System Identification Toolbox™

Estimating a Second-Order Transfer Function (Process Model) with Complex Poles

Estimating a Second-Order Transfer Function Using Default Settings

In this portion of the tutorial, you estimate models with this structure:

You must have already processed the data for estimation, as described in Plotting and Processing Data.

If you have not performed this step, click here to complete it.

To identify a second-order transfer function:

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

   

2. In the Model Transfer Function area, specify the following options:

   • Under Poles, select 2 and Underdamped.

     This selection updates the Model Transfer Function to a second-order model structure that can contain complex poles.

     

     The Parameter area now shows four active parameters: K, Tw, Zeta, and Td. By default, the model Name (located at the bottom of the dialog box) is set to the acronym P2DU, which indicates two poles (P2), a delay (D), and underdamped modes (U).

   • Make sure that the Zero and Integrator check boxes are cleared to exclude a zero and an integrator (self-regulating ) from the model.

3. In the Initial Guess area, keep the default Auto-selected option to calculate the initial parameter values during the estimation. The Initial Guess column in the Parameter table displays Auto.

   

4. Keep the default Bounds values, which specify the minimum and maximum values of each parameter.

   Tip   If you know the range of possible values for a parameter, you can type these values into the corresponding Bounds fields to help the estimation algorithm.

5. Keep the default settings for the estimation algorithm:

   • Disturbance Model — None means that the algorithm does not estimate the noise model. This option also sets the Focus to Simulation.

   • Focus — Simulation means that the estimation algorithm does not use the noise model to weigh the relative importance of how closely to fit the data in various frequency ranges. Instead, the algorithm uses the input spectrum in a particular frequency range to weigh the relative importance of the fit in that frequency range.

     Tip   The Simulation setting is optimized for identifying models that you plan to use for output simulation. If you plan to use your model for output prediction or control applications, or to improve parameter estimates using a noise model, select Prediction.

   • Initial state — Auto means that the algorithm analyzes the data and chooses the optimum method for handling the initial state of the system. If you get poor results, you might try setting a specific method for handling initial states, rather than choosing it automatically.

   • Covariance — Estimate means that the algorithm computes parameter uncertainties that display as model confidence regions on plots.

6. Click Estimate. This selection adds the model P2DU to the System Identification Tool GUI.

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Tips for Specifying Known Parameters

If you know a parameter value exactly, you can type this value in the Initial Guess column of the Process Models dialog box.

If you know the approximate value of a parameter, you can help the estimation algorithm by entering an initial value in the Initial Guess column. In this case, keep the Known check box cleared to allow the estimation to fine-tune this initial guess.

For example, to fix the time-delay value Td at 2s, you can type this value into Value field of the Parameter table in the Process Models dialog box and select the corresponding Known check box.

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Validating the Model

You can analyze the following plots to evaluate the quality of the model:

• Comparison of the model output and the measured output on a time plot

• Autocorrelation of the output residuals, and cross-correlation of the input and the output residuals

You must have already estimated the model, as described in Estimating a Second-Order Transfer Function Using Default Settings.

If you have not performed this step, click here to complete it.

Examining Model Output

You can use the model-output plot to check how well the model output matches the measured output in the validation data set. A good model is the simplest model that best describes the dynamics and successfully simulates or predicts the output for different inputs.

To generate the model-output plot:

• Select the Model output check box in the System Identification Tool GUI.

  If the plot is empty, click the model icon in the System Identification Tool window to display the model on the plot.

The System Identification Toolbox product uses input validation data as input to the model, and plots the simulated output on top of the output validation data. The preceding plot shows that the model output agrees well with the validation-data output.

The Best Fits area of the Model Output plot shows the agreement (in percent) between the model output and the validation-data output.

Recall that the data was simulated using the following second-order system with underdamped modes (complex poles), as described in Data Description, and has a peak response at 1 rad/s:

Because the data includes noise at the input during the simulation, the estimated model cannot exactly reproduce the model used to simulate the data.

Examining Model Residuals

You can validate a model by checking the behavior of its residuals.

To generate a Residual Analysis plot, select the Model resids check box in the System Identification Tool GUI.

The top axes show the autocorrelation of residuals for the output (whiteness test). The horizontal scale is the number of lags, which is the time difference (in samples) between the signals at which the correlation is estimated. Any fluctuations within the confidence interval are considered to be insignificant. A good model should have a residual autocorrelation function within the confidence interval, indicating that the residuals are uncorrelated. However, in this example, the residuals appear to be correlated, which is natural because the noise model is used to make the residuals white.

The bottom axes show the cross-correlation of the residuals with the input. A good model should have residuals uncorrelated with past inputs (independence test). Evidence of correlation indicates that the model does not describe how a portion of the output relates to the corresponding input. For example, when there is a peak outside the confidence interval for lag k, this means that the contribution to the output y(t) that originates from the input u(t-k) is not properly described by the model. In this example, there is no correlation between the residuals and the inputs.

Thus, residual analysis indicates that this model is good, but that there might be a need for a noise model.

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Preparing Data for System Identification

Estimating a Transfer Function with a Noise Model