The urgent need for industrial efficient solutions allowing to reduce the environmental footprint of modern manmade systems is nowadays acknowledged by all and becomes a central issue. This reality is all the more true for cyber-physical systems where the integrations of computation, networking, and physical processes are intensively realized. These intelligent systems which combine several computing resources interacting with the physical environment using sensors and actuators are known to be energy-intensive. So, the management of the energy becomes one of the major concerns driven by the increasing number of connected systems.This paper addresses the challenge of estimating least-cost firing sequences within P-Time Labeled Petri Net systems, incorporating both energy considerations and temporal constraints into the modeling framework. By extending the existing models with a dynamic programming algorithm, the proposed method calculates optimal sequences that minimize energy consumption for a given set of actions over time. An industrial example is presented to illustrate the practical application of the approach, demonstrating the potential for significant energy savings and efficiency improvements in industrial systems.

Energy considerations become a critical issue for modern man-made systems and the need for efficient ecoresponsible solutions ensuring energy savings is crucial. This paper develops an energy aware method allowing to optimize the energy consumption of data centers systems thanks to the introduction of Inhibitor P-Time Petri nets (IP-TPN) and Deep Reinforcement Learning techniques. Indeed, thanks to a schedulability analysis method and being given an energy cost function, the global energy consumed for a particular behavior of the system considered can be computed. Furthermore, Petri nets are a good framework for deep reinforcement learning, on the one hand because the cost function we introduce in this paper will naturally train an agent to minimize the total cost according to random inputs, and on the other hand because the observation space containing the tokens as well as their age in the square is high dimensional, which makes traditional algorithms less efficient than deep reinforcement learning.

This paper deals with the problem of estimating the Minimum Initial Marking (MIM) of Labeled Petri Nets (L-PN). By the observation of a sequence of labels, we determine the set of possible MIMs related to a given L-PN through an approach based on GRASP (Greedy Randomized Adaptive Search Procedure) inspired method – GMIM. The objective is to get the maximum of feasible MIMs by exploring the search space and giving best solutions for real time cyber systems in short time. We consider four basic assumptions during the reasoning: (i) the L-PN structure is known; (ii) for each transition of L-PN, a label is associated, (iii) the label sequence is known, and (iv) all transitions of L-PN are observable. We show the validity and efficiency of our approach by applying the proposed GMIM metaheuristic to two validation examples: Initialization of two parallel machines (example widely cited in literature) and resources allocation in a monitoring problem via mobile robot network.

This work develops a technique for estimating the minimum initial marking (an initial marking with the minimum total token number) of a real-time system modelled by a P-Time labelled Petri net system under partial observation. Indeed, the set of events is partitioned into a set of observable events, which can be detected by an external agent and a set of unobservable ones. Furthermore, some observable events can be associated to the same observation, i.e., they produce the same output signal and are called undistinguishable. In addition, the obtained marking must be consistent with a given label occurrence vector (LOV). The proposed method is based on an iterative procedure combined with a schedulability analysis technique for particular behaviors of the studied system.

This paper deals with a state estimation technique in a decentralized context for discrete events dynamic systems modeled by P-Time Petri nets (P-TPN). Indeed, the observation is distributed over a set of distinct sites, each of which has its own local view of the system. Thus, thanks to a coordinator the local information transmitted by the different sites will be used to determine the set of states consistent with the current observation. The proposed method relies on an iterative procedure and the schedulability analysis of particular firing sequences called time explanations.

This paper focuses on the decentralized diagnosis problem of discrete event systems modeled by P-time Petri net with partial information. Indeed, the observation is realized via a set of local sites which can communicate and transmit information to a global coordinator. The proposed method is based on a state observer synthesis and a schedulability analysis technique of particular occurrence sequences. A procedure allowing to determine the diagnosis state of the firing sequences consistent with the current observation is provided and the diagnosability issue is also discussed.

The aim of the paper is the estimation of sequences in Timed Petri nets. We propose a general strategy composed of two phases: The first phase considers the logical aspect only and suggests candidate count vectors where the second one checks the existence of a relevant time sequence for a Timed Petri net and generate a subspace of time sequences for a given candidate vector.

This paper presents a diagnosability technique of a P-time Petri net under partial observation. Indeed, the set of transitions is partitioned into those labeled with the empty string ϵ called silent (as their firing cannot be detected) including the faulty transitions and the observable ones. The diagnosability can be defined as the ability to detect the type of a failure within a finite number of steps after its occurrence - the system is then said to be diagnosable. The proposed approach is based on the synthesis of a modified state observer where the fault transitions are considered as observable allowing the construction of a Sampath-like diagnoser. The novelty of the developed approach resides in the fact that, although the time factor is considered as intervals, the diagnoser is computed thanks to the underlying untimed Petri net structure of the P-time model considered. Furthermore, the method relies on linear programming techniques and the schedulability analysis of particular firing sequences exhibited by the analysis of the obtained diagnoser and does not require the construction of the state class graph.

This paper focuses on the problem of estimating the marking of an unlabeled P-time Petri net with partial information modeled by unobservable transitions. The proposed approach is based on a novel state observer synthesis method under partial observation. A procedure that, given a sequence of observable transitions with their firing time instants, allows to determine the set of markings consistent with the considered observation is given. The method relies on the feasibility in time (called schedulability) of particular firing sequences, namely the candidates firing sequences. Moreover, although time is taken into consideration, the proposed technique is not hampered by the state space explosion problem as it relies on the underlying untimed structure of the P-time model considered-the building of the state class graph is not necessary.

This paper focuses on the problem of estimating the marking of a P-time labeled Petri net with partial information. The set of transitions is partitioned into those labeled with the empty string e called silent, as their firing cannot be detected, and the observable ones. The proposed approach is based on the observation of transitions labels thanks to the synthesis of a state observer obtained from the behavior of the untimed underlying Petri net. A procedure that, given an observed word (sequence of labels defined over a given alphabet) with their occurrence dates, allows to determine the set of states consistent with the considered observation is provided, under the assumption of non-Zenoness of the subnet induced by the set of unobservable transitions. Furthermore, a source of nondeterminism is added as the case of indistinguishable transitions (transitions sharing the same label) is also considered. The method relies on the schedulability analysis of particular firing sequences, namely the candidates firing sequences and does not require the building of the state classes graph although the time is modeled as intervals.

The aim of this paper is to reconstruct the least/greatest sequence of unobservable transitions in timed Petri nets based on the online observation of firing occurrences of some transitions on a sliding horizon. The Petri net, which can be unbounded and can contain self-loops and circuits, is described under an algebraic form composed of A.x ≤ b which expresses the possible time sequence x and the fundamental marking relation. Under the assumption of Backward/Forward Conflict Freeness of the unobservable-induced subnet, we show the existence of a finite least/greatest sequence with respect to the data known on a given horizon. A technique of computation using linear programming is given.

This paper deals with a novel state observer synthesis method of real-time systems under partial observation. Indeed, the developed approach aims at estimating the marking of a P-time Petri net system based on the observation of the occurrence of particular transitions called observable transitions. As opposed to many existing approaches the application of the method does not require strong assumptions on the structure of the model considered. In addition, although the time factor is taken into consideration, the proposed technique is not hampered by the state space explosion problem as it relies on the underlying untimed structure of the P-time model studied - the building of the state class graph is not necessary.

Petri nets are a powerful formalism for the specification and verification of concurrent systems, such as sequential systems and manufacturing systems where many processes can compete for limited and constrained resources. To deal with systems whose time issues become essential, different extensions of Petri nets with time have been proposed in the literature, each one being dependent on the application considered. In this paper, a new analysis technique for P-time Petri nets is proposed. It consists of simple operations on time intervals to determine a superset of behaviors of the modelled system. Then, once a desired behavior is extracted, thanks to a linear programming problem, its feasibility is verified via the existence of a solution.

Petri nets are a powerful formalism for the specification and verification of concurrent systems including sequential systems and manufacturing systems. To deal with systems whose time issues become fundamental, different time Petri nets extensions have been developed in the literature, each one being dependent on the application considered. For the time critical systems their correctness depends not only on the logic correctness but also on the time constraints. In this paper, a new enumerative and control technique for P-time Petri nets is proposed. It is based on the firing instant notion and a graph constraints theoretic approach.

The maintenance of systems involved in production processes must be carried out with maximum efficiency and minimum downtime, especially on time-critical systems (operation times are included between a minimum and a maximum value). This paper presents a proactive/reactive control to fit preventive maintenance into a busy production schedule. As preventive maintenance is regarded as a non-preemptive task, p-time Petri net is used as modelling tool to generate temporal and logical constraints to be considered. To update the tasks scheduling, flexibility on time and on task order is obtained by permitting negative instants in the used firing instant approach. The illustration of the method is made on a graphite ceramic cell

Time-critical systems are characterized by operation times included between a minimum and a maximum value. A transgression of these specifications acts again the quality of service. In this paper, a dynamic robust control is proposed to cancel as quickly as possible the effects of a delay or an advance on a planned task. P-time Petri Net is used as modeling tool and the control is based on an extension of the firing instant approach permitting negative instants to update the tasks scheduling. The illustration of the method is made on a graphite ceramic cell.

A production system can be seldom considered in isolation or only in its nominal functioning mode. Indeed, due to its environment or its parameters drifts, it can be subject to a various kind of disturbances. The disturbances considered are internal ones (operation duration variations due to parameter drifts) or external ones (fluctuation on the input flow of the parts). The problem addresses in this paper consists of the control of time-critical systems where strict deadlines must be satisfied. Thus, in a disturbed environment preserving the whole system specifications requires a robust control. A formal method, using p-time Petri nets as modeling tool, based on an actualization procedure of particular firing instants of the model transitions is proposed. This approach is then applied to an electroplating line where severe requirements on the quality of the whole process are imposed.

Time-critical systems are characterized by operation times included between a minimum and a maximum value. An acceptable behaviour depends not only on the logical correctness of the task sequence but also on the time at which the operations are performed. So, a specific control is required. In this paper, the robustness property is used to build a time schedule of when the manufacturing system is to process several kinds of parts. This approach uses a p-time Petri net as its modeling tool and considers the multi-processing case as a disturbed mono-processing one. To make a more succinct and manageable description of the obtained control, p-time coloured Petri nets are defined. A typical illustration of an electroplating line is given.