Residuation of tropical series: rationality issues

Eric Badouel; Anne Bouillard; Philippe Darondeau; Jan Komenda

inria-00567390, version 1

Residuation of tropical series: rationality issues

Eric Badouel ()^b^, 1, Anne Bouillard ()^a^, 2, 3, Philippe Darondeau ()^b^, 1, Jan Komenda ⁴

N° RR-7547 (2011)

Résumé : Decidability of existence, rationality of delay controllers and robust delay controllers are investigated for systems with time weights in the tropical and interval semirings. Depending on the (max,+) or (min,+)-rationality of the series specifying the controlled system and the control objective, cases are identified where the controller series defined by residuation is rational, and when it is positive (i.e., when delay control is feasible). When the control objective is specified by a tolerance, i.e. by two bounding rational series, a nice case is identified in which the controller series is of the same rational type as the system specification series.

a – ENS
b – INRIA
1 : S4 (INRIA - IRISA)
CNRS : UMR6074 – INRIA – Institut National des Sciences Appliquées (INSA) - Rennes – Université de Rennes 1
2 : Laboratoire d'informatique de l'école normale supérieure (LIENS)
CNRS : UMR8548 – Ecole normale supérieure de Paris - ENS Paris
3 : TREC (INRIA Rocquencourt)
INRIA – Ecole normale supérieure de Paris - ENS Paris
4 : Institute of Mathematics [République Tchèque]
Academy of Sciences of the Czech Republic

inria-00567390, version 1
http://hal.inria.fr/inria-00567390
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Contributeur : Anne Bouillard <>
Soumis le : Lundi 21 Février 2011, 10:33:53
Dernière modification le : Lundi 12 Décembre 2011, 16:03:44

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%F inria-00567390, version 1
%0 Report
%U http://hal.inria.fr/inria-00567390
%2 INFO:INFO_DM
%T Residuation of tropical series: rationality issues
%A Badouel, Eric
%A Bouillard, Anne
%A Darondeau, Philippe
%A Komenda, Jan
%+ S4 - INRIA - IRISA , Laboratoire d'informatique de l'école normale supérieure - LIENS , TREC - INRIA Rocquencourt , Institute of Mathematics [République Tchèque]
%X Decidability of existence, rationality of delay controllers and robust delay controllers are investigated for systems with time weights in the tropical and interval semirings. Depending on the (max,+) or (min,+)-rationality of the series specifying the controlled system and the control objective, cases are identified where the controller series defined by residuation is rational, and when it is positive (i.e., when delay control is feasible). When the control objective is specified by a tolerance, i.e. by two bounding rational series, a nice case is identified in which the controller series is of the same rational type as the system specification series.
%2 Les questions abordées dans ce rapport concernent l'existence et la rationalité d'un contrôleur (retardateur) pour des systèmes représentés par des automates à poids, dans des semi-anneaux tropicaux. Selon la (min,plus) ou (max,plus)-rationalité des séries spécifiant le système à contrôler et l'objectif du contrôle, on identifie des cas où le contrôleur défini par résiduation de séries est rationel et où il est positif (le système est alors contrôlable). Quand l'objectif de contrôle est spécifié par un intervalle de tolérance (encadrement par deux séries), un cas est identifié pour lequel la série du contrôleur a le même type de rationalité, (max,plus) ou (min,plus), que la spécification du système.
%2 F.: Theory of Computation/F.4: MATHEMATICAL LOGIC AND FORMAL LANGUAGES/F.4.3: Formal Languages/F.4.3.3: Decision problems, F.: Theory of Computation/F.4: MATHEMATICAL LOGIC AND FORMAL LANGUAGES/F.4.3: Formal Languages/F.4.3.4: Operations on languages, B.: Hardware/B.1: CONTROL STRUCTURES AND MICROPROGRAMMING/B.1.2: Control Structure Performance Analysis and Design Aids/B.1.2.1: Formal models
%G Anglais
%2 Rapport de recherche
%P 19
%8 2011-02-21
%K Control theory; (max,plus) automata; residuation.
%Z RR-7547
%2 16911

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