So far, H2 ⊗ L2-optimal model order reduction (MOR) of linear time-invariant systems, preserving the affine parameter dependence, was only considered for special cases by Baur et al in 2011. In this contribution, we present necessary conditions for an H2 ⊗ L2-optimal parametric reduced order model, for general affine parametric systems resembling the special case investigated by Baur et al.
We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation, we show a connection to a rational Krylov subspace, and thus to moment matching. This theoretical link between the MOR techniques is illustrated by three numerical examples. For linear time-invariant systems, the link also motivates that the time domain approach can be at best as accurate as moment matching, since the expansion points are fixed by the choice of the polynomial basis, while in moment matching they can be adapted to the system. KEYWORDS time domain model order reduction, moment matching, Sylvester equation arXiv:1801.07085v3 [math.NA] 6 Jul 2018 ∞ := G(s) − G r (s) H∞ .Applying this method, the system (1) is first balanced, i.e. the observability and controllability Gramians P O and P C , given as the solutions of two Lyapunov equationsare made equal and diagonal, such that P O = P C = diag(σ 1 · · · σ n ) and σ 1 ≥ · · · ≥ σ n > 0 are the systems invariant Hankel singular values (HSVs). The discardable portions are identified and truncated according to the magnitude of the HSVs. More details about this method can be found, e.g. in [4, Chapter 7].The above MOR techniques are motivated and derived by frequency domain considerations.
We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [4] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation we show a connection to a rational Krylov subspace, and thus moment matching. This theoretical link between the MOR techniques is illustrated by one numerical example.
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