A Subspace Framework for ${\mathcal H}_\infty$-Norm Minimization
This work addresses the computationally challenging problem of H∞-norm minimization for large-scale parameter-dependent systems, which is important in robust control theory.
The paper proposes greedy interpolatory subspace frameworks for minimizing the H∞-norm of large-scale parameter-dependent descriptor systems, achieving superlinear convergence under smoothness assumptions and demonstrating fast convergence on several large-scale examples.
We deal with the minimization of the ${\mathcal H}_\infty$-norm of the transfer function of a parameter-dependent descriptor system over the set of admissible parameter values. Subspace frameworks are proposed for such minimization problems where the involved systems are of large order. The proposed algorithms are greedy interpolatory approaches inspired by our recent work [Aliyev et al., SIAM J. Matrix Anal. Appl., 38(4):1496--1516, 2017] for the computation of the ${\mathcal H}_\infty$-norm. In this work, we minimize the ${\mathcal H}_\infty$-norm of a reduced-order parameter-dependent system obtained by two-sided restrictions onto certain subspaces. Then we expand the subspaces so that Hermite interpolation properties hold between the full and reduced-order system at the optimal parameter value for the reduced order system. We formally establish the superlinear convergence of the subspace frameworks under some smoothness assumptions. The fast convergence of the proposed frameworks in practice is illustrated by several large-scale systems.