The minimal realization problem in the max-plus algebra: An overview

In this overview report we present known results and open problems in connection with the minimal state space realization problem in the max-plus algebra, which is a framework that can be used to model a class of discrete event systems. 1 Description of the problem Given an n×n matrix A, an n×1 vector b and a 1×n vector c one can construct the sequence gi, i = 1, 2, . . ., where gi is defined by gi = cA b . (1) If instead of the starting point of given A, b and c, the starting point is an arbitrary sequence gi, i = 1, 2, . . ., then necessary and sufficiency conditions are known under which appropriate A, b and c exist such that (1) is valid for i = 1, 2, . . .. An additional requirement is that n, which determines the sizes of A, b and c, must be as small as possible. Efficient algorithms to calculate such A, b and c are known. The problem of this chapter is to reformulate these necessary and sufficiency conditions when the underlying algebra is the so-called max-plus algebra rather than the conventional algebra tacitly used above. One obtains the max-plus algebra from the conventional algebra by replacing addition by maximization and multiplication by addition. These operations are Senior research assistant with the F.W.O. (Fund for Scientific Research-Flanders)