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24
Hybrid Petri nets
 European Control Conference Grenoble
, 1991
"... Abstract. Petri nets (PNs) are widely used to model discrete event dynamic systems (computer systems, manufacturing systems, communication systems, etc). Continuous Petri nets (in which the markings are real numbers and the transition firings are continuous) were defined more recently; such a PN ma ..."
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Cited by 139 (2 self)
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Abstract. Petri nets (PNs) are widely used to model discrete event dynamic systems (computer systems, manufacturing systems, communication systems, etc). Continuous Petri nets (in which the markings are real numbers and the transition firings are continuous) were defined more recently; such a PN may model a continuous system or approximate a discrete system. A hybrid Petri net can be obtained if one part is discrete and another part is continuous. This paper is basically a survey of the work of the authors ’ team on hybrid PNs (definition, properties, modeling). In addition, it contains new material such as the definition of extended hybrid PNs and several applications, explanations and comments about the timings in Petri nets, more on the conflict resolution in hybrid PNs, and connection between hybrid PNs and hybrid automata. The paper is illustrated by many examples.
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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Cited by 71 (19 self)
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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.
Modeling and Analysis of Timed Petri Nets Using Heaps of Pieces
, 1997
"... We show that safe timed Petri nets can be represented by special automata over the (max,+) semiring, which compute the height of heaps of pieces. This extends to the timed case the classical representation a la Mazurkievicz of the behavior of safe Petri nets by trace monoids and trace languages. Fo ..."
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Cited by 56 (18 self)
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We show that safe timed Petri nets can be represented by special automata over the (max,+) semiring, which compute the height of heaps of pieces. This extends to the timed case the classical representation a la Mazurkievicz of the behavior of safe Petri nets by trace monoids and trace languages. For a subclass including all safe Free Choice Petri nets, we obtain reduced heap realizations using structural properties of the net (covering by safe state machine components). We illustrate the heapbased modeling by the typical case of safe jobshops. For a periodic schedule, we obtain a heapbased throughput formula, which is simpler to compute than its traditional timed event graph version, particularly if one is interested in the successive evaluation of a large number of possible schedules. Keywords Timed Petri nets, automata with multiplicities, heaps of pieces, (max,+) semiring, scheduling. I. Introduction The purpose of this paper 1 is to prove the following result: Timed safe Pe...
A spectral theorem for convex monotone homogeneous maps
 In Proceedings of the Satellite Workshop on MaxPlus Algebras, IFAC SSSC’01
, 2001
"... Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point ..."
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Cited by 27 (15 self)
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Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex infsubsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group
A nonlinear hierarchy for discrete event dynamical systems
, 1998
"... Dynamical systems of monotone homogeneous functions appear in Markov decision theory, in discrete event systems and in PerronFrobenius theory. We consider the case when these functions are given by finite algebraic expressions involving the operations max, min, convex hull, translations, and an inf ..."
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Cited by 21 (8 self)
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Dynamical systems of monotone homogeneous functions appear in Markov decision theory, in discrete event systems and in PerronFrobenius theory. We consider the case when these functions are given by finite algebraic expressions involving the operations max, min, convex hull, translations, and an infinite family of binary operations, of which max and min are limit cases. We set up a hierarchy of monotone homogeneous functions that reflects the complexity of their defining algebraic expressions. For two classes of this hierarchy, we show that the trajectories of the corresponding dynamical systems admit a linear growth rate (cycle time). The first class generalizes the minmax functions considered previously in the literature. The second class generalizes both maxplus linear maps and ordinary nonnegative linear maps.
Fuzzy hideal of hemirings
 Inform. Sci
, 2007
"... Abstract: We introduce the notion of intuitionistic fuzzy (left) hideals of hemirings and investigate their properties connected with the corresponding level subsets. Methods of constructions of such intuitionistic fuzzy ideals from given sequences of left hideals of a hemiring R are presented. So ..."
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Cited by 21 (5 self)
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Abstract: We introduce the notion of intuitionistic fuzzy (left) hideals of hemirings and investigate their properties connected with the corresponding level subsets. Methods of constructions of such intuitionistic fuzzy ideals from given sequences of left hideals of a hemiring R are presented. Some natural classification of such intuitionistic fuzzy hideals is given. Key–Words: Hemiring, fuzzy set, intuitionistic fuzzy left hideal, descending chain. 1
Asymptotic Throughput of Continuous Timed Petri Nets
, 1995
"... We set up a connection between Continuous Timed Petri Nets (the fluid version of usual Timed Petri Nets) and Markov decision processes. We characterize the subclass of Continuous Timed Petri Nets corresponding to undiscounted average cost structure. This subclass satisfies consetration laws and show ..."
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Cited by 19 (6 self)
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We set up a connection between Continuous Timed Petri Nets (the fluid version of usual Timed Petri Nets) and Markov decision processes. We characterize the subclass of Continuous Timed Petri Nets corresponding to undiscounted average cost structure. This subclass satisfies consetration laws and shows a linear growth: one obtains as mere application of existing results for Dynamic Programming the existence of an asymptotic throughput. This rate can be computed using Howardtype 'algorithms, or by an extension of the well known cycle time formula for timed event graphs. We present an illustrating example and briefly sketch the relation with the discrete case.
On fluidization of discrete event models: observation and control of continuous Petri nets
, 2011
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Optimal Control of (Min,+) Linear TimeVarying Systems, Petri Nets and Performance Models
 Proceedings of PNPM’99
, 1999
"... The class of discrete event dynamic systems involving only synchronization phenomena can be seen as linear timeinvariant systems in a particular algebraic structure called (�, ) algebra. In the same framework, this paper deals with linear timevarying systems, that is, systems whose parameters may ..."
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Cited by 5 (5 self)
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The class of discrete event dynamic systems involving only synchronization phenomena can be seen as linear timeinvariant systems in a particular algebraic structure called (�, ) algebra. In the same framework, this paper deals with linear timevarying systems, that is, systems whose parameters may change as functions of time. For example, in a manufacturing system the number of working machines, or the number of trains running in a closed network of railway connections, can vary as functions of time. For such systems, the output tracking problem is optimally solved under justintime criterion. 1.