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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...
The extended linear complementarity problem and its applications in the maxplus algebra
, 1995
"... ..."
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.
Minimal Realization in the Max Algebra is an Extended Linear Complementarity Problem
 SYSTEMS & CONTROL LETTERS
, 1993
"... In this paper we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an Extended Linear Complementarity Problem (ELCP). We use an algorithm that finds all solutions of an ELCP to find all equivalent minimal state space realizations of a single inpu ..."
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Cited by 29 (26 self)
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In this paper we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an Extended Linear Complementarity Problem (ELCP). We use an algorithm that finds all solutions of an ELCP to find all equivalent minimal state space realizations of a single input single output (SISO) discrete event system. We also give a geometrical description of the set of all minimal realizations of a SISO maxlinear discrete event system.
A Method to Find All Solutions of a System of Multivariate Polynomial Equalities and Inequalities in the Max Algebra
, 1996
"... In this paper we show that finding solutions of a system of multivariate polynomial equalities and inequalities in the max algebra is equivalent to solving an Extended Linear Complementarity Problem. This allows us to find all solutions of such a system of multivariate polynomial equalities and ine ..."
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Cited by 17 (15 self)
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In this paper we show that finding solutions of a system of multivariate polynomial equalities and inequalities in the max algebra is equivalent to solving an Extended Linear Complementarity Problem. This allows us to find all solutions of such a system of multivariate polynomial equalities and inequalities and provides a geometrical insight in the structure of the solution set. We also demonstrate that this enables us to solve many important problems in the max algebra and the maxminplus algebra such as matrix decompositions, construction of matrices with a given characteristic polynomial, state space transformations and the (minimal) state space realization problem.
Rational Series over Dioids and Discrete Event Systems
 In Proc. of the 11th Conf. on Anal. and Opt. of Systems: Discrete Event Systems, number 199 in Lect. Notes. in Control and Inf. Sci, Sophia Antipolis
, 1994
"... 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 ..."
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Cited by 16 (6 self)
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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
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 14 (6 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 ...
Finding All Essential Terms Of A Characteristic Maxpolynomial
, 2002
"... Let us denote a b = max(a; b) and b = a + b for a; b 2 R = R [ f1g and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n ) algorithm for nding all essential terms of the maxalgebraic characteristic polynomial of an n n matrix o ..."
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Cited by 11 (4 self)
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Let us denote a b = max(a; b) and b = a + b for a; b 2 R = R [ f1g and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n ) algorithm for nding all essential terms of the maxalgebraic characteristic polynomial of an n n matrix over R : In the cases when all terms are essential this algorithm also solves the following problem: Given an nn matrix A and k 2 f1; :::; ng, nd a kk principal submatrix of A whose assignment problem value is maximum.
On the Boolean Minimal Realization Problem in the MaxPlus Algebra
, 1998
"... One of the open problems in the maxplusalgebraic system theory for discrete event systems is the minimal realization problem. In this paper we present some results in connection with the minimal realization problem in the maxplus algebra. First we characterize the minimal system order of a maxli ..."
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Cited by 10 (7 self)
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One of the open problems in the maxplusalgebraic system theory for discrete event systems is the minimal realization problem. In this paper we present some results in connection with the minimal realization problem in the maxplus algebra. First we characterize the minimal system order of a maxlinear discrete event system. We also introduce a canonical representation of the impulse response of a maxlinear discrete event system. Next we consider a simpli#ed version of the general minimal realization problem: the boolean minimal realization problem, i.e., we consider models in which the entries of the system matrices are either equal to the maxplusalgebraic zero element or to the maxplusalgebraic identity element. We give a lower bound for the minimal system order of a maxplusalgebraic boolean discrete event system. We show that the decision problem that corresponds to the boolean realization problem (i.e., deciding whether or not a boolean realization of a given order exists) ...
Resource Optimization and (min,+) Spectral Theory
 IEEE Trans. on Automat. Contr
, 1995
"... We show that certain resource optimization problems relative to Timed Event Graphs reduce to linear programs. The auxiliary variables which allow this reduction can be interpreted in terms of eigenvectors in the (min,+) algebra. KeywordsResource Optimization, Timed Event Graphs, (max,+) algebra, ..."
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Cited by 9 (2 self)
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We show that certain resource optimization problems relative to Timed Event Graphs reduce to linear programs. The auxiliary variables which allow this reduction can be interpreted in terms of eigenvectors in the (min,+) algebra. KeywordsResource Optimization, Timed Event Graphs, (max,+) algebra, spectral theory. I. INTRODUCTION Timed Event Graphs (TEGs) are a subclass of timed Petri nets which can be used to model deterministic discrete event dynamic systems subject to saturation and synchronization phenomena: typically, flexible manufacturing systems, multiprocessor systems, transportation networks [5], [1], [3], [2], [16], [17]. The most remarkable result about TEGs [4], [3], [1] is certainly the following: a TEG functioning at maximal speed reaches after a finite time a periodic regime. More precisely, let x denote the counter function of a given transition of the graph. That is, x(t) represents the number of firings of the transition up to time t, usually the number of parts...