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When working with state-space models, proper scaling is important for accurate computations. A state-space model is well scaled when the following conditions exist:

The entries of the

*A*,*B*, and*C*matrices are homogenous in magnitude.The model characteristics are insensitive to small perturbations in

*A*,*B*, and*C*(in comparison to their norms).

Working with poorly scaled models can cause your model a severe
loss of accuracy and puzzling results. An example of a poorly scaled
model is a dynamic system with two states in the state vector that
have units of light years and millimeters. Such disparate units may
introduce both very large and very small entries into the *A* matrix.
Over the course of computations, this mix of small and large entries
in the matrix could destroy important characteristics of the model
and lead to incorrect results.

For more information on the harmful affects of a poorly scaled model, see Scaling Models to Maximize Accuracy.

You can avoid scaling issues altogether by carefully selecting units to reduce the spread between small and large coefficients.

In general, you do not have to perform your own scaling when using the Control System Toolbox™ software. The algorithms automatically scale your model to prevent loss of accuracy. The automated scaling chooses a frequency range to maximize accuracy based on the dominant dynamics of the model.

In most cases, automated scaling provides high accuracy without
your intervention. For some models with dynamics spanning a wide frequency
range, however, it is impossible to achieve good accuracy at *all* frequencies
and some tradeoff of accuracy in different frequency bands is necessary.
In such cases, a warning alerts you of potential inaccuracies. If
you receive this warning, evaluate the tradeoffs and consider manually
adjusting the frequency interval where you most need high accuracy.
For information on how to manually scale your model, see Manually Scaling Your Model.

For models with satisfactory scaling, you can bypass automated
scaling in the Control System
Toolbox software. To do so, set the `Scaled`

property
of your state-space model to `1`

(true). For information
on how to set this property, see the `set`

reference
page.

If automatic scaling produces a warning, you can use the `prescale`

command
to manually scale your model and adjust the frequency interval where
you most need high accuracy.

The `prescale`

command includes a Scaling Tool
GUI, which you can use to visualize accuracy tradeoffs and to adjust
the frequency interval where this accuracy is maximized.

To scale your model using the Scaling Tool GUI, you perform the following steps:

Specifying the Frequency Axis Limits in the Scaling Tool GUI

Specifying the Frequency Band for Maximum Accuracy in the Scaling Tool GUI

For an example of using the Scaling Tool GUI on a real model, see Scaling Models to Maximize Accuracy.

For more information about scaling models from the command line,
see the `prescale`

reference
page.

To open the Scaling Tool GUI for a state-space model named `sys`

,
type

prescale(sys)

The Scaling Tool GUI resembles one shown in the following figure.

The Scaling Tool GUI contains the following plots:

The

**Frequency Response Gain**plot helps you determine the frequency band over which you want to maximize scaling.For SISO systems, this plot shows the gain of your model. For MIMO systems, the plot shows the principle gain (largest singular value) of your model.

The

**Frequency Response Accuracy**plot allows you to view the accuracy tradeoffs for your model when maximizing accuracy in a particular frequency bands.This plot shows the following information:

Relative accuracy of the response of the original unscaled model in red

Relative accuracy of the response of the scaled model in blue

Best achievable accuracy when using independent scaling at each frequency in brown

When you compute some model characteristics, such as the frequency response or the system zeros, the software produces the exact answer for some perturbation of the model you specified. The

*relative accuracy*is a measure of the worst-case relative gap between the frequency response of the original and perturbed models. The perturbation accounts for rounding errors during calculation. Any relative accuracy value greater than`1`

implies poor accuracy.### Tip

If the blue Scaled curve is close to the brown Pointwise Optimal curve in a particular frequency band, you already have the best possible accuracy in that frequency band.

You can change the limits of the plot axis to view a particular
frequency band of interest in the Scaling Tool GUI. To view a particular
frequency band, specify the band in the **Show response in
the frequency band** fields.

This action updates the frequency axis of the Scaling tool to show the specified frequency band.

To return to the default display, select the **Auto** check
box.

To adjust the frequency band where you want maximum accuracy,
set a new frequency band in the **Maximize accuracy in the
frequency band** fields. You can visualize accuracy tradeoffs
by trying out different frequency bands and viewing the resulting
relative accuracy across the frequency band of interest.

You can use the **Frequency Response Gain** plot,
which plots the gain of your model, to view the dynamics in your model
to help determine the frequency band to maximize accuracy.

Each time you specify a new frequency band, the **Frequency
Response Accuracy** plot updates with the result of the new
scaling. Compare the Scaled curve (blue) to the Pointwise Optimal
curve (brown) to determine where the new scaling is nearly optimal
and where you need more accuracy.

To return to the default scaling, select the **Auto** check
box.

When you find a good scaling for your model, save the scaled model as follows:

Click

**Save Scaling**.This action opens the

**Save to Workspace**dialog box.In the

**Save to Workspace**dialog box, verify that any of the following items you want to save are selected, and specify variable names for these items.Scaled model

Scaling information, including:

Scaling factors

Frequencies used to test accuracy

Relative accuracy at each test frequency

For details about the scaling information, see the

`prescale`

reference page.

Click

**OK**.This action sets the State-Space (@ss) object

`Scaled`

property of your model to true. When you set this property to`True`

, the Control System Toolbox algorithms skip the automated scaling of the model.