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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 con- siderable 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. 1. Introduction. The theory of switched linear systems has received consider- able attention in the systems theory literature in the last two decades and there is a wealth of solid results concerning the stability and stabilizability of such systems. A recent survey by Lin and Antsaklis, (14), gives a rather complete overview of the field. The main theorem cited there on stability for discrete time switched linear sys- tems, (their Theorem 6, (13)), requires that 2 n matrix norms must be calculated. In this paper we give a simpler calculation but one that will only hold for "almost all" (5) switching sequences. An excellent and readable source for the general theory of switched linear systems is the monograph by Liberzon, (12). In his book the primary focus is on continuous dynamics but the concepts are much the same. However, for discrete time switched systems the literature is much older than the systems theory literature, going back to at least the early 1960's with the publication of the papers of Furstenberg and Kesten, (9, 10), in the area of products of random matrices, or, if you like, the random products of matrices. In the statistical literature on random products "almost all" is the usual condition. We approach the problem in the following way. We study 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.

Switched linear systems: stability and the convergence of random products