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Reduce complexity of linear time-invariant (LTI) models

The **Model Reducer** app lets you compute reduced-order approximations
of high-order models. Working with lower-order models can simplify analysis and control
design. Simpler models are also easier to understand and manipulate. You can reduce a
plant model to focus on relevant dynamics before designing a controller for the plant.
Or, you can use model reduction to simplify a full-order controller.

Using any of the following methods, **Model Reducer** helps you reduce model
order while preserving model characteristics that are important to your
application:

Balanced Truncation — Remove states with relatively small energy contributions.

Mode Selection — Select modes by specifying a frequency range of interest.

Pole-Zero Simplification — Eliminate canceling or near-canceling pole-zero pairs.

**Model Reducer** provides response plots and error plots to help ensure that
the reduced-order model preserves important dynamics. For more information on model
reduction and why it is useful, see Model Reduction Basics.

For an alternative to the **Model Reducer** app that lets you interactively
perform model reduction and generate code for a live script, see the Reduce Model Order task in the Live
Editor.

MATLAB

^{®}Toolstrip: On the**Apps**tab, under**Control System Design and Analysis**, click the app icon.MATLAB command prompt: Enter

`modelReducer`

.

`Model`

— Currently selected model for reductionmodel name

Specify the model you want to reduce by selecting from the
**Model** drop-down list. The list includes all models
currently in the **Data Browser**. To get a model from the
MATLAB workspace into the **Data Browser**, on the
**Model Reducer** tab, click
**Import Model**. You can
import any:

`tf`

,`ss`

, or`zpk`

model that is proper. The model can be SISO or MIMO, and continuous or discrete.Continuous-time models must not have time delays. To reduce a continuous-time model with time delays, first use

`pade`

to approximate the time delays as model dynamics.Discrete-time models can have time delays. For the Balanced Truncation reduction method, the app uses

`absorbDelay`

to convert the delay into poles at*z*= 0 before reducing the model. The additional states are reflected in the response plot and Hankel singular-value plot.

Generalized model such as a

`genss`

model. The**Model Reducer**app uses the current or nominal value of all control design blocks in`model`

(see`getValue`

).

**Note**

**Model Reducer** assumes that the model time unit (specified
in the `TimeUnit`

property of the model) is seconds. If
your model does not have `TimeUnit = 'seconds'`

, use
`chgTimeUnit`

to convert
the model to seconds.

`Reduced orders`

— Number of states in reduced modelinteger | integer array

Specify the number of states in the reduced-order model. Any value is
permitted that falls between the number of unstable states in the model and
the number of states in the original model. If you specify a single value,
**Model Reducer** computes and displays the responses of a model
of that order. If you specify multiple values, **Model Reducer**
computes models of all specified orders and displays their responses on the
same plot. To store reduced models in the **Data Browser**,
click .

For more information, see Balanced Truncation Model Reduction.

**Example: **`5`

**Example: **`4:7`

**Example: **`[3,7,10]`

`Error Bound`

— Type of error bound`absolute`

(default) | `relative`

You can choose between absolute and relative errors by selecting the
appropriate option in **Error Bound**. Setting it to
`absolute`

controls the absolute error $${\Vert G-{G}_{r}\Vert}_{\infty}$$ while setting it to `relative`

controls the relative error $${\Vert {G}^{-1}\left(G-{G}_{r}\right)\Vert}_{\infty}$$. Relative error gives a better match across frequency
while absolute error emphasizes areas with most gain.

For more information, see Balanced Truncation Model Reduction.

`Preserve DC Gain`

— Match DC gain of reduced model to original modelchecked (default) | unchecked

When **Preserve DC Gain** is checked, the DC gain of the
reduced model equals the DC gain of the original model. When the DC behavior
of the model is important in your application, leave this option checked.
Uncheck the option to get better matching of higher-frequency
behavior.

For more information, see Balanced Truncation Model Reduction.

`Focus on range`

— Limit analysis to specified frequenciesunchecked (default) | checked

By default, **Model Reducer** analyzes Hankel singular values
across all frequencies. Restricting this analysis to a particular frequency
range is useful when you know the model has modes outside the region of
interest to your particular application. When you apply a frequency limit,
**Model Reducer** determines which states are the low-energy
states to truncate based on their energy contribution within the specified
frequency range only. **Focus on range** is only available
when **Error Bound** is set to
`absolute`

.

To limit the analysis of state contributions to a particular frequency
range, check **Focus on range**. Then, drag the vertical
cursors on the response plot to specify the frequency range of interest.
Alternatively, enter a frequency range in the text box as a vector of the
form `[fmin,fmax]`

. Units are
`rad/TimeUnit`

, where `TimeUnit`

is
the `TimeUnit`

property of the model you are
reducing.

`Model`

— Currently selected model for reductionmodel name

Specify the model you want to reduce by selecting from the
**Model** drop-down list. The list includes all models
currently in the **Data Browser**. To get a model from the
MATLAB workspace into the **Data Browser**, on the
**Model Reducer** tab, click
**Import Model**. You can
import any:

`tf`

,`ss`

, or`zpk`

model that is proper. The model can be SISO or MIMO, and continuous or discrete.Continuous-time models must not have time delays. To reduce a continuous-time model with time delays, first use

`pade`

to approximate the time delays as model dynamics.Discrete-time models can have time delays. For the Balanced Truncation reduction method, the app uses

`absorbDelay`

to convert the delay into poles at*z*= 0 before reducing the model. The additional states are reflected in the response plot and Hankel singular-value plot.

Generalized model such as a

`genss`

model. The**Model Reducer**app uses the current or nominal value of all control design blocks in`model`

(see`getValue`

).

For more information, see Mode-Selection Model Reduction.

**Note**

**Reduce Model Order** assumes that the
model time unit (specified in the `TimeUnit`

property
of the model) is seconds. If your model does not have ```
TimeUnit
= 'seconds'
```

, use `chgTimeUnit`

to convert
the model to seconds.

`Lower Cutoff`

— Lowest mode frequencypositive scalar

Enter the frequency of the slowest dynamics to preserve in the reduced model. Poles with natural frequency below this cutoff are eliminated from the reduced model.

`Upper Cutoff`

— Highest mode frequencypositive scalar

Enter the frequency of the fastest dynamics to preserve in the reduced model. Poles with natural frequency above this cutoff are eliminated from the reduced model.

`Model`

— Currently selected model for reductionmodel name

Specify the model you want to reduce by selecting from the
**Model** drop-down list. The list includes all models
currently in the **Data Browser**. To get a model from the
MATLAB workspace into the **Data Browser**, on the
**Model Reducer** tab, click
**Import Model**. You can import any:

`tf`

,`ss`

, or`zpk`

model that is proper. The model can be SISO or MIMO, and continuous or discrete.Continuous-time models must not have time delays. To reduce a continuous-time model with time delays, first use

`pade`

to approximate the time delays as model dynamics.Discrete-time models can have time delays. For the Balanced Truncation reduction method, the app uses

`absorbDelay`

to convert the delay into poles at*z*= 0 before reducing the model. The additional states are reflected in the response plot and Hankel singular-value plot.

Generalized model such as a

`genss`

model. The**Model Reducer**app uses the current or nominal value of all control design blocks in`model`

(see`getValue`

).

`Simplification of Pole-Zero Pairs`

— Tolerance for pole-zero cancellationpositive scalar

Set the tolerance for pole-zero cancellation by using the slider or
entering a value in the text box. The value determines how close together a
pole and zero must be for **Model Reducer** to eliminate them from
the reduced model. Moving the slider to the left or entering a smaller value
in the text box simplifies the model less, by cancelling fewer poles and
zeros. Moving the slider to the right, or entering a larger value,
simplifies the model more by cancelling poles and zeros that are further
apart.

For more information, see Pole-Zero Simplification.