This example shows how to estimate regularized ARX models using automatically generated regularization constants in the System Identification app.
filename = fullfile(matlabroot,'help','toolbox',... 'ident','examples','ex_arxregul.sid'); systemIdentification(filename)
The session imports the following data and model into the System Identification app:
The data is collected by simulating a system with the following known transfer function:
Transfer function model
trueSys is the transfer function model used
to generate the estimation data
previously. You also use the impulse response of this model later
to compare the impulse responses of estimated ARX models.
In the System Identification app, select Estimate > Polynomial Models to open the Polynomial Models dialog box.
Verify that ARX is selected in the Structure list.
In the Orders field, specify [0 50 0] as the ARX model order and delay.
Click Estimate to estimate the model.
arx0500 is added to the System Identification
In the Polynomial Models dialog box, click Regularization.
In the Regularization Options dialog box, select
the Regularization Kernel drop-down list.
Specifying this option automatically determines regularization
constants using the
TC regularization kernel. To
learn more, see the
Click Close to close the dialog box.
In the Name field in the Polynomial
Models dialog box, type
arx0500reg is added to the System
Select the Model output check box in the System Identification app.
The Measured and simulated model output plot shows that both the models have an 84% fit with the data.
Because the model fit to the estimation data is similar with
and without using regularization, compare the impulse response of
the ARX models with the impulse responses of
the system used to collect the estimation data.
trueSys icon in the model
board of the System Identification app.
Select the Transient resp check box to open the Transient Response plot window.
By default, the plot shows the step response.
In the Transient response plot window, select Options > Impulse response to change to plot to display the impulse response.
Select Options > Show 99% confidence intervals to plot the confidence intervals.
The plot shows that the impulse response of the unregularized
arx0500 is far off from the true system and
has huge uncertainties.
To get a closer look at the model fits to the data and the variances, magnify a portion of the plot.
The fit of the regularized ARX model
matches the impulse response of the true system and the variance is
greatly reduced as compared to the unregularized model.