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This example shows how to estimate lookup table values from time-domain input-output (I/O) data in the Parameter Estimation tool.

In this example, use the I/O data in `lookup_regular.mat`

to
estimate the values of a lookup table. The MAT-file includes the following
variables:

`xdata1`

— Consists of 63 uniformly-sampled input data points in the range [0,6.5]`ydata1`

— Consists of output data corresponding to the input data samples`time1`

— Time vector

Use the I/O data to estimate the lookup table values in the `lookup_regular`

Simulink^{®} model.
The lookup table in the model contains ten values, which are stored
in the MATLAB^{®} variable `table`

. The initial
table values comprise a vector of 0s. To learn more about how to model
a system using lookup tables, see Guidelines for Choosing a Lookup Table (Simulink) in the Simulink documentation.

To estimate the lookup table values, open a Parameter Estimation session.

Open the lookup table model by typing the following command at the MATLAB prompt:

lookup_regular

This command opens the Simulink model, and loads the estimation data into the MATLAB workspace.

In the Simulink model, select

**Analysis**>**Parameter Estimation**to open a new session with name**lookup_regular**in the Parameter Estimation tool.

Use the following steps to estimate the lookup table values.

Create a new experiment by clicking

**New Experiment**on the**Parameter Estimation**tab. Name it`EstimationData`

. Then import the I/O data,`xdata1`

and`ydata1`

, and the time vector,`time1`

, into the experiment. To do this open the experiment editor by right-clicking`EstimationData`

and selecting**Edit...**. Type`[time1,ydata1]`

in the output dialog box and`[time1,xdata1]`

in the input dialog box in the experiment editor. For more information, see Import Data for Parameter Estimation. After you import the data the experiment looks as follows:Run an initial simulation to view the I/O data, simulated output, and the initial table values. To do so, type the following commands at the MATLAB prompt:

sim('lookup_regular') figure(1); plot(xdata1,ydata1, 'm*', xout, yout,'b^') hold on; plot(linspace(0,6.5,10), table, 'k', 'LineWidth', 2); legend('Measured data','Initial simulation data','Initial table values');

The x-axis and y-axis of the figure represent the input and output data, respectively. The figure shows the following plots:

Measured data — Represented by the magenta stars (*).

Initial table values — Represented by the black line.

Initial simulation data — Represented by the blue deltas (Δ).

You can see that the initial table values and simulated data do not match with the measured data.

To select the table values to estimate, on the

**Parameter Estimation**tab, click the**Select Parameters**button to open the**Edit:Estimated Parameters**dialog. In the**Parameters Tuned for all Experiments**panel, click**Select parameters**to launch the Select Model Variables dialog. Check the box next to table, and click**OK**.The

**Edit:Estimated Parameters**window now looks as follows. The table values are selected for estimation by default.On the

**Parameter Estimation**tab, click**Select Experiment**.`EstimationData`

is selected for estimation by default. If not, check the box under the**Estimation**column, and click**OK**.To estimate the table values using the default settings, on the

**Parameter Estimation**tab, click**Estimate**to open the**Parameter Trajectory**plot and**Estimation Progress Report**window. The**Parameter Trajectory**plot shows the change in the parameter values at each iteration.After the estimation converges, the

**Parameter Trajectory**plot looks like this:The

**Estimation Progress Report**shows the iteration number, number of times the objective function is evaluated, and the value of the cost function at the end of each iteration. After the estimation converges, the**Estimation Progress Report**looks like this:The estimated parameters are saved in

`EstimatedParams`

in the**Results**section of the**Data Browser**pane on the left. To view the results, right-click on`EstimatedParams`

and then select**Open**. The report resembles the following.This report includes the estimated parameter values, the final value of the cost function, and other optimization results. You can see that the optimization stopped when the size of the gradient, 1.18e-14 was less than the criteria value, 1e-3.

After you estimate the table values, as described in Estimate the Table Values Using Default Settings, you must use another data set to validate that you have not over-fitted the model. You can plot and examine the following plots to validate the estimation results:

Residuals plot

Measured and simulated data plots

To validate the estimation results:

Create a new experiment to use for validation. Name it

`ValidationData`

. Import the validation I/O data,`xdata2`

and`ydata2`

, and time vector,`time2`

in the`ValidationData`

experiment. To do this open the experiment editor by right-clicking`ValidationData`

and selecting**Edit...**. Then, type`[time2,ydata2]`

in the output dialog box and`[time2,xdata2]`

in the input dialog box in the experiment editor. For more information, see Import Data for Parameter Estimation.To select the experiment for validation, on the

**Parameter Estimation**tab, click**Select Experiments**. The`ValidationData`

experiment is selected for estimation by default. Deselect the box for estimation and check it for validation.To select results to use, on the

**Validation**tab, click**Select Results to Validate**. Deselect`Use current parameter values`

and select`EstimatedParams`

, and click**OK**.The Parameter Estimation tool, by default, displays the experiment plot after validation. Add the residuals plot by checking the corresponding box on the

**Validation**tab.To start validation, on the

**Validation**tab, click**Validate**.Examine the plots

Experiment plot

You can see that the data simulated using the estimated parameters agrees with the measured validation data.

Click Residual plot: ValidationData to open the residuals plot.

The residuals, which show the difference between the simulated and measured data, lie in the range [-0.15,0.15]— within 15% of the maximum output variation. This indicates a good match between the measured and the simulated table data values.

Plot and examine the estimated table values against the validation data set and the simulated table values by typing the following commands at the MATLAB prompt.

sim('lookup_regular') figure(2); plot(xdata2,ydata2, 'm*', xout, yout,'b^') hold on; plot(linspace(0,6.5,10), table, 'k', 'LineWidth', 2)

The plot shows that the table values, displayed as the black line, match both the validation data and the simulated table values. The table data values cover the entire range of input values, which indicates that all the lookup table values have been estimated.

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