Dominant Subspaces

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Dominant Subspace Projection Model Reduction Method

The Dominant Subspace Projection Model Reduction Method (DSPMR), introduced in , or short Dominant Subspaces method is closely related to Balanced Truncation, and can be seen as a simplification. This is projection-based method is designed for linear time-invariant systems:

$\begin{array}{rcl}\dot{x}(t)& =& Ax(t)+Bu(t),\\ y(t)& =& Cx(t).\end{array}$

Similar to Balanced Truncation, the controllability Gramian matrix $W_C$ and the observability Gramian matrix $W_O$ are computed:

$A{W}_C+W_C{A}^T=-B{B}^T,$

$A^T{W}_O+W_{O}A=-C^{T}C.$

As a second step, instead of balancing controllability and observability subspaces, the respective subspaces are conjoined. Typically, (low-rank) Cholesky factors of the system Gramians are computed directly,

$W_C=L_C{L}_C^T,$

$W_O=L_O{L}_O^T,$

which are then combined using a singular value decomposition of the concatenated factors:

$[L_C,L_O]=UDV$.

The principal left (or right) singular vectors, based on the magnitude of the singular values, then serve as a Galerkin projection.

Alternatively, different approaches for the combination of subspaces can be used. For example in and , the singular vectors of the singular value decompositions of the system Gramians,

$W_C=U_C{D}_C{V}_C,$

$W_O=U_O{D}_O{V}_O,$

are concatenated and orthogonalized by a rank-revealing QR decomposition:

$[U_C,U_O]=QR$,

with $Q$ being the resulting Galerkin projection.

Furthermore, the singular vectors of the system Gramians can also be combined by a second singular value decomposition as in .

Variants

Refined DSPMR

In , a variant called Refined Dominant Subspace Projection Model Reduction Method is presented. For this variant the subspaces are weighted by their Frobenius norm before combination, so for example:

$[\frac{1}{\Vert L_C{\Vert }_{\text{Fro}}}{L}_C,\frac{1}{\Vert L_O{\Vert }_{\text{Fro}}}{L}_O]=UDV$.

$[\frac{1}{\Vert U_C{\Vert }_{\text{Fro}}}{U}_C,\frac{1}{\Vert U_O{\Vert }_{\text{Fro}}}{U}_O]=QR$.

Cross-Gramian-Based DSPMR

Instead of using controllability and observability Gramians, also the cross Gramian $W_X$ can be used for the DSPMR procedure ,

$A{W}_X+W_{X}A=-BC,$.

The cross Gramian encodes controllability and observability, hence a singular value decompostion of the (the generally not-symmetric) cross Gramian matrix reveals dominant controllability and observability subspaces,

$W_X=U_C{D}_X{V}_O,$

which can then be combined by another singular value decomposition:

$[U_C,V_O]=UDV$.

The principal left (or right) singular values then represent the Galerkin projection.

This variant can also be refined, as proposed in , by scaling the left and right singular vectors by their associated singular values:

$[U_C{D}_X,V_O{D}_X]=UDV$.