Controllability in the max-algebra

Jean-Michel Prou; Edouard Wagneur

Kybernetika (1999)

  • Volume: 35, Issue: 1, page [13]-24
  • ISSN: 0023-5954

Abstract

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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 ≥ 1 , there is a matrix Γ 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 Γ k for the condition to be sufficient. A similar result also holds by duality on the observability side.

How to cite

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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 -

References

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  2. Baccelli F., Cohen G., Olsder G. J., Quadrat J. P., Synchronization and Linearity, Wiley, Chichester 1992 Zbl0824.93003MR1204266
  3. Cunninghame–Green R. A., Minimax Algebra, (Lecture Notes in Economics and Mathematical Systems 83.) Springer–Verlag, Berlin 1979 MR0580321
  4. Gaubert S., Théorie des Systèmes linéaires dans les Dioïdes, Thèse. Ecole Nationale Supérieure des Mines de Paris 1992 
  5. Gazarik M. J., Kamen E. W., Reachability and observability of linear system over Max–Plus, In: 5th IEEE Mediterranean Conference on Control and Systems, Paphos 1997, revised version: Kybernetika 35 (1999), 2–12 (1997) MR1705526
  6. Gondran M., Minoux M., Valeurs propres et vecteurs propres dans les dioïdes et leur interprétation en théorie des graphes, EDF Bull. Direction Études Rech. Sér. C Math. Inform. 2 (1977), 25–41 (1977) 
  7. Prou J.-M., Thèse, Ecole Centrale de Nantes 1997 
  8. Wagneur E., 10.1016/0012-365X(91)90412-U, 1. Dimension Theory. Discrete Math. 98 (1991), 57–73 (1991) Zbl0757.06008MR1139597DOI10.1016/0012-365X(91)90412-U

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