Théorême de Cayley-Hamilton dans les dioïdes et application à l’étude des sytèmes à évènements discrets

  • Pierre Moller
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 83)


Discrete-event systems, when studied from a control-theorist’s point of view, can be represented by a linear dynamic system in the so-called max-algebra, or dioïd.

Some methods used in the usual linear-system theory still work in this algebra: z-transform, duality,... Problems arise when trying to reduce the state-dimension, or to define a canonical state-representation. This is due to the lack of an adequate theory of the rank, for matrixes, families of vectors, or linear operators in this algebra.

The Cayley-Hamilton theorem, which is a consequence only of combinatorial properties of matrix-calculus, is quite easy to prove in the max-algebra. Thus, a recurrent equation can be defined, which is satisfied by the transfer function of the system.

A conjecture is proposed:

In the max-algebra, a necessary condition for the state-representation of a SISO linear system to be minimal, is:

All non-decreasing solutions of the recurrent equation, deduced from the state-representation by applying the Cayley-hamilton theorem, have the same asymptotic growth-rate.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. COHEN, D. DUBOIS, J.P. QUADRAT, M. VIOT: Analyse du comportement périodique des sytstèmes de production par la théorie des Rapport I.N.R.I.A n° 191, Le Chesnay, 1983.Google Scholar
  2. [2]
    G. COHEN, D. DUBOIS, J.P. QUADRAT, M. VIOT: A linear-System-Theoretic view of Discrete-Event Processes. 22nd IEEE Conf. on Decision and Control, San Antonio, Texas, 1983.Google Scholar
  3. [3]
    G. COHEN, D. DUBOIS, J.P. QUADRAT, M. VIOT: A linear-System-Theoretic view of Discrete-Event Processesands its use for Performance evaluation in Manufacturing. IEEE Trans. on Aut. Control, Vol AC-30, n° 3, pp. 210–220.Google Scholar
  4. [4]
    G. COHEN, P. MOLLER, J.P. QUADRAT, M. VIOT: Une théorie linéaire des systèmes évènements discrets. Rapport I.N.R. I A n° 362, Le Ohesnay, Jernrier 1985.Google Scholar
  5. [5]
    G. COHEN, P. MOLLER, J.P. QUADRAT, M. VIOT: Limer System Theory for Discrete Event Systems. 23rd IEEE conf. on Decision and Control, las Vegas, Nevada, December 1984.Google Scholar
  6. [5]
    C.V. RAMAMOORTHY, O.S. HO: Performance evaluation of asynchronous concurrent systems using Petri nets. IEEE Trans. on Software Ena.. 6. n° 5, 1980.Google Scholar
  7. [6]
    G.W. BRAMS: Réseaux de PETRI, TOME 1: Théorie et analyse. MASSON, Paris, 1982. TOME 2: Théorie et Pratiaue. MASSON, Paris, 1983.Google Scholar
  8. [7]
    J.L. PETERSON: PETRI Net Theory and the modellino of systems. PRENTICE HALL, 1981.Google Scholar
  9. [8]
    P. CHRETIENNE: Les réseaux de PETRI temporisés. Thèse, Université Pierre et Marie Curie (Paris VI)1983.Google Scholar
  10. [9]
    R.A. CUNINOHAME-OREEN: Minimax Aoebra. Lecture Notes in Economics and Mathématical Systems,vol. 166, Springer Verlag, 1979.Google Scholar
  11. [10]
    H. OONDRAN, M. MINOUX: L’indépendance linéaire dans les didides. Bulletin de la clirection Etudes et Recherches. EDF, Série C, n°1, pp. 67–90.Google Scholar
  12. [11]
    D. ZEILBEROER: A combinatorial approach to matrix algebra. Discrete Mathematics 56. pp 61–72, NORTH-HOLLAND, 1985.Google Scholar
  13. [12]
    H. STRAUBINO: A combinatorial proof of the Cayley-Hamilton Theorem. Discrete Mathematics 43. pp. 273–279, NORTH-HOLLAND, 1983.Google Scholar
  14. [13]
    G.J. OLSDER, C. ROOS: Cramer and Cayley-Hamilton in the max-algebra. Delft University of technology Report re 85–30, 1985.Google Scholar
  15. [14]
    G.J. OLSDER: Some results on the minimal real ization of discrete-event dvnamic systems. Seventh international conference on enalysis and optimisation of systems. 1986 Antibes FRANCE.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1986

Authors and Affiliations

  • Pierre Moller
    • 1
  1. 1.I.N.R.I.A. RoaquencourtLe Chesnay CedexFrance

Personalised recommendations