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19
Synchronization and linearity: an algebra for discrete event systems
, 2001
"... The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific ..."
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Cited by 250 (10 self)
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The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX crossreferences are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The
Methods and Applications of (max,+) Linear Algebra
 STACS'97, NUMBER 1200 IN LNCS, LUBECK
, 1997
"... Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; g ..."
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Cited by 73 (26 self)
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Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, HamiltonJacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities) . Despite this apparent profusion, there is a small set of common, nonnaive, basic results and problems, in general not known outside the (max, +) community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of (max, +) results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of...
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 42 (18 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.
Duality and separation theorems in idempotent semimodules
 Linear Algebra and its Applications 379 (2004), 395–422. Also arXiv:math.FA/0212294
"... Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to sep ..."
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Cited by 35 (19 self)
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Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and halfspaces over the maxplus semiring. 1.
The Minkowski Theorem for Maxplus Convex Sets
, 2006
"... We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones a ..."
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Cited by 22 (11 self)
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We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones and closed unbounded maxplus convex sets. In particular, we show that a closed maxplus convex set can be decomposed as a maxplus sum of its recession cone and of the maxplus convex hull of its extreme points.
Generators, Extremals and Bases of Max Cones
, 2006
"... We give simple algebraic proofs of results on generators and bases of max cones, some of which are known. We show that every generating set S for a cone in max algebra can be partitioned into two parts: the independent set of extremals E in the cone and a set F every member of which is redundant in ..."
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Cited by 15 (6 self)
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We give simple algebraic proofs of results on generators and bases of max cones, some of which are known. We show that every generating set S for a cone in max algebra can be partitioned into two parts: the independent set of extremals E in the cone and a set F every member of which is redundant in S. We exploit the result that extremals are minimal elements under suitable scalings of vectors. We also give an algorithm for finding the (essentially unique) basis of a finitely generated cone.
Kernels, Images And Projections In Dioids
 PROCEEDINGS OF WODES’96
, 1996
"... We consider the projection problem for linear spaces and operators over dioids such as the (max, +) semiring. We give existence and uniqueness conditions for the projection onto the image of an operator, parallel to the kernel of another one, together with an explicit formula for the projector. Th ..."
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Cited by 15 (12 self)
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We consider the projection problem for linear spaces and operators over dioids such as the (max, +) semiring. We give existence and uniqueness conditions for the projection onto the image of an operator, parallel to the kernel of another one, together with an explicit formula for the projector. The theory is not limited to linear operators: the result holds more generally for residuated operators over complete dioids. Illustrative examples are provided.
Supervisory Control of Realtime Discrete Event Systems Using Lattice Theory
 In Proceedings of the 33rd IEEE Conference on Decision and Control
, 1994
"... The behavior of timed DES can be described by sequences of event occurrence times. These sequences can be ordered to form a lattice. Since logical (untimed) DES behaviors described by regular languages also form a lattice, questions of controllability for timed DES may be treated in much the same ma ..."
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Cited by 11 (4 self)
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The behavior of timed DES can be described by sequences of event occurrence times. These sequences can be ordered to form a lattice. Since logical (untimed) DES behaviors described by regular languages also form a lattice, questions of controllability for timed DES may be treated in much the same manner as they are for untimed systems. In this paper we establish conditions for the controllability of timed DES performance specifications which are expressed as inequations on the lattice of sequences. These specifications may take the form of sets of acceptable event occurrence times, maximum or minimum occurrence times, or limits on the separation times between events. Optimal behaviors are found as extremal solutions to these inequations using fixed point results for lattices. Keywords: Discrete event systems, supervisory control, maxalgebra, lattices. I. Introduction Discrete event systems (DES) are characterized by a collection of events, such as the completion of a job in a manufac...
HahnBanach Separation Theorem for MaxPlus Semimodules
"... We introduce maxplus analogues of basic Euclidian geometry notions: scalar product is replaced by a scalar division, and the associated distance is essentially Hilbert's projective distance. We introduce an orthogonal projection and prove a HahnBanach type theorem: a point can be separated from a ..."
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Cited by 7 (6 self)
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We introduce maxplus analogues of basic Euclidian geometry notions: scalar product is replaced by a scalar division, and the associated distance is essentially Hilbert's projective distance. We introduce an orthogonal projection and prove a HahnBanach type theorem: a point can be separated from a semimodule by a hyperplane orthogonal to the direction of projection. We use these results to separate maxplus convex sets, and illustrate this new geometry by twodimensional examples. 1 Introduction In the last twenty years, the maxplus algebra and related structures (called "dioids" or idempotent semirings) have emerged as a natural setting for various areas of applied mathematics and mathematical models (e.g. the socalled "discrete event systems" area [1], optimization theory [13], etc.). As vector spaces are built up from fields of scalars, like C or R, semimodules with an idempotent addition can be built up from the maxplus semiring. Such algebraic structures share several common f...