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A structured pseudospectral method for \(\mathcal {H}_{\infty}\)-norm computation of large-scale descriptor systems. (English) Zbl 1290.93083
Summary: In this paper, we discuss the problem of computing the \(\mathcal {H}_{\infty}\)-norm of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the \(\mathcal {H}_{\infty}\)-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 pseudo-eigenvalue. 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.

93B60 Eigenvalue problems
93B36 \(H^\infty\)-control
93A15 Large-scale systems
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory
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