## Tune 2-DOF PID Controller (PID Tuner)

This example shows how to design a two-degree-of-freedom (2-DOF) PID controller using PID Tuner. The example also compares the 2-DOF controller performance to the performance achieved with a 1-DOF PID controller.

In this example, you represent the plant as an LTI model. For information about using PID Tuner to tune a PID Controller (2DOF) block in a Simulink® model, see Design Two-Degree-of-Freedom PID Controllers (Simulink Control Design).

2-DOF PID controllers include setpoint weighting on the proportional and derivative terms. Compared to a 1-DOF PID controller, a 2-DOF PID controller can achieve better disturbance rejection without significant increase of overshoot in setpoint tracking. A typical control architecture using a 2-DOF PID controller is shown in the following diagram.

For this example, first design a 1-DOF controller for the plant given by:

`$G\left(s\right)=\frac{1}{{s}^{2}+0.5s+0.1}.$`
```G = tf(1,[1 0.5 0.1]); pidTuner(G,'PID')```

Suppose for this example that your application requires a faster response than the PID Tuner initial design. In the text box next to the Response Time slider, enter 2.

The resulting response is fast, but has a considerable amount of overshoot. Design a 2-DOF controller to improve the overshoot. First, set the 1-DOF controller as the baseline controller for comparison. Click the Export arrow and select ```Save as Baseline```.

Design the 2-DOF controller. In the Type menu, select `PID2`.

PID Tuner generates a 2-DOF controller with the same target response time. The controller parameters displayed at the bottom right show that PID Tuner tunes all controller coefficients, including the setpoint weights `b` and `c`, to balance performance and robustness. Compare the 2-DOF controller performance (solid line) with the performance of the 1-DOF controller that you stored as the baseline (dotted line).

Adding the second degree of freedom eliminates the overshoot in the reference tracking response. Next, add a step response plot to compare the disturbance rejection performance of the two controllers. Select Add Plot > Input Disturbance Rejection.

You can move the plots in the PID Tuner such that the disturbance-rejection plot side by side with the reference-tracking plot.

The disturbance-rejection performance is identical with both controllers. Thus, using a 2-DOF controller eliminates reference-tracking overshoot without any cost to disturbance rejection.

You can improve disturbance rejection too by changing the PID Tuner design focus. First, click the Export arrow and select ```Save as Baseline``` again to set the 2-DOF controller as the baseline for comparison.

Change the PID Tuner design focus to favor reference tracking without changing the response time or the transient-behavior coefficient. To do so, click Options, and in the Focus menu, select `Input disturbance rejection`.

PID Tuner automatically retunes the controller coefficients with a focus on disturbance-rejection performance.

With the default balanced design focus, PID Tuner selects a `b` value between 0 and 1. For this plant, when you change design focus to favor disturbance rejection, PID Tuner sets `b` = 0 and `c` = 0. Thus, PID Tuner automatically generates an I-PD controller to optimize for disturbance rejection. (Explicitly specifying an I-PD controller without setting the design focus yields a similar controller.)

The response plots show that with the change in design focus, the disturbance rejection is further improved compared to the balanced 2-DOF controller. This improvement comes with some sacrifice of reference-tracking performance, which is slightly slower. However, the reference-tracking response still has no overshoot.

Thus, using 2-DOF control can improve disturbance rejection without sacrificing as much reference tracking performance as 1-DOF control. These effects on system performance depend strongly on the properties of your plant and the speed of your controller. For some plants and some control bandwidths, using 2-DOF control or changing the design focus has less or no impact on the tuned result.