On Gramian-Based Techniques for Minimal Realization of Large-Scale Mechanical Systems

A note on the cross Gramian for non-symmetric systems

“…Furthermore, since the procedure for cross-Gramian-based Petrov-Galerkin projections suggested in Rahrovani, Vakilzadeh, and Abrahamsson (2014) does generally not yield stable reduced-order models in this setting, an investigation of two-sided and stability-preserving projections for the (non-symmetric) cross Gramian may yield errors comparable to balanced truncation.…”

Cross-Gramian-Based Model Reduction: A Comparison

“…Alternatively, a singular value decomposition of the cross Gramian, W X SVD = U ΣV, can be utilized. Similarly, S := U and R := V can truncated based on the associated singular values σ i = Σ ii ; yet this projection is only approximately balancing [27,23] and the reduced order model's stability is not guaranteed to be preserved.…”

“…Various approaches for cross-Gramian-based model reduction have been studied [2,26,27,23,10], of which this work compares a small selection using a procedural benchmark based on a method to generate random systems introduced in [25]. In this setting, a linear time-invariant input-output system is the central object of interest:ẋ (t) = Ax(t) + Bu(t),…”

Cross-Gramian-based dominant subspaces

“…Hence, both subspaces should be incorporated in the reducing and lifting operator. Balanced truncation, for example, determines a suitable Petrov-Galerkin projection 3 , where U 1 = V 1 , by simultaneous diagonalization of the controllability and observability Gramians, while approximate balancing applies the left and right singular vectors of the cross Gramian as oblique projections directly [34]. For the proposed variant of the dominant subspace method (for an algorithmic description see Section 3.3.1), the left and right singular vectors are conjoined as before, but also scaled column-wise by the associated singular values:…”

“…Hence, both subspaces should be incorporated in the reducing and lifting operator. Balanced truncation, for example, determines a suitable Petrov-Galerkin projection 3 , where U 1 = V 1 , by simultaneous diagonalization of the controllability and observability Gramians, while approximate balancing applies the left and right singular vectors of the cross Gramian as oblique projections directly [34].…”