A. B. Anderson, M. Deistler, L. Farina, and L. Benvenuti, Nonnegative realization of a linear system with nonnegative impulse response, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.43, issue.2, pp.134-142, 1996.
DOI : 10.1109/81.486435

M. Akian, S. Gaubert, and C. Walsh, Discrete max-plus spectral theory, Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, pp.19-51, 2005.
DOI : 10.1090/conm/377/06984
URL : https://hal.archives-ouvertes.fr/inria-00000968

L. Benvenuti and L. Farina, An example of how positivity may force realizations of ???large??? dimension, Systems & Control Letters, vol.36, issue.4, pp.261-266, 1999.
DOI : 10.1016/S0167-6911(98)00098-X

G. Basile and G. Marro, Controlled and conditioned invariant subspaces in linear system theory, Journal of Optimization Theory and Applications, vol.3, issue.5, 1991.
DOI : 10.1007/BF00931370

N. [. Blondel and . Portier, The presence of a zero in an integer linear recurrent sequence is NP-hard to decide, Linear Algebra and its Applications, vol.351, issue.352, pp.351-35291, 2002.
DOI : 10.1016/S0024-3795(01)00466-9

R. [. Basu, M. Pollack, and . Roy, On the combinatorial and algebraic complexity of quantifier elimination, Journal of the ACM, vol.43, issue.6, pp.1002-1045, 1996.
DOI : 10.1145/235809.235813

J. Berstel and C. Reutenauer, Rational Series and their Languages, 1988.
DOI : 10.1007/978-3-642-73235-5
URL : https://hal.archives-ouvertes.fr/hal-00619791

J. Berstel and C. Reutenauer, Noncommutative Rational Series With Applications New edition of [BR88], available on the first author's web page, 2010.

D. [. Bonnier-rigny and . Krob, A complete system of identities for one-letter rational expressions with multiplicities in the tropical semiring, Second International Colloquium on Words, Languages and Combinatorics, pp.27-50, 1992.
DOI : 10.1016/0304-3975(94)90276-3
URL : https://hal.archives-ouvertes.fr/hal-00017324

D. [. Cohen, J. Dubois, M. Quadrat, and . Viot, Analyse du comportement périodique des systèmes de production par la théorie des dio¨?desdio¨?des, 1983.

D. [. Cohen, J. Dubois, M. Quadrat, and . Viot, A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing, IEEE Transactions on Automatic Control, vol.30, issue.3, pp.210-220, 1985.
DOI : 10.1109/TAC.1985.1103925

. [. Cuninghame-green, Algebraic realization of discrete dynamic systems, Proceedings of the 1991 IFAC Workshop on Discrete Event System Theory and applications in manufacturing and social phenonmena, 1991.

P. [. Cuninghame-green and . Butkovi?, Discrete-event dynamic systems: The strictly convex case, Annals of Operations Research, vol.101, issue.1, pp.45-63, 1995.
DOI : 10.1007/BF02099690

]. G. Cgq99a, S. Cohen, J. P. Gaubert, and . Quadrat, Max-plus algebra and system theory: where we are and where to go now, Annual Reviews in Control, vol.23, pp.207-219, 1999.

]. G. Cgq99b, S. Cohen, J. Gaubert, and . Quadrat, Max-plus algebra and system theory : where we are and where to go now, Annual Reviews in Control, vol.23, pp.207-219, 1999.

P. [. Cohen, J. Moller, M. Quadrat, and . Viot, Une théorie linéaire des systèmessystèmes`systèmesàsystèmesàévénements discrets, 1985.

P. [. Cohen, J. Moller, M. Quadrat, and . Viot, Algebraic tools for the performance evaluation of discrete event systems, IEEE Proceedings: Special issue on Discrete Event Systems, 1989.
DOI : 10.1109/5.21069

]. J. Con71 and . Conway, Regular algebra and finite machines, 1971.

G. Duchamp and C. Reutenauer, Un crit??re de rationalit?? provenant de la g??om??trie non commutative, Inventiones mathematicae, vol.13, issue.3, pp.613-622, 1997.
DOI : 10.1007/s002220050154

[. Schutter and . Ku-leuven, Max-algebraic system theory for discrete-event systems, 1996.

[. Schutter, B. De, and . Moor, Minimal realization in the max algebra is an extended linear complementarity problem, Systems & Control Letters, vol.25, issue.2, pp.103-111, 1995.
DOI : 10.1016/0167-6911(94)00062-Z

]. S. Eil74 and . Eilenberg, Automata, Languages and Machines, volume A, 1974.

]. H. End72 and . Enderton, An introduction to mathematical logic, 1972.

M. [. Eilenberg and . Schützenberger, Rational sets in commutative monoids, Journal of Algebra, vol.13, issue.2, pp.173-191, 1969.
DOI : 10.1016/0021-8693(69)90070-2

M. Fliess, Séries rationnelles positives et processus stochastiques

A. [. Flajolet and . Odlyzko, Singularity Analysis of Generating Functions, SIAM Journal on Discrete Mathematics, vol.3, issue.2, pp.216-240, 1990.
DOI : 10.1137/0403019
URL : https://hal.archives-ouvertes.fr/inria-00075725

S. Gaubert, Théorie des systèmes linéaires dans les dio¨?desdio¨?des, Thèse, ´ Ecole des Mines de Paris, 1992.

]. S. Gau94a and . Gaubert, On rational series in one variable over certain dioids, 1994.

]. S. Gau94b and . Gaubert, Rational series over dioids and discrete event systems, Proc. of the 11th Conf. on Anal. and Opt. of Systems: Discrete Event Systems, number 199 in Lect. Notes. in Control and Inf. Sci, 1994.

P. [. Gaubert, R. Butkovi?, and . Cuninghame-green, Minimal (max,+) Realization of Convex Sequences, SIAM Journal on Control and Optimization, vol.36, issue.1, pp.137-147, 1998.
DOI : 10.1137/S036301299528534X

M. [. Gaubert and . Plus, Methods and applications of (max,+) linear algebra, STACS'97 number 1200, 1997.
DOI : 10.1007/BFb0023465
URL : https://hal.archives-ouvertes.fr/inria-00073603

]. B. Hovdw06, G. J. Heidergott, J. Olsder, and . Van-der-woude, Max Plus at Work: Modeling and Analysis of Synchronized Systems, A Course on Max-Plus Algebra and Its Applications, 2006.

J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation, 1979.

I. Itenberg, G. Mikhalkin, and E. Shustin, Tropical algebraic geometry . Oberwolfach seminars. Birkhäuser, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00143239

]. G. Jac75 and . Jacob, Représentations et Substitutions Matricielles dans la théorie matricielle des semigroupes, Thèse, 1975.

]. R. Kat07 and . Katz, Max-plus (A,B)-invariant spaces and control of timed discrete event systems, IEEE Trans. Automat. Control, vol.52, issue.2, pp.229-241, 2007.

S. [. Loreto, R. D. Gaubert, J. J. Katz, and . Loiseau, Duality Between Invariant Spaces for Max-Plus Linear Discrete Event Systems, SIAM Journal on Control and Optimization, vol.48, issue.8, 2009.
DOI : 10.1137/090747191
URL : https://hal.archives-ouvertes.fr/hal-00562586

H. Maeda and S. Kodama, Reachability, observability and realizability of linear systems with possitivity constraints, IECE Trans, vol.63, pp.688-694, 1980.

]. P. Mol88 and . Moller, Théorie algébrique des SystèmesSystèmesàSystèmesà´SystèmesàÉvénements Discrets, Thèse, ´ Ecole des Mines de Paris, 1988.

G. Olsder and B. Schutter, The minimal realization problem in the max-plus algebra, Appears in [BSVW99], pp.157-162, 1999.
DOI : 10.1007/978-1-4471-0807-8_32

]. G. Ols86 and . Olsder, On the characteristic equation and minimal realizations for discrete-event dynamic systems, Analysis and Optimizaton of Systems, number 83 in Lecture notes in Control and Information Sciences, pp.189-201, 1986.

]. D. Per92 and . Perrin, On positive matrices, Theor. Comp. Sci, vol.94, pp.357-366, 1992.

J. Pin, Tropical semirings, 1998.
DOI : 10.1017/CBO9780511662508.004
URL : https://hal.archives-ouvertes.fr/hal-00113779

B. [. Richter-gebert, T. Sturmfels, and . Theobald, First steps in tropical geometry, Idempotent mathematics and mathematical physics, pp.289-317, 2005.
DOI : 10.1090/conm/377/06998

A. [. Stockmeyer and . Meyer, Word problems requiring exponential time(Preliminary Report), Proceedings of the fifth annual ACM symposium on Theory of computing , STOC '73, 1973.
DOI : 10.1145/800125.804029

M. Soittola, Positive rational sequences, Theoretical Computer Science, vol.2, issue.3, pp.317-322, 1976.
DOI : 10.1016/0304-3975(76)90084-0