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.

Abstract—This paper formally models and studies latencyinsensitive systems (LISs) through max-plus algebra. We introduce state traces to model behaviors of LISs and obtain a formally proved performance upper bound achievable by latencyinsensitive design. An implementation of the latency-insensitive protocol that can provide robust communication through backpressure is also proposed. The intrinsic performance of the proposed implementation is acquired based on state traces. It is also proved that the proposed implementation can always reach the best performance achievable by latency-insensitive design. Index Terms—Back-pressure, latency-insensitive system, maxplus algebra, performance analysis, state trace.

Piecewise linear (PL) systems provide one systematic approach to discrete-time hybrid systems. They blend switching mechanisms with classical linear components, and model arbitrary interconnections of finite automata and linear systems. Tools from automata theory, logic, and related areas of computer science and finite mathematics are used in the study of PL systems, in conjunction with linear algebra techniques, all in the context of a "PL algebra" formalism. PL systems are of interest as controllers as well as identification models. Basic questions for any class of systems are those of equivalence, and, in particular, if state spaces are equivalent under a change of variables. This paper studies this state-space equivalence problem for PL systems. The problem was known to be decidable, but its computational complexity was potentially exponential; here it is shown to be solvable in polynomial-time. 1 Introduction Hybrid systems theory has recently become the focus of increased resear...

Abstract The chapter describes the modeling of a material handling system with the production of individual units in a scheduled order. The units represent the agents in the model and are transported in the system which is abstracted as a directed graph. Since the hindrances of units on their path to the destination can lead to inefficiencies in the production, the blockages of units are to be reduced. Therefore, the units operate in the system by means of local interactions in the conveying elements and indirect interactions based on a measure of possible hindrances. If most of the units behave cooperatively (“socially”), the blockings in the system are reduced. A simulation based on the model shows the collective behavior of the units in the system. The transport processes in the simulation can be compared with the processes in a real plant, which gives conclusions about the consequencies for the production based on the superordinate planning. 1