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, design a 2-DOF controller for the plant given by:

Suppose that your target bandwidth for the system is 1.5 rad/s.

wc = 1.5;特遣部队(G = 1, 0.5 - 0.1 [1]);C2 = pidtune(G,'PID2',wc)

C2 = 1 u = Kp (b*r-y) + Ki --- (r-y) + Kd*s (c*r-y) s with Kp = 1.26, Ki = 0.255, Kd = 1.38, b = 0.665, c = 0 Continuous-time 2-DOF PID controller in parallel form.

Using the type'PID2'causespidtuneto generate a 2-DOF controller, represented as apid2object. The display confirms this result.pidtunetunes all controller coefficients, including the setpoint weightsbandc, to balance performance and robustness.

To compute the closed-loop response, note that a 2-DOF PID controller is a 2-input, 1-output dynamic system. You can resolve the controller into two channels, one for the reference signal and one for the feedback signal, as shown in the diagram. (SeeContinuous-Time 2-DOF PID Controller Representationsfor more information.)

Decompose the controller into the componentsCrandCy, and use them to compute the closed-loop response fromrtoy.

C2tf = tf(C2); Cr = C2tf(1); Cy = C2tf(2); T2 = Cr*feedback(G,Cy,+1);

To examine the disturbance-rejection performance, compute the transfer function fromdtoy.

S2 = feedback(G,Cy,+1);

For comparison, design a 1-DOF PID controller with the same bandwidth and compute the corresponding transfer functions. Then compare the step responses.

C1 = pidtune(G,'PID',wc); T1 = feedback(G*C1,1); S1 = feedback(G,C1); subplot(2,1,1) stepplot(T1,T2) title('Reference Tracking') subplot(2,1,2) stepplot(S1,S2) title('Disturbance Rejection') legend('1-DOF','2-DOF')

The plots show that adding the second degree of freedom eliminates the overshoot in the reference-tracking response without any cost to disturbance rejection. You can improve disturbance rejection too using theDesignFocusoption. This option causespidtuneto favor disturbance rejection over setpoint tracking.

opt = pidtuneOptions('DesignFocus','disturbance-rejection'); C2dr = pidtune(G,'PID2',wc,opt)

C2dr = 1 u = Kp (b*r-y) + Ki --- (r-y) + Kd*s (c*r-y) s with Kp = 1.72, Ki = 0.593, Kd = 1.25, b = 0, c = 0 Continuous-time 2-DOF PID controller in parallel form.

With the default balanced design focus,pidtuneselects abvalue between 0 and 1. For this plant, when you change design focus to favor disturbance rejection,pidtunesetsb= 0 andc= 0. Thus,pidtuneautomatically 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.)

Compare the closed-loop responses using all three controllers.

C2dr_tf = tf(C2dr); Cdr_r = C2dr_tf(1); Cdr_y = C2dr_tf(2); T2dr = Cdr_r*feedback(G,Cdr_y,+1); S2dr = feedback(G,Cdr_y,+1); subplot(2,1,1) stepplot(T1,T2,T2dr) title('Reference Tracking') subplot(2,1,2) stepplot(S1,S2,S2dr); title('Disturbance Rejection') legend('1-DOF','2-DOF','2-DOF rejection focus')

The plots show that 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. 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.