Switched Linear Systems: Stability and the Convergence of Random Products

In this paper we provide conditions for the stability of discrete time switched linear systems. We accomplish this by calculating the mean and covariance of the set of matrices obtained by using all possible switching sequences. 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 exist in the statistics literature.