Stability-preserving parametric model reduction by matrix interpolation

Abstract

In this article, a method to preserve stability in parametric model reduction by matrix interpolation is presented. Based on the matrix measure approach, sufficient conditions on the original system matrices are derived. Once they are fulfilled, the stability of each of the reduced models is guaranteed as well as that of the parametric model resulting from interpolation. In addition, it is shown that these sufficient conditions are met by port-Hamiltonian systems and by a relevant set of second-order systems obtained by the finite element method. The new approach is illustrated by two numerical examples.

A Black-Box method for parametric model order reduction based on matrix interpolation with application to simulation and control

“…In [11], stable interpolation on matrix manifolds is proposed for systems of ODE in second-order form. Stability-preserving matrix interpolation for dissipative systems is suggested in [55], for Port-Hamiltonian systems in [78] and for passive systems in [61,58]. All these methods are efficient because they can make use of Corollary 2.2 as the structure of the considered high-order systems meets the associated requirements.…”

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

“…First, reapply the transformation (8). Second, write every x r, in coordinates of the large-scale system by multiplying it with V from the left:x = V T −1 x * r, (10) In the following, we will refer to this as 'embedding'-step. Third, every system is orthogonally projected into the subspace spanned by the columns of U , which we will subsequently call 'reprojection'-step:…”

“…Remark: Note that M = (V ⊤ U ) −1 and T = U ⊤ V are the same as in Section 2.2, if W = V . Therefore, the pMOR approach presented in [5] includes port-Hamiltonian systems in co-energy representation as special case, which was already highlighted in [10].…”

Robust Passivity Preserving Parametric Model Order Reduction using Matrix Interpolation

“…In Ferranti et al [2010Ferranti et al [ , 2011 the overall passivity of parametric reduced order models (ROMs) is guaranteed over the design space of interest. In Panzer et al [2010], Eid et al [2011] a matrix interpolation-based PMOR is presented. A set of reduced system matrices in a common subspace is computed and interpolated to generate a parametric reduced order model.…”

“…The passivity of parametric reduced order model is not guaranteed. This paper proposes a novel PMOR technique that enhances and improves the method of Panzer et al [2010], Eid et al [2011] by using common projection matrices over the entire design space and passivity preserving parameterization schemes. The modified Smith technique, Gugercin et al [2001], Wong et al [2006], is used to compute efficiently the grammians of the large system and then similarity transformation matrices are generated using the projection-based passive truncated balanced realization (TBR), Yan et al [2007].…”