Interactively Estimate Plant Parameters from Response Data

This example shows how to use PID Tuner to fit a linear model to measured SISO response data.

If you have System Identification Toolbox™ software, you can use PID Tuner to estimate the parameters of a linear plant model based on time-domain response data measured from your system. PID Tuner then tunes a PID controller for the resulting estimated model. PID Tuner gives you several techniques to graphically, manually, or automatically adjust the estimated model to match your response data. This example illustrates some of those techniques.

In this example, you load measured response data from a data file into the MATLAB^® workspace you represent the plant as an LTI model. For information about generating simulated data from a Simulink^® model, see Interactively Estimate Plant from Measured or Simulated Response Data (Simulink Control Design).

Load Measured Response Data

Open Live Script

Load the measured response data for this example into the MATLAB® workspace.

load PIDPlantMeasuredIOData

When you import response data, PID Tuner assumes that your measured data represents a plant connected to the PID controller in a negative-feedback loop. In other words, PID Tuner assumes the following structure for your system. PID Tuner assumes that you injected a step signal at the plant input u and measured the system response at y, as shown.

The sample data file for this example, contains three variables, each of which is a 501-by-1 array. inputu is the unit step function injected at u to obtain the response data. outputy is the measured response of the system at y. The time vector t, runs from 0 to 50 s with a 0.1 s sample time. Comparing inputu to t shows that the step occurs at t = 5 s.

You can import response data stored as a numeric array (as in this example), a timeseries object, or an iddata object.

Import Response Data for Identification

Open PID Tuner.
```
pidTuner(tf(1),'PI')
```
In PID Tuner, in the Plant menu, select Identify New Plant.
In the Plant Identification tab, click Get I/O data and select Step Response. This action opens the Import Step Response dialog box.
Enter information about the response data. The output signal is the measured system response, outputy. The input step signal is parametrized as shown in the diagram in the dialog box. Here, enter 5 for the Onset Lag, and 0.1 for Sample Time. Then, click Import.
The Plant Identification plot displays the response data and the response of an initial estimated plant.

Preprocess Data

Depending on the quality and features of your response data, you might want to perform some preprocessing on the data to improve the estimated plant results. PID Tuner provides several options for preprocessing response data, such as removing offsets, filtering, or extracting a subset of the data. In this example, the response data has an offset. It is important for good identification results to remove data offsets. Use the Preprocess menu to do so. (For information about other data preprocessing options, see Preprocess Data.)

On the Plant Identification tab, click Preprocess and select Remove Offset. The Remove Offset tab opens, displaying time plots of the response data and corresponding input signal.
Select Remove offset from signal and choose the response, Output (y). In the Offset to remove text box, specify a value of –2. You can also select the signal initial value or signal mean, or enter a numerical value. The plot updates with an additional trace showing the signal with the offset applied.
Click Apply to save the change to the signal. Click Close Remove Offset to return to the Plant Identification tab.
PID Tuner automatically adjusts the plant parameters to create a new initial guess for the plant based on the preprocessed response signal.

Adjust Plant Structure and Parameters

PID Tuner allows you to specify a plant structure, such as One Pole, Underdamped Pair, or State-Space Model. In the Structure menu, choose the plant structure that best matches your response. You can also add a transport delay, a zero, or an integrator to your plant. For this example, the one-pole structure gives the qualitatively correct response. You can make further adjustments to the plant structure and parameter values to make the estimated system’s response a better match to the measured response data.

PID Tuner gives you several ways to adjust the plant parameters:

Graphically adjust the response of the estimated system by dragging the adjustors on the plot. In this example, drag the red x to adjust the estimated plant time constant. PID Tuner recalculates system parameters as you do so. As you change the estimated system’s response, it becomes apparent that there is some time delay between the application of the step input at t = 5 s, and the response of the system to that step input.
To add a transport delay to the estimated plant model, in the Plant Structure section, check Delay. A vertical line appears on the plot, indicating the current value of the delay. Drag the line left or right to change the delay, and make further adjustments to the system response by dragging the red x.
Adjust the numerical values of system parameters such as gains, time constants, and time delays. To numerically adjust the values of system parameters, click Edit Parameters.
Suppose that you know from an independent measurement that the transport delay in your system is 1.5 seconds. In the Plant Parameters dialog box, enter 1.5 for τ. Check Fix to fix the parameter value. When you check Fix for a parameter, neither graphical nor automatic adjustments to the estimated plant model affect that parameter value.
Automatically optimize the system parameters to match the measured response data. Click Auto Estimate to update the estimated system parameters using the current values as an initial guess.

You can continue to iterate using any of these methods to adjust plant structure and parameter values until the response of the estimated system adequately matches the measured response.

Save Plant and Tune PID Controller

When you are satisfied with the fit, click Apply. Doing so saves the estimated plant, Plant1, to the PID Tuner workspace. PID Tuner automatically designs a PI controller for Plant1 and, in the Step Plot: Reference Tracking plot, displays a new closed-loop response. The Plant List table reflects that Plant1 is selected for the current controller design.

Tip

To examine variables stored in the PID Tuner workspace, view the Plant List.

You can now use the PID Tuner tools to refine the controller design for the estimated plant and examine tuned system responses.

You can also export the identified plant from the PID Tuner workspace to the MATLAB workspace for further analysis. On the PID Tuner tab, click Export. Check the plant model you want to export to the MATLAB workspace. For this example, export Plant1, the plant you identified from response data. You can also export the tuned PID controller. Click OK. The models you selected are saved to the MATLAB workspace.

Identified plant models are saved as identified LTI models, such as idproc (System Identification Toolbox) or idss (System Identification Toolbox).

Tip

Alternatively, right-click a plant in the Data Browser to select it for tuning or export it to the MATLAB workspace.