Modelling, analysis and control of electroplating line modelled by P-time event graphs

Becha, T.; Kara, Redouane; Dutilleul, Simon Collart; Loiseau, Jean Jacques

doi:10.3182/20130911-3-br-3021.00027

Cited by 15 publications

(11 citation statements)

References 7 publications

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“…Events that do not occur in the specified time window correspond, in the real system modeled by the P-TEG, to failure of meeting time specifications. P-TEGs have been applied for modeling and analyzing time schedules in manufacturing systems such as electroplating lines, baking processes, cluster tools [2], [17], [7], [13].…”

Section: Introductionmentioning

confidence: 99%

“…In [19], we introduced a stronger property than consistency that is even more desirable for P-TEGs applications; a P-TEG is boundedly consistent if it admits at least one consistent trajectory such that the delay between the k th firings of every pair of transitions is bounded for all k. Moreover, we proved that a P-TEG is boundedly consistent if and only if there exists at least one 1-periodic trajectory that is consistent for the P-TEG. Since the problem of checking the existence of 1-periodic trajectories can be solved using linear programming [2], [6], bounded consistency can be checked in weakly polynomial time (on the contrary, no algorithm that checks standard consistency has been found yet). However, the use of linear programming for finding 1-periodic trajectories implies two issues that we will try to overcome in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Periodic trajectories in P-time event graphs and the non-positive circuit weight problem

Zorzenon,

Komenda,

Raisch

2021

Preprint

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P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers d, the existence of consistent d-periodic trajectories for a given P-TEG. Let us consider a different, seemingly unrelated problem in graph theory: given three arbitrary square matrices P, I and C with elements in R∪{−∞}, find all real values of parameter λ such that the parametric directed graph having arcs weights of the form w(( j, i)) = max(P i j + λ , I i j − λ ,C i j ) (for all arcs ( j, i)) does not contain circuits with positive weight. In a related paper, we have proposed a strongly polynomial algorithm that solves the latter problem faster than other algorithms reported in literature. In the present paper, we show that the first problem can be formulated as an instance of the second; consequently, we prove that the same algorithm can be used to find d-periodic trajectories in P-TEGs faster than with previous approaches.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Periodic trajectories in P-time event graphs and the non-positive circuit weight problem

Zorzenon,

Komenda,

Raisch

2021

Preprint

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“…In [26], it has been shown that, for all d ∈ N, a P-TEG with initially 0 or 1 token per place, characterized by four square matrices A 0 , A 1 , B 0 , B 1 , admits consistent d-periodic trajectories of period λ if and only if the precedence graph G(λB 1 ⊕ λ −1 (A 1 ⊕ E ⊗ ) ⊕ (A 0 ⊕ B 0 )) does not contain circuits with positive weight. The problem of finding the periods of all admissible 1-periodic trajectories has been studied in [2,8,18] but an explicit formula for the admissible periods has not yet been found. Moreover, in [25], we proved that a P-TEG is boundedly consistent (i.e., there exists a consistent trajectory for the P-TEG in which the delay of the k-th firing of every pair of transitions is bounded for all k) if and only if it admits a 1-periodic trajectory.…”

Section: Introductionmentioning

confidence: 99%

The non-positive circuit weight problem in parametric graphs: a solution based on dioid theory

Zorzenon¹,

Komenda²,

Raisch³

2021

Preprint

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Let us consider a parametric weighted directed graph in which every arc (j, i) has weight of the form w((j, i)) = max(P ij + λ, I ij − λ, C ij ), where λ is a real parameter and P , I and C are arbitrary square matrices with elements in R ∪ {−∞}. In this paper, we design an algorithm that solves the Non-positive Circuit weight Problem (NCP) on this class of parametric graphs, which consists in finding all values of λ such that the graph does not contain circuits with positive weight. This problem, which generalizes other instances of the NCP previously investigated in the literature, has applications in the consistency analysis of a class of discrete-event systems called P-time event graphs. The proposed algorithm is based on max-plus algebra and formal languages and runs faster than other existing approaches, achieving strongly polynomial time complexity O(n 4 ) (where n is the number of nodes in the graph).

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“…P-time Petri nets are Petri nets with nondeterministic timing of places that have been proven useful in analysing manufacturing systems. For instance, in electroplating industry [23,4], food or chemical industry both upper and lower bounds on the operation times do matter. The operation times are not given exactly, but only by their lower and upper bounds.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, linear programming techniques based on classical linear algebra have been used in the cycle time analysis of P-time event graphs in [4] and in [9]. The approach based on classical linear algebra has been successfully applied to the computation of extremal cycle times, namely 1-periodic trajectories, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Analysis of P-time Event Graphs in (Max,+) and (Min,+) Algebras

Špaček,

Komenda,

Lahaye

2020

Preprint

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In this work we investigate the behavior of P-time event graphs, a class of time Petri nets with nondeterministic timing of places. Our approach is based on combined linear descriptions in both (max,+) and (min,+) semirings, where lower bounds on the state vector are (max,+)linear and upper bounds are (min,+)-linear. We present necessary and sufficient conditions for the existence of extremal (fastest and slowest) periodic trajectories that are derived from the new description. The results are illustrated by a realistic example of an electroplating process.

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Modelling, analysis and control of electroplating line modelled by P-time event graphs

Cited by 15 publications

References 7 publications

Periodic trajectories in P-time event graphs and the non-positive circuit weight problem

Periodic trajectories in P-time event graphs and the non-positive circuit weight problem

The non-positive circuit weight problem in parametric graphs: a solution based on dioid theory

Analysis of P-time Event Graphs in (Max,+) and (Min,+) Algebras

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