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Maxplus algebra and system theory: Where we are and where to go now
 Annu. Rev. Control
, 1999
"... Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison ..."
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Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to systemtheoretic and controlsynthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the maxplus algebra and their use for system theory is proposed in the last part of the paper. Résumé: Plus de seize ans après le début d’une théorie linéaire de certains systèmes à événements discrets dans laquelle l’algèbre maxplus et autres outils algébriques assimilés jouent un rôle central, ce papier cherche àdécrire quelques uns des principaux résultats obtenus de façon informelle, en s’appuyant sur des exemples. Par comparaison avec la théorie classique des systèmes linéaires, il existe des domaines pratiquement vierges, surtout en raison du fait que les outils mathématiques correspondants restent à forger. C’est en particulier le cas de l’approche géométrique des systèmes qui, dans la théorie classique, est connue pour apporter un autre regard important sur les questions de théorie des systèmes et de synthèse de lois de commandes àcôté de la machinerie purement algébrique. Une discussion préliminaire sur les aspects géométriques de l’algèbre maxplus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.
Algebraic System Analysis of Timed Petri Nets
, 1997
"... We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (inputoutput) properti ..."
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Cited by 24 (7 self)
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We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (inputoutput) properties, algebraic representations, asymptotic regime. A particular attention is paid to the subclass of stable systems (with asymptotic linear growth). 1 Introduction The fact that a subclass of Discrete Event Systems equations write linearly in the (min,+) or in the (max,+) semiring is now almost classical [9, 2]. The (min,+) linearity allows the presence of synchronization and saturation features but unfortunately prohibits the modeling of many interesting phenomena such as "birth" and "death" processes (multiplication of tokens) and concurrency. The purpose of this paper is to show that after some simplifications, these additional features can be represented by polynomial recurrences in the ...
Interval Analysis and Dioid: Application to Robust Controller Design for Timed Event Graphs ⋆
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On visualization scaling, subeigenvectors and Kleene stars in max algebra
 Linear Algebra Appl
"... The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given n ..."
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The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X −1 AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.
cycle time and plant control using dioid algebra, Chapter 6
 in Supply Chain Optimisation, Series Applied Optimization, A. Dolgui J. Soldek
, 2004
"... Abstract: Using an industrial process from the car sector, we show how dioid algebra may be used for the performance evaluation, sizing, and control of this discreteevent dynamic system. Based on a Petri net model as an event graph, maxplus algebra and minplus algebra permit to write linear equat ..."
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Cited by 4 (3 self)
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Abstract: Using an industrial process from the car sector, we show how dioid algebra may be used for the performance evaluation, sizing, and control of this discreteevent dynamic system. Based on a Petri net model as an event graph, maxplus algebra and minplus algebra permit to write linear equations of the behavior. From this formalism, the cycle time is determined and an optimal sizing is characterized for a required cyclic behavior. Finally, a strict temporal constraint the system is subject to is reformulated in terms of inequalities that the (min, +) system should satisfy, and a control law is designed so that the controlled system satisfies the constraint.
The Analytic Hierarchy Process, Max Algebra and Multiobjective Optimisation. arXiv:1207.6572v1
, 2012
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CYCLE TIME OF PTIME EVENT GRAPHS
"... Abstract: The dater equalities constitutes an appropriate tool which allows a linear description of Timed Event Graphs in the field of (max, +) algebra. This paper proposes an equivalent model in the usual algebra which can describe Timed and Ptime Event Graphs. Considering 1periodic behavior, the ..."
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Abstract: The dater equalities constitutes an appropriate tool which allows a linear description of Timed Event Graphs in the field of (max, +) algebra. This paper proposes an equivalent model in the usual algebra which can describe Timed and Ptime Event Graphs. Considering 1periodic behavior, the application of a variant of Farkas’ lemma allows the determination of upper and lower bounds of the production rate and necessary conditions of consistency. 1
On Stabilization of MinMax Systems
"... This note studies minmax systems which are dynamic systems involving three operations (min,max,+). We present a feedback stabilization policy such that the closedloop systems have a global cycle time which is the same as the maximal Lyapunov exponent of the openloop systems for a class of min ..."
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This note studies minmax systems which are dynamic systems involving three operations (min,max,+). We present a feedback stabilization policy such that the closedloop systems have a global cycle time which is the same as the maximal Lyapunov exponent of the openloop systems for a class of minmax systems. Our results are based on structural properties of minmax systems.
Timed Event Graphs with variable resources: asymptotic behavior, representation in (min,+) algebra
, 2013
"... ABSTRACT. Our aim is to demonstrate that the approach developed for Timed Event Graphs over ��algebra may be extended to a broader subclass of Petri Nets. The graphs considered can be seen as Timed Event Graphs on which some source and/or sink transitions are added to some places. Elements of perfor ..."
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ABSTRACT. Our aim is to demonstrate that the approach developed for Timed Event Graphs over ��algebra may be extended to a broader subclass of Petri Nets. The graphs considered can be seen as Timed Event Graphs on which some source and/or sink transitions are added to some places. Elements of performance evaluation and the linear representation of these systems over the (min,+) algebra (state model with variable parameters and inputoutput relationship) are proposed. RÉSUMÉ. Notre but est de montrer que l’approche développée dans l’algèbre��pour les graphes d’événements temporisés peut s’étendre à une sousclasse plus large de réseaux de Petri. Les graphes considérés peuvent être vus comme des graphes d’événements temporisés sur lesquels des transitions source et/ou puits sont adjointes à certaines places. Des éléments d’évaluation de performance, et la représentation linéaire de ces graphes dans l’algèbre (min,+) (modèle d’état à paramètres variables et relation entréesortie) sont donnés.
Networked Conflicting Timed Event Graphs Representation in (Max,+) Algebra
, 2013
"... Abstract: Timed Event Graphs (TEGs) are a specific class of Petri nets that have been thoroughly studied given their useful linear state representation in (Max,+) algebra. Unfortunately, TEGs are generally not suitable for modeling systems displaying resources sharing (or conflicts). In this paper, ..."
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Abstract: Timed Event Graphs (TEGs) are a specific class of Petri nets that have been thoroughly studied given their useful linear state representation in (Max,+) algebra. Unfortunately, TEGs are generally not suitable for modeling systems displaying resources sharing (or conflicts). In this paper, we show that if a system with conflicts is modeled using a NCTEG (Networked Conflicting Timed Event Graphs), it is quite possible to obtain an equivalent (Max,+) representation. More precisely, we prove that the evolution of a NCTEG satisfies linear timevarying (Max,+) equations. In case of cyclic NCTEGs, which are a natural model of many repetitive systems, we provide a standard timeinvariant (Max,+) representation. As an application of the proposed approach to exhibit its interest, we consider the case of Jobshops. We first propose a generic NCTEGbased model of these systems and subsequently apply the corresponding (Max,+) representation to evaluate some of their performances.