## Specify PI-D and I-PD Controllers

PI-D and I-PD controllers are used to mitigate the influence of changes in the reference signal on the control signal. These controllers are variants of the 2DOF PID controller.

The general formula of a parallel-form 2DOF PID controller is:

$$u=P(br-y)+I\frac{1}{s}(r-y)+D\frac{N}{1+N\frac{1}{s}}(cr-y).$$

Here, *r* and *y* are the reference input and measured
output, respectively. *u* is the controller output, also called the
*control signal*. *P*, *I*, and
*D* specify the proportional, integral, and derivative gains, respectively.
*N* specifies the derivative filter coefficient. *b* and
*c* specify setpoint weights for the proportional and derivative
components, respectively. For a 1DOF PID controller, *b* and
*c* are equal to 1.

If *r* is nonsmooth or discontinuous, the derivative and proportional
components can contribute large spikes or offsets in *u*, which can be
infeasible. For example, a step input can lead to a large spike in *u*
because of the derivative component. For a motor actuator, such an aggressive control signal
could damage the motor.

To mitigate the influence of *r* on *u*, set
*b* or *c*, or both, to 0. Use one of the following
setpoint-weight-based forms:

PI-D (

*b*= 1 and*c*= 0) — Derivative component does not directly propagate changes in*r*to*u*, whereas the proportional component does. However, the derivative component, which has a greater impact, is suppressed. Also referred to as the*derivative of output controller*.The general formula for this controller form is:

$$u=P(r-y)+I\frac{1}{s}(r-y)-D\frac{N}{1+N\frac{1}{s}}y.$$

I-PD (

*b*= 0 and*c*= 0) — Proportional and derivative components do not directly propagate changes in*r*to*u*.The general formula for this controller form is:

$$u=-Py+I\frac{1}{s}(r-y)-D\frac{N}{1+N\frac{1}{s}}y.$$

You can tune the *P*, *I*, *D*, and
*N* coefficients of a PI-D or I-PD controller to achieve the desired
disturbance rejection and reference tracking.

### Simulate PI-D and I-PD Controllers in Simulink

To specify a PI-D or I-PD controller using the PID Controller (2DOF) or Discrete PID Controller (2DOF) blocks, open the block and set the **Controller** parameter to `PID`

.

For a PI-D controller, set the

**Setpoint weight (b)**parameter to`1`

and the**Setpoint weight (c)**parameter to`0`

.For an I-PD controller, set the

**Setpoint weight (b)**parameter to 0 and the**Setpoint weight (c)**parameter to`0`

.

Open the `pid2dof_setpoint_based_controllers`

model, which compares the performance of a 1DOF PID, a PI-D, and an I-PD controller. The model uses the same P, I, and D parameters for all three controllers

```
mdl = "pid2dof_setpoint_based_controllers";
open_system(mdl)
```

Simulate the model.

`sim("pid2dof_setpoint_based_controllers")`

All of the controllers reject the disturbance equally well. The 1DOF PID controller results in a large spike when the reference changes from 0 to 1. The PI-D form results in a smaller jump. In contrast, the I-PD form does not react as much to the reference change.

### Automatic Tuning of PI-D and I-PD Controllers

You can use PID Tuner to automatically tune PI-D and I-PD controllers while
preserving the fixed *b* and *c* values. To do so:

In the model, open the block. In the block dialog box, in the

**Controller**menu, select`PID`

.Click

**Tune**. PID Tuner opens.In PID Tuner, in the

**Type**menu, select`PI-DF`

or`I-PDF`

. PID Tuner retunes the controller gains, fixing*b*= 1 and*c*= 0 for PI-D, and*b*= 0 and*c*= 0 for I-PD.

You can now analyze system responses as described in Analyze Design in PID Tuner.

## See Also

PID Controller (2DOF) | Discrete PID Controller (2DOF)