14 research outputs found
A relaxation scheme for computation of the joint spectral radius of matrix sets
The problem of computation of the joint (generalized) spectral radius of
matrix sets has been discussed in a number of publications. In the paper an
iteration procedure is considered that allows to build numerically Barabanov
norms for the irreducible matrix sets and simultaneously to compute the joint
spectral radius of these sets.Comment: 16 pages, 2 figures, corrected typos, accepted for publication in
JDE
On global asymptotic stability of with bounded and not persistently exciting
International audienceWe study global convergence to zero of the solutions of the th order differential equation . We are interested in the case when the vector is not persistently exciting, which is a necessary and sufficient condition for global exponential stability. In particular, we establish new necessary conditions on for global asymptotic stability of the zero equilibrium of the “unexcited” system. A new sufficient condition, that is strictly weaker than the ones reported in the literature, is also established. Unfortunately, it is also shown that this condition is not necessary
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 (p) over cap 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 [(p) over cap/ rootn (p) over cap] 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 most m (m is the number of matrices) of the exact value
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 di#cult to compute and to approximate. We introduce in this paper the first polynomial time approximations of guaranteed precision. We provide an approximation # 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 [ #/ # n, #], 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 most m (m is the number of matrices) of the exact value