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

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
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