Calculating the H∞-norm of large sparse systems via Chandrasekhar iterations and extrapolation

Younes Chahlaoui, Kyle A. Gallivan, Paul Van Dooren, Mohammed-Najib Benbourhim, Patrick Chenin, Abdelhak Hassouni, Jean-Baptiste Hiriart-Urruty
2007 ESAIM: Proceedings and Surveys  
We describe an algorithm for estimating the H ∞ -norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm uses Chandrasekhar iterations to obtain an estimate of the H ∞ -norm and then uses extrapolation to improve these estimates. Résumé. Nous décrivons un algorithme pour estimer la norme H ∞ d'un système dynamique linéairè a temps invariant de grande dimension décrit par un modèle d'espace d'état discret. L'algorithme emploie des
more » ... me emploie des récurrences de Chandrasekhar pour obtenir une estimation de la norme H ∞ puis emploie l'extrapolation pour améliorer ces estimations. G(z) We therefore assume that the given quadruple {A, B, C, D} is a real and minimal realization of a stable transfer function G(z). The stability of G(z) implies that all of the eigenvalues of A are strictly inside the unit circle, and hence that ρ(A) < 1, where ρ(A) is the spectral radius of A. An important result upon which we rely is the bounded real lemma, which states that γ > G(z) ∞ if and only if there exists a positive definite solution P (denoted as P 0) to the linear matrix inequality (LMI) [1] : * This paper presents research supported
doi:10.1051/proc:072008 fatcat:37qcszgklfd6lbiw2c7vdso24i