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:

1. In the model, open the block. In the block dialog box, in the Controller menu, select PID.

2. Click Tune. PID Tuner opens.

3. 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.