Estimating Hammerstein-Wiener Models

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Estimating Hammerstein-Wiener Models with Default Settings Plotting the Nonlinearities and Linear Transfer Function Changing the Hammerstein-Wiener Model Input Delay Changing the Nonlinearity Estimator in a Hammerstein-Wiener Model Selecting the Best Model

Estimating Hammerstein-Wiener Models with Default Settings

In this portion of the tutorial, you estimate nonlinear Hammerstein-Wiener models using default model structure and estimation options.

You must have already prepared the data, as described in Preparing Data. For more information about nonlinear ARX models, see What Is a Hammerstein-Wiener Model?

1. In the System Identification Tool GUI, select Estimate > Nonlinear models to open the Nonlinear Models dialog box.

2. In the Configure tab, select Hammerstein-Wiener in the Model type list.

   

   The I/O Nonlinearity tab is open. The default nonlinearity estimator is Piecewise Linear with 10 units for Input Channels and Output Channels, which corresponds to 10 breakpoints for the piecewise linear function.

3. Select the Linear Block tab to view the model orders and input delay.

   

   By default, the model orders and delay of the linear output-error (OE) model are n_b=2, n_f=3, and n_k=1.

4. Click Estimate.

   This action adds the model nlhw1 to the System Identification Tool GUI.

5. In the System Identification Tool GUI, select the Model output check box.

   This action simulates the model using the input validation data as input to the model and plots the simulated output on top of the output validation data.

   The Best Fits area of the Model Output window shows the agreement between the model output and the validation-data output.

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Plotting the Nonlinearities and Linear Transfer Function

You can plot the input/output nonlinearities and the linear transfer function of the model on a Hammerstein-Wiener plot.

1. In the System Identification Tool GUI, select the Hamm-Wiener check box to view the Hammerstein-Wiener model plot.

   The plot displays the input nonlinearity, as shown in the following figure.

   

2. Click the y_NL rectangle in the top portion of the Hammerstein-Wiener Model Plot window.

   The plot updates to display the output nonlinearity.

   

3. Click the Linear Block rectangle in the top portion of the Hammerstein-Wiener Model Plot window.

   The plot updates to display the step response of the linear transfer function.

   

4. In the Choose plot type list, select Bode. This action displays a Bode plot of the linear transfer function.

   

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Changing the Hammerstein-Wiener Model Input Delay

In this portion of the tutorial, you estimate a Hammerstein-Wiener model with a specific model order and nonlinearity settings. Typically, you select model orders and delays by trial and error until you get a model that produces a satisfactory fit to the data.

You must have already estimated the Hammerstein-Wiener model with default settings, as described in Estimating Hammerstein-Wiener Models with Default Settings.

1. In the Nonlinear Models dialog box, click the Configure tab, and select the Linear Block tab.

2. For the Voltage input channel, double-click the corresponding Input Delay (nk) cell, change the value to 3, and press Enter.

3. Click Estimate.

   This action adds the model nlhw2 to the System Identification Tool GUI and the Model Output window is updated to include this model, as shown in the following figure.

   

   The Best Fits area of the Model Output window shows the quality of the nlhw2 fit.

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Changing the Nonlinearity Estimator in a Hammerstein-Wiener Model

In this portion of the example, you modify the default Hammerstein-Wiener model structure by changing its nonlinearity estimator.

Tip   If you know that your system includes saturation or dead-zone nonlinearities, you can specify these specialized nonlinearity estimators in your model. Piecewise Linear and Sigmoid Network are nonlinearity estimators for general nonlinearity approximation.

1. In the Nonlinear Models dialog box, click the Configure tab, and click the Linear Block tab.

2. In the I/O Nonlinearity tab, for the Voltage input, click the Nonlinearity cell, and select Sigmoid Network from the list. Click the corresponding No. of Units cell and set the value to 20.

   

3. Click Estimate.

   This action adds the model nlhw3 to the System Identification Tool GUI. It also updates the Model Output window, as shown in the following figure.

   

4. In the Nonlinear Models dialog box, click the Configure tab.

5. In the I/O Nonlinearity tab, set the Voltage input Nonlinearity to Wavelet Network. This action sets the No. of Units to be determined automatically, by default.

6. Set the Height output Nonlinearity to One-dimensional Polynomial.

   

7. Click Estimate.

   This action adds the model nlhw4 to the System Identification Tool GUI. It also updates the Model Output window, as shown in the following figure.

   

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Selecting the Best Model

The best model is the simplest model that accurately describes the dynamics.

In this example, the best model fit was produced in Changing the Nonlinearity Estimator in a Hammerstein-Wiener Model.

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Estimating Nonlinear ARX Models

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