of advanced fault prognosis techniques proposed in the literature are based on vibration signals because they are rich in information. However, the noisy and nonlinear natures of the raw vibratory collected signals make the prognostic task hard and more challenging. In this sense, a new strategy for bearing state of health estimation is proposed in this work. The proposed strategy is based on the application of the Extended Kalman filter (EKF) with some advanced digital processing steps. The EKF is used to approximate the nonlinear variation of the selected feature. In fact, the EKF algorithm is based on two steps of online feature extraction and evaluation. These two steps ensure the online selection of the best feature considering some mathematical characteristics. This method has been validated on vibratory data from the full-scale test bench of the University of Cincinnati, USA. Experimental results show that the proposed approach illustrates good prediction capabilities even with a long horizon and it can be applied to the Prognostic and Health Management (PHM) of several other assets.

Recently, Prognostics and Health Management (PHM) solutions are increasingly implemented in order to complete maintenance activities. PHM predicts the future behavior of a system as well as its remaining useful life (RUL). One of the main approaches of the prognostic is data-driven approach who offer an advantage of being able to learn models based on empirical data and uses artificial intelligence methods. Present paper offers an implementation of PHM solution. We were interested by the estimation of the RUL of the aircraft engine by using historical data. We have implemented two technics: Artificial Neural Network and Neuro-Fuzzy System. To compare between these methods, we have studied the performance of the prognostic system according to the accuracy, precision, MSE (Mean Squared Error) and the training time. The best method was concluded finally.

This paper presents the data acquisition step of a Prognostics and Health Management (PHM) of Micro-Electro-Mechanical Systems (MEMS) application. The targeted MEMS device is an electro-thermally actuated MEMS valve. The data acquisition is performed during the accelerated lifetime tests. To perform tests, an experimental test bed is designed and built. Several test campaigns are performed where MEMS valves operated continuously and data acquired regularly. The obtained experimental results show that MEMS fabricated with the same micro-fabrication process and tested in the same conditions do not have the same behavior and the same evolution of degradation in time. Therefore, this supports the importance of applying PHM of MEMS rather than the predictive reliability.

Proton Exchange Membrane Fuel cells (PEMFC) are one of the most promising fuel cell technologies, which qualify for variety of applications as power generation source. The Prognostics & Health Management of fuel cell is an emerging field, which is paving the way for large scale industrial deployment of PEMFC technology. More precisely, prognostics of PEMFC become a major area of focus nowadays that enables predicting the behavior of PEMFC to produce actionable information to extend its life span. This paper contributes the first application on data-driven prognostics of PEMFC stack under variable load for combined heat and power generation (μCHP). In brief, an ensemble structure of Summation Wavelet-Extreme Learning Machine models is proposed with a new incremental learning scheme, to achieve long-term predictions on stack state of health (SOH) and to give confidence for better decisions. The proposed prognostics model is validated on data from PEMFC stack used for a μCHP application under variable load profile for a complete year. A thorough comparison on SOH predictions results clearly shows the significance of proposed prognostics model, which can predict with few learning data for a long-term prognostics horizon around 650 hours with high accuracy and low uncertainty.

In condition-based maintenance, real-time observations are crucial for on-line health assessment. When the monitoring system is a wireless sensor network (WSN), data loss becomes highly probable and this affects the quality of the remaining useful life prediction. In this paper, we present a fully distributed algorithm that ensures fault tolerance and recovers data loss in WSNs. We first theoretically analyze the algorithm and give correctness proofs, then provide simulation results and show that the algorithm is (i) able to ensure data recovery with a low failure rate and (ii) preserves the overall energy for dense networks.

Proton Exchange Membrane Fuel Cells (PEMFC) are promising energy converters, but still suffer from a short life duration. Applying Prognostics and Health Management seems to be a great solution to overcome that issue. But developing prognostics to anticipate and try to avoid failures is a critical challenge. To tackle this problem, a hybrid prognostics approach is proposed. It aims at predicting the power aging of a PEMFC stack working at a constant operating condition and a constant current solicitation. The main difficulties to overcome are the lack of adapted modeling of the aging for prognostics, and the occurrence of disturbances creating recovery phenomena through aging. Consequently, this work proposes a new empirical model for power aging that takes into account these recoveries based on different features extracted from the data. These models are used in a joint particle filter framework directly initialized by an automatic parameter estimate process. When sufficient data are available, the prognostics can give accurate behavior predictions compared to experimentation. Remaining useful life estimates can be given with an error smaller than 5% for a horizon of 500 hours on a life duration of 1750 hours, which is clearly long enough for decision making.

Prognostics is a core process of prognostics and health management (PHM) discipline, that estimates the remaining useful life (RUL) of a degrading machinery to optimize its service delivery potential. However, machinery operates in a dynamic environment and the acquired condition monitoring data are usually noisy and subject to a high level of uncertainty/unpredictability, which complicates prognostics. The complexity further increases, when there is absence of prior knowledge about ground truth (or failure definition). For such issues, data-driven prognostics can be a valuable solution without deep understanding of system physics. This paper contributes a new data-driven prognostics approach namely, an "enhanced multivariate degradation modeling," which enables modeling degrading states of machinery without assuming a homogeneous pattern. In brief, a predictability scheme is introduced to reduce the dimensionality of the data. Following that, the proposed prognostics model is achieved by integrating two new algorithms namely, the summation wavelet-extreme learning machine and subtractive-maximum entropy fuzzy clustering to show evolution of machine degradation by simultaneous predictions and discrete state estimation. The prognostics model is equipped with a dynamic failure threshold assignment procedure to estimate RUL in a realistic manner. To validate the proposition, a case study is performed on turbofan engines data from PHM challenge 2008 (NASA), and results are compared with recent publications.

Estimating remaining useful life (RUL) of critical machinery is a challenging task. It is achieved through essential steps of data acquisition, data pre-processing and prognostics modeling. To estimate RUL of a degrading machinery, prognostics modeling phase requires precise knowledge about failure threshold (FT) (or failure definition). Practically, degrading machinery can have different levels (states) of degradation before failure, and prognostics can be quite complicated or even impossible when there is absence of prior knowledge about actual states of degrading machinery or FT. In this paper a novel approach is proposed to improve failure prognostics. In brief, the proposed prognostics model integrates two new algorithms, namely, a Summation Wavelet Extreme Learning Machine (SWELM) and Subtractive-Maximum Entropy Fuzzy Clustering (S-MEFC) to predict degrading behavior, automatically identify the states of degrading machinery, and to dynamically assign FT. Indeed, for practical reasons there is no interest in assuming FT for RUL estimation. The effectiveness of the approach is judged by applying it to real dataset in order to estimate future breakdown of a real machinery.

One remaining technological bottleneck to develop industrial Fuel Cell (FC) applications resides in the system limited useful lifetime. Consequently, it is important to develop failure diagnostic and prognostic tools enabling the optimization of the FC. Among all the existing prognostics approaches, datamining methods such as artificial neural networks aim at estimating the process' behavior without huge knowledge about the underlying physical phenomena. Nevertheless, this kind of approach needs huge learning dataset. Also, the deployment of such an approach can be long (trial and error method), which represents a real problem for industrial applications where real-time complying algorithms must be developed. According to this, the aim of this paper is to study the application of a reservoir computing tool (the Echo State Network) as a prognostics system enabling the estimation of the Remaining Useful Life of a Proton Exchange Membrane Fuel Cell. Developments emphasize on the prediction of the mean voltage cells of a degrading FC. Accuracy and time consumption of the approach are studied, as well as sensitivity of several parameters of the ESN. Results appear to be very promising.

Forecasting the future states of a complex system is a complicated challenge that is encountered in many industrial applications covered in the community of prognostics and health management. Practically, states can be either continuous or discrete: Continuous states generally represent the value of a signal while discrete states generally depict functioning modes reflecting the current degradation. For each case, specific techniques exist. In this paper, we propose an approach based on case-based reasoning that jointly estimates the future values of the continuous signal and the future discrete modes. The main characteristics of the proposed approach are the following: 1) It relies on the K-nearest neighbor algorithm based on belief function theory; 2) belief functions allow the user to represent his/her partial knowledge concerning the possible states in the training data set, particularly concerning transitions between functioning modes which are imprecisely known; and 3) two distinct strategies are proposed for state prediction, and the fusion of both strategies is also considered. Two real data sets were used in order to assess the performance in estimating future breakdown of a real system.

Job Shop Scheduling is one of the most difficult problems in industry and it is the main interest of the major researchers in the manufacturing research area. This problem becomes crucial when the production planning and maintenance have to be jointly solved. Several heuristics and intelligent methods have been so far proposed in the literature and applied. This work deals with a Hopfield Neural Network (HNN) method used for solving the JSP taking into account the maintenance tasks. While this method had been already proposed in the literature to solve the JSP alone, our main improvement of this method is to take into account the maintenance periods by extending the Hopfield net to handle the joint problem. Experimental study shows that the proposed HNN algorithm gives efficient results for the resolution of the joint job shop scheduling problem.

Prognostics and Health Management aims at estimating the remaining useful life of a system (RUL) , i.e. the remaining time before a failure occurs. It benefits thereby from an increasing interest: prognostic estimates (and related decision-making processes) enable increasing availability and safety of industrial equipment while reducing costs. However, prognostics is generally based on a prediction step which, in the context of data-driven approaches as considered in this paper, can be hard to achieve because future outcomes are in essence difficult to estimate. Also, a prognostic system must perform sufficient long term estimates, whereas many works focus on short term predictions. Following that, the aim of this paper is to formalize and discuss the connexionist-systems-based approaches to ensure multi-step ahead predictions for prognostics. Five approaches are pointed out: the Iterative, Direct, DirRec, Parallel, and MISMO approaches. Conclusions of the paper are based, on one side, on a literature review; and on the other side, on simulations among 111 time series prediction problems, and among a real engine fault prognostics application. These experiments are performed using the exTS (evolving extended Takagi-Sugeno system). As for comparison purpose, three types of performances measures are used: prediction accuracy, complexity (computational time), and implementation requirements. Results show that all three criteria are never optimized at the same time (same experiment), and best practices for prognostics application are finally pointed out.

Prognostics activity deals with the estimation of the Remaining Useful Life (RUL) of physical systems based on their current health state and their future operating conditions. RUL estimation can be done by using two main approaches, namely model-based and data-driven approaches. The first approach is based on the utilization of physics of failure models of the degradation, while the second approach is based on the transformation of the data provided by the sensors into models that represent the behavior of the degradation. This paper deals with a data-driven prognostics method, where the RUL of the physical system is assessed depending on its critical component. Once the critical component is identified, and the appropriate sensors installed, the data provided by these sensors are exploited to model the degradation's behavior. For this purpose, Mixture of Gaussians Hidden Markov Models (MoG-HMMs), represented by Dynamic Bayesian Networks (DBNs), are used as a modeling tool. MoG-HMMs allow us to represent the evolution of the component's health condition by hidden states by using temporal or frequency features extracted from the raw signals provided by the sensors. The prognostics process is then done in two phases: a learning phase to generate the behavior model, and an exploitation phase to estimate the current health state and calculate the RUL. Furthermore, the performance of the proposed method is verified by implementing prognostics performance metrics, such as accuracy, precision, and prediction horizon. Finally, the proposed method is applied to real data corresponding to the accelerated life of bearings, and experimental results are discussed.

This paper addresses a data-driven prognostics method for the estimation of the Remaining Useful Life (RUL) and the associated confidence value of bearings. The proposed method is based on the utilization of the Wavelet Packet Decomposition (WPD) technique, and the Mixture of Gaussians Hidden Markov Models (MoG-HMM). The method relies on two phases: an off-line phase, and an on-line phase. During the first phase, the raw data provided by the sensors are first processed to extract features in the form of WPD coefficients. The extracted features are then fed to dedicated learning algorithms to estimate the parameters of a corresponding MoG-HMM, which best fits the degradation phenomenon. The generated model is exploited during the second phase to continuously assess the current health state of the physical component, and to estimate its RUL value with the associated confidence. The developed method is tested on benchmark data taken from the “NASA prognostics data repository” related to several experiments of failures on bearings done under different operating conditions. Furthermore, the method is compared to traditional time-feature prognostics and simulation results are given at the end of the paper. The results of the developed prognostics method, particularly the estimation of the RUL, can help improving the availability, reliability, and security while reducing the maintenance costs. Indeed, the RUL and associated confidence value are relevant information which can be used to take appropriate maintenance and exploitation decisions. In practice, this information may help the maintainers to prepare the necessary material and human resources before the occurrence of a failure. Thus, the traditional maintenance policies involving corrective and preventive maintenance can be replaced by condition based maintenance.

In this paper, we introduce a new energy efficient medium access control (MAC) protocol for event-driven sensor networks. The event-driven wireless sensor networks do not often have much data to send. When an alarm occurs, many sensors in the network send the same message at the same time. These redundant communications lead to an energy waste due to many transmissions of redundant data and collisions in the network. By using over-hearing message, we propose a new method to solve this problem in order to save energy and to maximize the lifetime of the sensor networks

The remote maintenance of the industrial processes knows an important competitiveness with the technological advance which returns the performance of a machine accessible from any place in the world. The industrial processes become increasingly complex, this complexity makes only increase the overload of information and the risk of errors, which involves forcing an immense difficulty to supervise them by the human operator, and an important cost of the maintenance action. The development of a telemaintenance system offers to the industrialists and users a great flexibility in industrial activities control, it must support distant facilities to ensure the industrial plant performance and the quality of the operations. We can say that the operator or the technician will be able to work in a virtual operational environment. In this direction the approach by powerful multi agents systems with their characteristics in the resolution of problems tends to be widespread in all the fields of research in particular in those of remote maintenance. In front of the vastness of information, no genius can remember everything, nor to solve any problem. Thus a cooperative working group proves necessary even essential. However the capacities of problems resolution do not rest solely on the group knowledge, its experiment, but still depend on its capacities to make an effective search for an assistance of this working group. Within this framework of idea our modeling of the activities of a remote maintenance system is by choosing an interaction protocol between expert agents making it possible to ensure an efficient cooperation.

We study in this paper the combined maintenance and production scheduling. Such problem is traditionally treated independently specially for the multi product environment (like the job shop). We look for optimizing jointly two criteria, one for production which is the makespan, and one for maintenance, which is the total cost. These two criteria are antagonistic, that is why we develop a multiobjective optimization and genetic algorithm based method that determines simultaneously maintenance and production schedules for a job shop. The genetic algorithm uses a Pareto optimal selection keeping only the more adapted solutions. The Pareto optimal solutions given by the genetic algorithm are validated using some lower bounds. For the purpose of reducing the frequency of breakdowns, we apply the systematic preventive maintenance to the various resources of the workshop. We compare in this paper two manners of choosing periods of maintenance: the first consists in maintaining machines each a fix period of time. The second applies maintenance according to a fix work load of each machine.

In this paper, we propose a unitary tool for modeling and analysis of discrete event systems monitoring. Uncertain knowledge of such tasks asks specific reasoning and adapted fuzzy logic modeling and analysis methods. In this context, we propose a new fuzzy Petri net called Fuzzy Reasoning Petri Net: the FRPN. The modeling consists in a set of two collaborative FRPN. The first is used for the fault dynamic state of the system by temporal spectrum of the marking. A monitoring fuzzy Petri net (MFPN) represents the fault tree. The second model, the recovery fuzzy Petri net (RFPN) corresponds to recovering activities. The two proposed models form a dynamic loop for production system monitoring and recovery. Production system is supposed to be modelized using temporal Petri nets, with the assumption that the primary fault symptoms are detected. These symptoms are considered in the MFPN and evolve according to all other derived faults of the system. Synchronizing signals, corresponding to different warning levels and alarms, form the interface between these two tools.

This paper introduces a modular modeling approach for distributed production systems, considering the production and maintenance processes synchronization. The production job shop, preventive and curative specialized tele-maintenance actions are studied. Stochastic synchronized Petri nets are used for modeling. This tool was adapted to the tele-maintenance distributed case and to integrate distant communications, synchronization problems and execution scheduling. Performance evaluation was performed using Markov Processes and Monte Carlo simulation.

Job shop scheduling (JSS) is a strongly NP-hard problem of combinatorial optimisation and one of the most well known machine scheduling problems. We propose a method based on a genetic algorithm and data mining to resolve this problem. The developed genetic algorithm generates a learning population of good solutions, which are mined by the mean of See5 classifier systems. The mining step produces decision rules which are transformed in to a meta-heuristic allowing the affectation of operations on machines.

This paper introduces the Recurrent Radial Basis Function networks (RRBF) for recognition of simple temporal sequence. The RRBF combines features from recurrent neural network and Radial Basis Function networks (RBF). An application has been developed by implementation on the IBM/ZISC (Zero Instruction Set Computer) card for temporal sequences recognition.

We study a job shop scheduling problem using a Genetic Algorithm and Data Mining. The developed Genetic Algorithm generates a learning population of optimal solutions. We used C4/5 decision tree to mind the population. Some decision rules finding for each machine an affectation order of operations were finally induced.

Stochastic Petri nets are a powerful tool for performance evaluation of concurrent systems like parallel computing, communication network and production systems. In many practical applications, performance evaluation using this model is very difficult because of the great dimension of the marking space. We present a graphical method for the reduction of stochastic Petri nets, applied to safe production system modeling. The approach is based on the principle of places interactivity in the model and the function of transition firing rates. The reduction of the Petri net is applied directly to the graphical model after a simple analysis of places efficiency by using mathematical techniques of birth-death processes. Thus, the problem of the model dimension is solved since our method is independent of the marking graph.

The purpose of the paper is to consider some special types of Petri nets, introduced by Lien (1976), and to propose a complete and unified approach for the study of their structural properties by using techniques of linear algebra of matrices. We distinguish four subclasses: forward-conflict-free, backward-conflict-free, forward-concurrent-free, and backward-concurrent-free Petri nets. A modification of the classical incidence matrix results in a square matrix, called a modified incidence matrix, with nonpositive (nonnegative) off-diagonal elements when backward-(forward-) conflict-free or concurrent-free Petri nets are considered. The modified incidence matrix eigenvalues are computed and theorems on matrices of this type are used to prove several sufficient and/or necessary conditions for structural boundedness, liveness, repetitiveness, conservativeness, and consistency of these four subclasses of Petri nets.

In this paper we are interested in a mathematical description of the class of hybrid systems, which can be modelled by hybrid Petri nets (HPN). A state space model using the conventional algebra for the continuous subsystem and the minplus algebra for the discrete subsystem is given, This model is used in the analysis of a special class of HPN.

In this paper we are interested in a mathematical description of the class of hybrid systems, which can be modeled by hybrid Petri nets HPNs . A state space model using conventional algebra for the continuous subsystem and minplus algebra for the discrete subsystem is given. The interface between the discrete and continuous part appears in our model. This model is illustrated through an example. We introduce the concept of discrete control of HPNs.

In this work, we propose a methodology to study the control of a class of discrete event systems, more precisely of opened manufacturing lines. Our contribution is based on the variable speeds and the controlled maximal speeds continuous Petri net model. The computation of the control is made using standard linear programming methods. A study leading to a characterisation of the domain of accessible markings is presented.

In this paper, a method for the calculation of the transfer function of timed event graphs is developed. The transfer function has the form of a formal series in one or two formal variables. The knowledge of this formal series plays a crucial role in the max-plus system theory. It is very important in the investigation of the problem of controllability of a system, because it provides us with the input-output relation as an equation in the dioid of formal series. The direct approach to this calculation requires the computation of a star operation of a matrix with entries in this dioid,which seems to be difficult for matrices of large dimensions. For this reason, we propose another approach using only the max-plus modeling of a system, that is, the dater's description. We have considered the multivariable case with several input and output transitions. We can obtain the transfer function as the output corresponding to a special control. This control follows from the dater's description of the zero and identity elements of this dioid. To illustrate this approach we give two examples. The purpose of the first one is to compare different approaches to this problem; the second one may serve as an application. We use our approach to determine the transfer function of a manufacturing system.

In this paper, a method for the calculation of transfer function of timed event graphs is developed. The transfer function has form of a formal series in one or two formal variables. The knowledge of this formal series plays a crucial role in the max-plus system theory. Namely, it is very important in the investigation of the problem of controllability of a system, because it provides the input-output relation as an equation in the dioid of formal series. The direct approach to this calculation requires the computation of a star operation of a matrix with entries in this dioid which seems to be difficult for matrices of large dimensions. This is why the authors propose another approach using only the max-plus modelling of a system. The authors have considered the multivariable case with several input and output transitions. The authors can obtain the transfer function as the output corresponding to a special control. To illustrate this approach the authors give two examples. The purpose of the first one is to compare different approaches to this problem, whereas the second one may serve as an application. The authors use their approach to determine the transfer function of a manufacturing system.

In this article, finite nonautonomous discrete event systems are studied in the dioid of formal series. If such a system can be described by a timed event Petri net, a max algebra model of the system can be written. Furthermore, the input-output relation of the nonautonomous system appears in the dioid of formal series. The greatest subsolution of a system of linear equations in the dioid of formal series is found. This subsolution can be used to calculate the latest admissible input of the nonautonomous discrete event system, such that the output of the system is achieved. The application of the theory to an assembly line is given.

Our work concerns the adaptation of the singular perturbation method jointly to the homographic transformation to the category of ergodic Markov chains which presents the two-time-scale property. For Markov chains, the two-time-scale property becomes a property of two-weighting-scale of the states in the system evolution. This lead us to call the slow and fast parts of a decomposed system strong and respectively weak. The limit resolution methodology of the Markov chains by the method of singular perturbation assumes firstly the detection of the irreducible classes of the chain, and secondly, the decomposition of each final ergodic classes presenting the two-weighting-scale property. In the resolution at the limit of the decomposed system, we struck the problem of the stochasticity of the subsystems obtained using directly the singular perturbation method. Indeed, the strong and weak submatrix are not stochastic matrix. In our method, we use the homographic transformation in order to make stochastic the strong part matrix.>

Continuous Petri nets (PNs) are presented, and continuous PN behavior is compared with the referential model (timed PN). A criterion is proposed to evaluate the approximation given by the continuous PN. The validity of this model is justified by comparison with the referential model (timed PN). This comparison permits the validation of the continuous model. The application of this tool for studying the dynamic behavior of manufacturing lines is discussed. Some properties of such lines are pointed out and then demonstrated using the analytical capacities of the continuous PN.>