  In this article, we introduce an exponential for tropical matrices and show
that this series is essential for the analysis of certain kinds of stability in
discrete event dynamic systems. A notion of a generalised eigenvector is
introduced to discuss this kind of stability and prove it exists at most in the
order of $1,p/2,p$, where $p$ is the period of the corresponding matrix. Thus
characterizing the generalised eigenvectors of all powers of the matrix. Also,
a sufficient condition is proved for the exponential of a matrix to be robust.
