\(\mathcal{H}_2\) model reduction for large-scale linear dynamical systems. (English) Zbl 1159.93318

Gugercin, S.; Antoulas, A. C.; Beattie, C.

SIAM J. Matrix Anal. Appl. 30, No. 2, 609-638 (2008).

Summary: The optimal \(\mathcal{H}_2\) model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal \(\mathcal{H}_2\) approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov- and interpolation-based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation-based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for \(\mathcal{H}_2\) model reduction. The formulation is based on finding a reduced order model that satisfies interpolation-based first-order necessary conditions for \(\mathcal H_2\) optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.

MSC:

93B11	System structure simplification
34C20	Transformation and reduction of ordinary differential equations and systems, normal forms
41A05	Interpolation in approximation theory
49K15	Optimality conditions for problems involving ordinary differential equations
49M05	Numerical methods based on necessary conditions
93A15	Large-scale systems
93C05	Linear systems in control theory
93C15	Control/observation systems governed by ordinary differential equations

Keywords:

model reduction; rational Krylov; \(\mathcal{H}_2\) approximation