## Specify Options for Balanced Truncation in Model Reducer

The Model Reducer app provides several options to configure model order
reduction using Balanced Truncation method for LTI and sparse LTI models. The following
sections describe the available options on the **Options**
dialog.

### Balanced Truncation Configuration

Use these options when your model has unstable or near-unstable dynamics to control the tolerance with which Model Reducer separates these dynamics.

When reducing a model that has unstable poles, Model Reducer separates
the unstable and stable dynamics to analyze the state contributions of stable modes.
For a model *G*, this stable/unstable decomposition is:

*G* → *GS* + *GU*.

The options in this dialog control numerical aspects of this decomposition.

**Apply balanced truncation to normalized coprime factors**— Apply balanced truncation to normalized coprime factors of the original model. This option requires Robust Control Toolbox™ software.**Preserve DC gain**— Preserve the DC gain (steady-state value of step response) to match the time response better.**Control relative error**— Choose between absolute and relative error bounds. The absolute error is $${\Vert G-{G}_{r}\Vert}_{\infty}$$ while the relative error is $${\Vert {G}^{-1}\left(G-{G}_{r}\right)\Vert}_{\infty}$$. Relative error gives a better match across frequency while absolute error emphasizes areas with most gain.**Specify Regularization**— Regularization level value that ensures a well-defined relative error at all frequencies, specified as a nonnegative scalar value. When you select**Control relative error**, Model Reducer reduces the model`[sys,alpha*I]`

instead of`sys`

. Uncheck this option to let Model Reducer pick a suitable`alpha`

value. Enable this option to specify a value`alpha >= 0`

to override this default.**Frequency Intervals**— Specify frequency intervals for computing frequency-limited Hankel singular values. Use the**Lower cutoff**and**Upper cutoff**parameters to specify the range. To enable this option, set**Focus**to**Use intervals**.**Time Intervals**— Specify time intervals for computing time-limited Hankel singular values. Use the**Lower cutoff**and**Upper cutoff**parameters to specify the range. To enable this option, set**Focus**to**Use intervals**.**Input Weight**and**Output Weight**— Input and output weights for input-output scaling and frequency weighting, specified as a matrix or a dynamic system model.These options control the frequency-weighted error $$\Vert {W}_{L}(G-{G}_{r}){W}_{R}\Vert $$, where

*G*and*G*are the full-order and reduced-order models, respectively._{r}*W*(output weight) and_{L}*W*(input weight) must be linear time-invariant models of compatible size and have high gain in frequency bands of interest and low gain elsewhere. Doing so emphasizes the accuracy of the reduced-order model in a particular frequency band._{R}**Offset**— Offset for the stable/unstable boundary, specified as a nonnegative scalar value. In the stable/unstable decomposition, the stable term includes only poles that satisfy:`Re(s) < -Offset * max(1,|Im(s)|)`

(Continuous time)`|z| < 1 - Offset`

(Discrete time)

The default value is

`1e-08`

. Increase the offset to treat poles close to the stability boundary as unstable. Model Reducer does not eliminate system modes that it treats as unstable.**SepTol**— Maximum loss of accuracy value in stable and unstable decomposition, specified as nonnegative scalar values.For models with unstable poles, Model Reducer first extracts the stable dynamics using

`stabsep`

. Use**SepTol**to control the decomposition accuracy. Increasing**SepTol**helps separate nearby stable and unstable modes at the expense of accuracy.

#### Related Links

### Sparse Balanced Truncation Configuration

For sparse models, the app requires you to specify some options before you can perform model order reduction. Use this dialog to specify balanced truncation options for sparse models.

**Frequency vector (rad/s)**— Frequencies at which to compute and plot frequency response, specified as a vector.**Preserve DC gain**— Preserve the DC gain (steady-state value of step response) to match the time response better.**Frequency focus**— Dynamic range of interest, specified as a vector of form`[fmix,fmax]`

. Use this option if you know all poles are in this range.**Maximum rank**— Maximum rank of Cholesky factors, specified as a positive integer. The algorithm terminates when the column size of the low-rank Gramian factors*L*and_{r}*L*reaches this limit. For more information, see Algorithms._{o}**Offset**— Spectral offset, specified as a positive scalar.Sparse balanced truncation is only supported for stable systems. For first-order models (

`sparss`

) with integral action, you can use this option to implicitly shift poles to enforce stability. The algorithm shifts poles as follows.Continuous time —

`p`

to`p-Offset`

Discrete time —

`p`

to`(1-Offset)p`

This option is not supported for

`mechss`

models.**Use Rayleigh damping**— Enforce stability for`mechss`

models.Sparse balanced truncation is supported only for stable systems. To enforce stability for undamped second-order models (

`mechss`

), you can use this option to implicitly add**Rayleigh damping**with minimum damping*ζ*at the**Rayleigh frequency***ω*. For best results, pick_{n}*ω*close to the dominant mode and_{n}*ζ*in the range [0.001,0.1].This option is not supported for

`sparss`

models.**Custom shifts**— Custom shifts to accelerate the convergence of the algorithm. Specify shifts based on prior knowledge of pole locations. The algorithm applies these shifts in addition to the default shifts. For more information, see Algorithms.**LyapTol**— Relative tolerance for Lyapunov residuals, specified as a positive scalar. Increasing`LyapTol`

helps speed up computation at the expense of reduced-order model accuracy. Decrease this value to capture more Hankel singular values.**RankTol**— Relative tolerance for rank decisions, specified as a positive scalar. Increasing this value reduces the ranks of*L*and_{r}*L*and results in less accurate reduced-order models. Decreasing this value helps compute small Hankel singular values more accurately and obtain more accurate reduced-order models._{o}