On the Characteristic Equation and Minimal Realizations for Discrete-Event Dynamic Systems

  • Geert Jan Olsder
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 83)


Recently an analogy between conventional linear system theory and the relatively new theory on discrete-event dynamic systems has been shown to exist. The system descrip-tion in the new theory resembles the one of the conventional theory, provided that the operations addition and multiplication are replaced by maximization and addition respectively. One also speaks of a system in the max-algebra, which is a semi-ring. In this paper we investigate, by pursuing the analogy mentioned above, whether mini-mal realizations exist for discrete-event dynamic system if only the input/output description is given by means of the impulse response. A constuction procedure is suggested. It turns out that the characteristic equation of a matrix in the max-al-gebra (to be defined) plays a crucial rôle.


Impulse Response Characteristic Equation Characteristic Polynomial Realization Theory Conventional Theory 
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  1. [1]
    Chen, C.T., Linear System Theory and Design, Holt, Rinehart and Winston, 1984.Google Scholar
  2. [2]
    Cohen, G., D. Dubois, J.P. Quadrat and M. Viot, A Linear-System-Theoretic View of Discrete-Event Processes and its Use for Performance Evaluation in Manufac-turing, IEEE Transaction on Automatic Control, Vol. AC-30, 1985, pp. 210–220.Google Scholar
  3. [3]
    Olsder, G.J., and C. Roos, Cramér and Cayley-Hamilton in the Max-Algebra sub-mitted for publication.Google Scholar
  4. [4]
    Berman, A., and R.J. Plemmons, Nonnegative Matrices in the Mathematical sciences, Academic Press, 1979.Google Scholar
  5. [5]
    Cuninghame Green, R., Minimax Algebra, Lecture Note no. 166 in Economics and Mathematical Systems, Springer Verslag, 1979.Google Scholar
  6. [6]
    Cohen, G., P. Moller, J.P. Quadrat and M. Viot, Linear System Theory for Dis-crete-Event Systems, Proceedings of the 23rd Conference on Decision and Control, Las Vegas, NV, December, 1984, pp. 539–544.Google Scholar
  7. [7]
    Olsder, G.J. Some results on the minimal realization of discrete event systems, Internal Report 85–35 Delft University of Technology, 1985.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1986

Authors and Affiliations

  • Geert Jan Olsder
    • 1
  1. 1.Dept. of Mathematics and InformaticsDelft University of TechnologyDelftThe Netherlands

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