An interpolation-based approach to ℋ<inf>∞</inf> model reduction of dynamical systems

Abstract

We introduce an interpolatory approach to ℋ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> model reduction for large-scale dynamical systems. Guided by the optimality conditions of [26] for best uniform rational approximants on the unit disk, our proposed method uses the freedom in choosing the d-term in the reduced order model to enforce 2r + 1 interpolation conditions in the right-half plane for any given reduction order, r. 2r of these points are initialized by the Iterative Rational Krylov Algorithm of [16]; and then the d-term is chosen to minimize the ℋ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> error for this initial set of interpolation points. Several numerical examples illustrate the effectiveness of the proposed method. It consistently yields better results than balanced truncation. In all cases examined its performance is very close to or better than that of Hankel norm approximation. For the special case of state-space symmetric systems, important properties are established. Finally, we examine ℋ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> model reduction from a potential theoretic perspective and present a second methodology for choosing interpolation points.