Stability of switched linear systems and the convergence of random products

In this paper we give conditions that a discrete time switched linear systems must satisfy if it is stable. We do this by calculating the mean and covariance of the set of matrices obtained by using all possible switches. The theory of switched linear systems has received considerable attention in the systems theory literature in the last two decades. However, for discrete time switched systems the literature is much older going back to at least the early 1960´s with the publication of the paper of Furstenberg and Kesten in the area of products of random matrices, or if you like the random products of matrices. The way that we have approached this problem is to consider the switched linear system as evolving on a partially ordered network that is, in fact, a tree. This allows us to make use of the developments of 50 years of study on random products that exists in the statistics literature. A nice byproduct of this research is that we use Konig´s theorem of finatary trees. This may be the first use of this theorem in systems and control.