Approximations of the Rate of Growth of Switched Linear Systems

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts, in particular it characterizes the growth rate of switched linear systems. The joint spectral radius is notoriously difficult to compute and to approximate. We introduce in this paper the first polynomial time approximations of guaranteed precision. We provide an approximation ρ̂ that is based on ellipsoid norms, that can be computed by convex optimization, and that is such that the joint spectral radius belongs to the interval [ρ̂/ √ n, ρ̂], where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of all the matrices are non-negative; in this case the approximation is proved to be within a factor at mostm (m is the number of matrices) of the exact value.