A structured pseudospectral method for $$\mathcal {H}_\infty $$H∞-norm computation of large-scale descriptor systems

In this paper, we discuss the problem of computing the $${\mathcal {H}}_\infty $$H∞-norm of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the $${\mathcal {H}}_\infty $$H∞-norm and the structured complex stability radius of a corresponding matrix 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 a matrix pencil. Finally, we analyze the performance of our algorithm by using real-world examples. In particular we compare our method with different other algorithms including a recently and independently derived method from Guglielmi, Gürbüzbalaban and Overton.

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