Minimal (max,+) realization of convex sequences

SIAM Journal on Control and Optimization

Volumn 36, Issue 1, 1998, Pages 137-147

  Gaubert, Stéphane ^a   Butkovič, Peter ^b   Cuninghame Green, Raymond ^b  

^a INRIA   (France)

^b UNIVERSITY OF BIRMINGHAM   (United Kingdom)

Author keywords

Discrete event systems; Max plus algebra; Minimal realization; Rational sequences

Indexed keywords

DISCRETE TIME CONTROL SYSTEMS; MATRIX ALGEBRA;

CONVEX SEQUENCES; HANKEL MATRIX; MINIMAL REALIZATION;

LINEAR CONTROL SYSTEMS;

EID: 0031674279     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S036301299528534X     Document Type: Article

References (34)

1. F. BACCELLI, G. COHEN, G. OLSDER, AND J. QUADRAT, Synchronization and Linearity, John Wiley, New York, 1992.
2. R. B. BAPAT, Permanents, max algebra and optimal assignment, Linear Algebra Appl., 226/228 (1995), pp. 73-86.
3. J. BERSTEL AND C. REUTENAUER, Rational Series and Their Languages, Springer-Verlag, New York, 1988.
4. G. BIRKHOFF AND S. MACLANE, A Survey of Modern Algebra, Macmillan, New York, 1965.
5. R. CUNINGHAME-GREEN, Minimax Algebra, Lecture Notes in Econom. and Math. Systems 166, Springer-Verlag, New York, 1979.
6. R. CUNINGHAME-GREEN, Algebraic realization of discrete dynamic systems, in Proc. of the 1991 IFAC Workshop on Discrete Event System Theory and Applications in Manufacturing and Social Phenomena, Shenyang, China, June 1991.
7. R. CUNINGHAME-GREEN AND P. BUTKOVIČ, Discrete-event dynamic systems: The strictly convex case, Ann. Oper. Res., 57 (1995), pp. 45-63.
8. R. CUNINGHAME-GREEN AND P. MEIJER, An algebra for piecewise-linear minimax problems, Discrete Appl. Math., 2 (1980), pp. 267-294.
9. B. DE SCHUTTER, Max-Algebraic System Theory for Discrete-Event Systems, Ph.D. thesis, Katholieke Univ. Leuven, Feb. 1996.
10. B. DE SCHUTTER AND B. DE MOOR, The characteristic equation and minimal state space realization of siso systems in the max algebra, in Proc. of the 11th Conf. on Anal. and Opt. of Systems: Discrete-Event Systems, Lect. Notes. in Control and Inform. Sci. 199, Springer-Verlag, New York, 1994.
11. B. DE SCHUTTER AND B. DE MOOR, Minimal realization in the max algebra is an extended linear complementarity problem, Sys. Control Lett., 25 (1995), pp. 103-111.
12. B. DE SCHUTTER AND B. DE MOOR, Minimal state space realization of mimo systems in the max algebra, in Proc. of the 3rd European Control Conference, Roma, Italy, Sept. 1995, pp. 411-416.
13. M. FLIESS, Matrices de Hankel, J. Math. Pures. Appl., 53 (1974), pp. 197-222.
14. M. FLIESS, Séries rationnelles positives et processus stochastiques, Ann. Inst. H. Poincaré, XI (1975), pp. 1-21.
15. S. GAUBERT, Théorie des systèmes linéaires dans les dioïdes, thèse, École des Mines de Paris, 1992.
16. S. GAUBERT, On Rational Series in One Variable over Certain Dioids, Rapport de Recherche 2162, INRIA, Le Chesnay, France, Jan. 1994.
17. S. GAUBERT, Rational series over dioids and discrete-event systems, in Proc. of the 11th Conf. on Anal. and Opt. of Systems: Discrete-Event Systems, Lect. Notes. in Control and Inform. Sci. 199, Springer-Verlag, New York, 1994.
18. J. S. GOLAN, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Pitman Monogr. Surveys Pure Appl. Math. 54, Longman Scientific and Technical Publishers, Harlow, U.K., 1992.
19. M. GONDRAN AND M. MINOUX, Graphes et algorithmes, Eyrolles, Paris, 1979. Engl. transl. Graphs and Algorithms, John Wiley, New York, 1984.
20. M. GONDRAN AND M. MINOUX, Graphes et algorithmes, Eyrolles, Paris, 1979. Engl. transl. Graphs and Algorithms, John Wiley, New York, 1984.
21. M. GONDRAN AND M. MINOUX, Linear algebra in dioids: A survey of recent results, Ann. Discrete Math., 19 (1984), pp. 147-164.
22. J. GUNAWARDENA, ED., Idempotency, Publications of the Newton Institute, Cambridge University Press, London, 1996.
23. G. HANSEL AND D. PERRIN, Mesures de probabilités rationnelles, in Mots, M. Lothaire, ed., Hermes, Paris, 1990.
24. G. HARDY, J. LITTLEWOOD, AND G. POLYA, Inequalities, Cambridge University Press, London, 1934, reprinted 1973.
25. K. KIM, Boolean Matrix Theory and Applications, Marcel Dekker, New York, 1982.
26. M. MARCUS AND H. MINC, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964.
27. A. MARSHALL AND I. OLKIN, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
28. V. MASLOV AND S. SAMBORSKIǏ, EDS., Idempotent analysis, Adv. in Soviet Math. 13, AMS, Providence, RI, 1992.
29. G. OLSDER, On the characteristic equation and minimal realizations for discrete-event dynamic systems, in Analysis and Optimization of Systems, A. Bensoussan and J. Lions, eds., in Lecture Notes in Control and Inform. Sci. 83, Springer-Verlag, New York, 1986, pp. 189-201.
30. G. OLSDER, Some results on the minimal realization of discrete-event systems, in 25th IEEE Conf. on Decision and Control, Athens, Greece, 1986.
31. M. PLUS, Linear systems in (max, +)-algebra, in Proceedings of the 29th Conference on Decision and Control, Honolulu, Dec. 1990.
32. X. QI AND W. CHEN, The minimal realization of discrete-event systems, in Proc. of the 1991 IFAC Workshop on Discrete Event System Theory and Applications in Manufacturing and Social Phenomena, Shenyang, China, June 1991, pp. 29-33.
33. C. REUTENAUER AND H. STRAUBING, Inversion of matrices over a commutative semiring, J. Algebra, 88 (1984), pp. 350-360.
34. D. ZEILBERGER, A combinatorial approach to matrix algebra, Discrete Math., 56 (1985), pp. 61-72.