B. De Schutter and B. De Moor, "Minimal realization in the max algebra," Tech. rep. 94-29, ESAT-SISTA, K.U.Leuven, Leuven, Belgium, 29 pp., May 1994.
The main topic of this paper is the minimal realization problem in the max algebra, which is one of the modeling frameworks that can be used to model discrete event systems. First we determine necessary and for some cases also sufficient conditions for a polynomial to be the characteristic polynomial of a matrix in the max algebra. Then we show how a system of multivariate max-algebraic polynomial equalities can be transformed into an Extended Linear Complementarity Problem (ELCP). Finally we combine these results to find all equivalent minimal state space realizations of a single input single output (SISO) discrete event system. We also give a geometrical description of the set of all minimal realizations of a SISO max-linear discrete event system.