Residuation of tropical series: Rationality issues

Abstract

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. Résiduation de séries tropicales : étude de la rationalitéRésumé : 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.

Supervisory control of (max,+) automata: extensions towards applications

“…This is because for a (max,+)-rational series y, the series with inverse coefficients Ây is also (max,+) rational if and only if y is at the same time (min,+) rational as shown in Lombardy and Mairesse (2006). Nevertheless, Theorem 3.1 in Badouel et al (2011) shows that (a) if y is (min,+) rational and y is (max,+) rational, then (H y ) (y ) is (max,+) rational; (b) if y is (max,+) rational and y is (min,+) rational, then (H y ) (y ) is (min,+) rational.…”

“…In Badouel et al (2011), authors consider the robust control of systems modelled by interval weighted automata. Their control objective is specified by a reference model defining a tolerance on the desired behaviour of the plant (instead of a trajectory tracking objective in our case).…”

“…Based on results from Badouel et al (2011), classes of (max,+) automata representing the system and the reference are identified to be sufficient to get rational supervisors.…”

“…Indeed, this last approach is made attractive by the fact that it combines concepts on automata with results on (max, +) algebra, to study at the same time the logical and timing aspects of DES. Recently, (max,+) automata have been applied to performance evaluation (Gaubert, 1995;Gaubert & Mairesse, 1999b), scheduling (Houssin, 2011) and control (Badouel, Bouillard, Darondeau, & Komenda, 2011;Komenda, Lahaye, & Boimond, 2009;Su, van Schuppen, & Rooda, 2012) problems for a large class of timed DES. 2 S. Lahaye et al discretisation, and which is based on parallel composition of (max,+) automata with their supervisors.…”