$\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems

Abstract

The optimal 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 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 Lyapunovand 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 H 2 model reduction. The formulation is based on finding a reduced order model that satisfies interpolation-based firstorder necessary conditions for 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.

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…Meier and Luenberger in [173] proved these conditions for SISO systems originally; generalizations to MIMO systems were given in [64,116,214]. The optimality conditions in (3.9) mean that an H 2 -optimal reduced model H r (s) is a bitangential Hermite interpolant to H(s).…”

“…Thus, the optimal points and associated tangent directions depend on the reduced model and are not known a priori. Gugercin, Antoulas, and Beattie [116] introduced the iterative rational Krylov algorithm (IRKA), which, using successive substitution, iteratively corrects the interpolation points and tangential directions until the optimality conditions in (3.9) are satisfied, i.e., until optimal interpolation points and tangential directions are reached. For details of IRKA, we refer the reader to [15,116].…”

“…Gugercin, Antoulas, and Beattie [116] introduced the iterative rational Krylov algorithm (IRKA), which, using successive substitution, iteratively corrects the interpolation points and tangential directions until the optimality conditions in (3.9) are satisfied, i.e., until optimal interpolation points and tangential directions are reached. For details of IRKA, we refer the reader to [15,116]. IRKA can be used in the context of parametric systems for computing locally optimal reduced models for the given parameter sample.…”

Structure-preserving model reduction of complex physical systems

“…by [8]. One is the splitting of the state variables x = (x 1 , x 2 ), which may be based on Krylov-type methods or on balancing methods.…”

“…The Effort-constraint and Flow-constraint reduction methods for linear port-Hamiltonian systems have a direct interpretation in terms of projection-based reduction methods [1], [8]. Consider a port-Hamiltonian system (9) with quadratic Hamiltonian H(x) = 1 2 x T Qx, Q = Q T > 0, and linear damping f R = −Re R given as…”