  We propose a fast method to approximate the real stability radius of a linear
dynamical system with output feedback, where the perturbations are restricted
to be real valued and bounded with respect to the Frobenius norm. Our work
builds on a number of scalable algorithms that have been proposed in recent
years, ranging from methods that approximate the complex or real pseudospectral
abscissa and radius of large sparse matrices (and generalizations of these
methods for pseudospectra to spectral value sets) to algorithms for
approximating the complex stability radius (the reciprocal of the $H_\infty$
norm). Although our algorithm is guaranteed to find only upper bounds to the
real stability radius, it seems quite effective in practice. As far as we know,
this is the first algorithm that addresses the Frobenius-norm version of this
problem. Because the cost mainly consists of computing the eigenvalue with
maximal real part for continuous-time systems (or modulus for discrete-time
systems) of a sequence of matrices, our algorithm remains very efficient for
large-scale systems provided that the system matrices are sparse.
