Prou, Jean-Michel, and Wagneur, Edouard. "Controllability in the max-algebra." Kybernetika 35.1 (1999): [13]-24. <http://eudml.org/doc/33406>.
@article{Prou1999,
abstract = {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 $\Gamma _k$ 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 $\Gamma _k$ for the condition to be sufficient. A similar result also holds by duality on the observability side.},
author = {Prou, Jean-Michel, Wagneur, Edouard},
journal = {Kybernetika},
keywords = {reachability; controllability; max-algebra; reachability; controllability; max-algebra},
language = {eng},
number = {1},
pages = {[13]-24},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Controllability in the max-algebra},
url = {http://eudml.org/doc/33406},
volume = {35},
year = {1999},
}
TY - JOUR
AU - Prou, Jean-Michel
AU - Wagneur, Edouard
TI - Controllability in the max-algebra
JO - Kybernetika
PY - 1999
PB - Institute of Information Theory and Automation AS CR
VL - 35
IS - 1
SP - [13]
EP - 24
AB - 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 $\Gamma _k$ 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 $\Gamma _k$ for the condition to be sufficient. A similar result also holds by duality on the observability side.
LA - eng
KW - reachability; controllability; max-algebra; reachability; controllability; max-algebra
UR - http://eudml.org/doc/33406
ER -
