Parametric Model Order Reduction by Matrix Interpolation

Abstract

In this paper, a new framework for model order reduction of LTI parametric systems is introduced. After generating and reducing several local original models in the parameter space, a parametric reduced-order model is calculated by interpolating the system matrices of the local reduced models. The main task is to find compatible system representations with optimal interpolation properties. Two approaches for this purpose are presented together with several numerical simulations.Zusammenfassung In diesem Beitrag wird ein neuer Rahmen zur Modellordnungsreduktion parametrischer LZI-Systeme vorgestellt. Er sieht zunächst die Reduktion des Originalmodells in einigen Stützstellen des Parameterraums vor. Anschließend wird ein reduziertes parametrisches System generiert, indem die Systemmatrizen der lokalen Reduktionen durch geeignete Transformationen kompatibel gemacht und interpoliert werden. Hierfür werden zwei alternative Verfahren beschrieben, deren Eigenschaften an drei numerischen Testfällen dargestellt werden.

Parametric Model Order Reduction of Port-Hamiltonian Systems by Matrix Interpolation

“…Therefore, it is mandatory to make the reduced system matrices compatible in some sense. As shown in [5], this can be achieved by embedding the locally reduced systems into the original coordinates of the The process of making two pairs of basis vectors of locally reduced systems { r,1 , r,1 } and { r,2 , r,2 } compatible with respect to a common subspace U . The subspace U unites the dominant directions of V 1 and V 2 .…”

“…In the following, we review how the matrices M and T are chosen in [5] such that a weighted interpolation of the locally reduced systems becomes meaningful.…”

“…Definition 1 [5]. (11) has to be chosen as T = U ⊤ V (12) in order to make the coordinate systems of all x * r, , = 1, 2, .…”

“…, compatible w. r. t. a subspace spanned by U . In [5], M is chosen as M = (W ⊤ U ) −1 in order to represent the state equations of all reduced models with respect to the same basis. Subsequently, the re-adjusted system matrices E * r, , A * r, , B * r, and C * r, from Equation (9) can be interpolated according to (7).…”

“…In this paper, the framework for parametric model order reduction of linear time-invariant systems as presented by Panzer et al in [5] is extended to linear timeinvariant (LTI) systems that are originally modeled in portHamiltonian form. Merging pMOR by matrix interpolation with port-Hamiltonian representation results in the advantage that stability is preserved in the interpolated system.…”