Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which max-plus 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 system-theoretic and control-synthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the max-plus 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 max-plus 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 max-plus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.

this paper is obviously too short for such a program, we have chosen to propose an introductive guided tour. A more detailed exposition will be found in our references and in a more complete paper to appear elsewhere. 1 Rational Series in a Single Indeterminate

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 (input-output) 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 ...

informs doi 10.1287/moor.1090.0434

We consider a system of uniform recurrence equations (URE) of dimension one. We show how its computation can be carried out using minimal memory size with several synchronous processors. This result is then applied to register minimization for digital circuits and parallel computation of task graphs.

Model predictive control (MPC) is a very popular controller design method in the process industry. Usually MPC uses linear discrete-time models. In this paper we extend MPC to a class of discrete-event systems with both hard and soft synchronization constraints. Typical examples of such systems are railway networks, subway networks, and other logistic operations. In general the MPC control design problem for these systems leads to a nonlinear non-convex optimization problem. We also show that the optimal MPC strategy can be computed using an extended linear complementarlty problem.

We will consider the modeling and analysis of public transportation networks which evolve according to a timetable. Some results are summarized. A way to control these networks is introduced. 1 Introduction Transportation networks are examples of what are known as Discrete Event Systems (DES). The evolution of these systems is determined by the occurrence of certain events. In a transportion network, e.g. a railway network, examples of discrete events are the departure from or arrival at a station of a train. The evolution of a class of DES, viz. those which involve synchronization constraints, can be described by linear models provided that the max-algebra structure is used. In transportation networks such constraints follow from the demand that trains should connect. The max-algebra consists of the real numbers and minus innity together with the operations maximization and addition. For an extensive discussion of the max-algebra and its applications in the modeling of DES we refer...

Q uantitative M odeling I n P arallel S ystems

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