In this paper we discuss the problem of computing the H∞-norms of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the H∞-norm and the structured complex stability radius of the corresponding system pencil. To compute the structured stability radius we consider so-called structured pseudospectra. Namely, we have to find the pseudospectrum touching the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudoeigenvalue. For that we use a new fast iterative scheme which is based on certain rank-1 perturbations of the system pencil. Finally, we analyze the performance of our algorithm by using real-world examples.