| Title:
|
Controllability in the max-algebra (English) |
| Author:
|
Prou, Jean-Michel |
| Author:
|
Wagneur, Edouard |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
35 |
| Issue:
|
1 |
| Year:
|
1999 |
| Pages:
|
[13]-24 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
reachability |
| Keyword:
|
controllability |
| Keyword:
|
max-algebra |
| MSC:
|
15A80 |
| MSC:
|
93B05 |
| MSC:
|
93B18 |
| MSC:
|
93C65 |
| idZBL:
|
Zbl 1274.93036 |
| idMR:
|
MR1705527 |
| . |
| Date available:
|
2009-09-24T19:22:54Z |
| Last updated:
|
2015-03-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135264 |
| . |
| Reference:
|
[1] G: Birkhoff: Lattice Theory.A.M.S. Coll. Pub. Vol. XXV, Providence 1967 MR 0227053 |
| Reference:
|
[2] Baccelli F., Cohen G., Olsder G. J., Quadrat J. P.: Synchronization and Linearity.Wiley, Chichester 1992 Zbl 0824.93003, MR 1204266 |
| Reference:
|
[3] Cunninghame–Green R. A.: Minimax Algebra.(Lecture Notes in Economics and Mathematical Systems 83.) Springer–Verlag, Berlin 1979 MR 0580321 |
| Reference:
|
[4] Gaubert S.: Théorie des Systèmes linéaires dans les Dioïdes.Thèse. Ecole Nationale Supérieure des Mines de Paris 1992 |
| Reference:
|
[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 MR 1705526 |
| Reference:
|
[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 |
| Reference:
|
[8] Prou J.-M.: Thèse.Ecole Centrale de Nantes 1997 |
| Reference:
|
[9] Wagneur E.: Moduloïds and Pseudomodules.1. Dimension Theory. Discrete Math. 98 (1991), 57–73 Zbl 0757.06008, MR 1139597, 10.1016/0012-365X(91)90412-U |
| . |
