SparseModalTruncation

Sparse modal truncation model order reduction specification

Since R2023b

Description

The SparseModalTruncation object stores model order reduction specifications for the modal truncation of sparse linear time-invariant (LTI) models.

For sparse modal truncation, unlike the ordinary modal truncation, you cannot compute all eigenvalues up front. The algorithm computes the N eigenvalues with smallest magnitude and then performs modal decomposition in this N-dimensional subspace. Hence, the modal decomposition is only approximate.

$$G(s)\approx D+{\displaystyle \sum _{k=1}^{n_c}{G}_k(s)}$$

By default, the algorithm computes the first 1000 modes, which can be very time consuming. But, you can limit the algorithm to compute a subset of modes based on the R.Options.Focus or R.Options.MaxOrder settings.

Creation

The reducespec function creates a modal truncation model order reduction object when you use this syntax.

R = reducespec(sys,"modal")

Here, sys is a sparse LTI model (sparss, mechss). The workflow uses this object to set up MOR tasks and store results. For the full model order reduction workflow, see Task-Based Model Order Reduction Workflow.

Tip

Computing all modes (poles) of sparse models is very time consuming. For best results, first set R.Options.Focus or R.Options.MaxOrder appropriately for your application to limit the number of modes to compute.

Properties

Mode — Average pole location in modal components

[] (default) | vector

This property is read-only.

Average mode (pole) value in the subset of computed modal components G_k, returned as a vector of length n_c-by-1, where n_c is the number of computed modal components.

Damping — Damping ratio of average poles

[] (default) | vector

This property is read-only.

Damping ratio of average poles in the modal component G_k, returned as a vector of length n_c-by-1, where n_c is the number of computed modal components.

Frequency — Natural frequency of average poles

[] (default) | vector

This property is read-only.

Natural frequency of average poles in the modal component G_k, returned as a vector of length n_c-by-1, where n_c is the number of computed modal components.

Component — Modal components

[] (default) | array of state-space models

This property is read-only.

Computed model components G_k, returned as an n_c-by-1 array of state-space models, where n_c is the number of computed modal components.

DCGain — Normalized DC contribution of modal components

[] (default) | vector

This property is read-only.

Normalized DC contribution of modal components, returned as a vector of length n_c-by-1, where n_c is the number of computed modal components.

These values are the DC gains of the computed modal components G_k(s) divided by the DC gain of the overall system. In some cases, the normalized DC contributions can exceed one if there are modal cancellations near DC (the overall DC gain is smaller than the sum of the DC contributions of each modal component).

TL — Left-side matrix of block-diagonalizing transformation

[] (default) | matrix

This property is read-only.

Left-side matrix of block-diagonalizing transformation, returned as a matrix of size N-by-n_c, where N is the number of states in the original model sys.

The modal truncation algorithm transforms the state-space realization (A, B, C, D, E) of a model to block diagonal matrices (A_m, B_m, C_m, D_m, E_m) given by:

• For explicit state-space models

  $$\begin{array}{ccccc}{A}_m=T_L^{T}A{T}_R,& B_m=T_L^{T}B,& C_m=C{T}_R,& D_m=D,& T_L^T{T}_R=I\end{array}$$

• For descriptor state-space models

  $$\begin{array}{ccccc}{A}_m=T_L^{T}A{T}_R,& B_m=T_L^{T}B,& C_m=C{T}_R,& D_m=D,& E_m=T_L^{T}E{T}_R\end{array}$$

Here, T_L and T_R are tall and skinny and project onto the invariant subspace associated with the computed modes.

TR — Right-side matrix of block-diagonalizing transformation

[] (default) | matrix

This property is read-only.

Right-side matrix of block-diagonalizing transformation, returned as a matrix of size N-by-n_c, where N is the number of states in the original model sys.

The modal truncation algorithm transforms the state-space realization (A, B, C, D, E) of a model to block diagonal matrices (A_m, B_m, C_m, D_m, E_m) given by:

• For explicit state-space models

  $$\begin{array}{ccccc}{A}_m=T_L^{T}A{T}_R,& B_m=T_L^{T}B,& C_m=C{T}_R,& D_m=D,& T_L^T{T}_R=I\end{array}$$

• For descriptor state-space models

  $$\begin{array}{ccccc}{A}_m=T_L^{T}A{T}_R,& B_m=T_L^{T}B,& C_m=C{T}_R,& D_m=D,& E_m=T_L^{T}E{T}_R\end{array}$$

Here, T_L and T_R are tall and skinny and project onto the invariant subspace associated with the computed modes.

Options — Options for sparse modal truncation

SparseModalTruncationOptions object

Options for modal truncation of sparse LTI models, specified as a SparseModalTruncationOptions object. Use dot notation to configure options for R. For example R.Options.Focus = [0,10000].

For more information about available options, see SparseModalTruncationOptions.

Object Functions

process	Run model order reduction algorithm
view (modal)	Plot mode information when using modal truncation method
getrom (modal)	Obtain reduced-order models when using modal truncation method

Examples

Modal Truncation of Sparse State-Space Model

This example shows how to obtain a reduced-order model of a structural beam using the modal truncation method. For this example, consider a SISO sparse state-space model of a cantilever beam. This example uses the linearized model from the Linear Analysis of Cantilever Beam example.

Load the beam model.

load linBeam.mat
size(sys)

Sparse second-order model with 1 outputs, 1 inputs, and 3303 degrees of freedom.

Plot the Bode response.

bode(sys,w)

Create a model order reduction task.

R = reducespec(sys,"modal");

Set the Focus option to compute the eigenvalues only in the specified frequency range.

R.Options.Focus = [0 5e5];

Analyze the model and compute the derived information.

R = process(R);
size(R.Mode)

ans = 1×2

    26     1

The original model contains 26 modes. View the DC contribution of the modal components using the view function.

view(R,"contrib")

Discard the modes with smaller contributions. For this example, discard all modes with DC contribution smaller than 1e-6. This results in a reduced model with 12 modes.

[rsys,info] = getrom(R,MinDC=1e-6);

Plot the bode responses.

bode(sys,rsys,w)

The reduced-order model provides a good approximation for the original sparse model.

Algorithms

Sparse modal truncation is applicable only to two types of symmetric semi-definite problems.

• First-order problems with A = A^T and E = E^T semi-definite (positive or negative)

• Second-order problems with K = K^T, M = M^T semi-definite (positive or negative), and Rayleigh damping C = αM + βK

For both problems, the software uses the Krylov-Schur algorithm [1] to compute a subset of modes (see eigs).

Once you decide which modes to keep and which modes to discard, the algorithm obtains the approximate modal decomposition G(s) = G₁(s) + G₂(s) + D, where G₁(s) contains the desired modes. Then, based on the specified method, getrom returns the reduced-order model as follows.

• "truncate" — G_r(s) = G₁(s) + D

• "matchDC" — G_r(s) = G₁(s) + D + G₂(0)

References

[1] Stewart, G. W. “A Krylov--Schur Algorithm for Large Eigenproblems.” SIAM Journal on Matrix Analysis and Applications 23, no. 3 (January 2002): 601–14. https://doi.org/10.1137/S0895479800371529.

Version History

Introduced in R2023b