Controllability in the max-algebra

Prou, Jean-Michel ; Wagneur, Edouard

Kybernetika, Tome 35 (1999) no. 1, p. [13].

Résumé

We are interested here in the reachability and controllability problems for DEDS in the max-algebra. Contrary to the situation in linear systems theory, where controllability (resp observability) refers to a (linear) subspace, these properties are essentially discrete in the $max$-linear dynamic system. We show that these problems, which consist in solving a $max$-linear equation lead to an eigenvector problem in the $min$-algebra. More precisely, we show that, given a $max$-linear system, then, for every natural number $k\ge 1$, there is a matrix ${\mathrm{\Gamma }}_k$ whose $min$-eigenspace associated with the eigenvalue $1$ (or $min$-fixed points set) contains all the states which are reachable in $k$ steps. This means in particular that if a state is not in this eigenspace, then it is not controllable. Also, we give an indirect characterization of ${\mathrm{\Gamma }}_k$ for the condition to be sufficient. A similar result also holds by duality on the observability side.

Détail

Classification : 15A80, 93B05, 93B18, 93C65
Mots-clés : reachability; controllability; max-algebra