The fifth controller discussed in this series of articles is PID+: a PID controller modified with the command filter (Figure 6-24 below). As with PI+, the goal for PID+ is to allow higher integral gains for improved DC stiffness. Again, the input filter cancels peaking caused by high integral gains; as with PI+, the command response suffers as the stiffness improves.

Figure 6-24. Experiment 6E, a PID+ controller

Tuning a PID+ controller is the same as tuning a PID controller except the value of KFR must be selected before tuning the integral gain (similar to PI+). The process is shown in Figure 6-25 below.

Figure 6-25. Tuning a PID+ controller.

The results of the tuning process of Figure 6-25 above are shown in Figures 6-24 and Figure 6-26 and Figure 6-27 below. The integral gain increased to 300, up from 120 in the PID controller.

The closed-loop Bode plot shows the bandwidth fell to 282 Hz, down from 359 Hz in the PID controller. However, the PID+ controller has only 140° phase lag, which is superior to the 170° phase lag of the closed-loop PID controller.

Figure 6-26. Closed-loop Bode plot of a PID+ controller (282-Hz bandwidth, 0.4 dB peaking).

Comparing the PID+ and PI+ controllers, introduction of D gain allows the PID+ controller to have higher bandwidth (282 Hz compared to 180 Hz) and similar DC stiffness, as indicated by the integral gain (300).

As shown in Figure 6-27 below, the GM for the PID+ controller is similar to that of the PID controller but with 45° PM, 10° less than the PID controller. This is expected; as with the PI+ controller, the command filter allows the controller to work with a lower PM.

Figure 6-27. Open-loop Bode plot of a PID+ controller (45° PM, 7.9 dB GM).

PD Control
The sixth controller covered in this series is a PD controller. Here the P controller is augmented with a D term to allow the higher proportional gain. The controller is shown in Figure 6-28 below. It is identical to the PID controller with a zero I gain.

Figure 6-28. Experiment 6F, a PD controller.

Tuning a PD controller (Figure 6-29 below) is the same as tuning a PID controller, but assume KI is zero. The effects of noise are the same as those experienced with the PID controller.

Figure 6-29. Tuning a PD controller.

The results of tuning are shown in Figures 6-28 earlier and Figures 6-30 and 6-31 below. The step response is square. The introduction of the D gain allowed the P gain to be raised from 1.2 to 1.7.

This allows much higher bandwidth (353 Hz for the PD controller compared to 186 Hz for the P controller), although the phase lag at that bandwidth is much higher (162° for the PD controller compared to 110° for the P controller). As with the PID controller, the PD controller is fast but more susceptible to stability problems.

Figure 6-30. Closed-loop Bode plot of a PD controller (353 Hz bandwidth, 0 dB peaking).

Figure 6-31. Open-loop Bode plot of a PD controller (63° PM, 8.8dB GM).

Also, the GM is smaller (8.8 dB, 3 dB lower than for the P controller). The PD controller is useful in the cases where the fastest response is required.

Choosing A Controller
The results of tuning each of the six controllers in this series are tabulated in Table 6-3 below. Each has its strengths and weaknesses. The simple P controller provides performance suitable for many applications.

Table 6-3. Comparison of the six controllers

The introduction of the I term provides DC stiffness but reduces PM. The command filter in PI+ and PID+ allows even higher DC stiffness but reduces bandwidth.

The D term provides higher responsiveness but erodes gain margin and adds phase shift, which is a disadvantage if this loop is to be enclosed in an outer loop.

Figure 6-32. Selecting the controller.

The chart in Figure 6-32 above provides a procedure for selecting a controller. First determine whether the application needs a D gain; if not, avoid it, because it adds complexity, increases noise susceptibility, and steals gain margin.

Next, make sure the application can support D gains; systems that are noisy may not work well with a differential gain. After that, examine the application for the needed DC stiffness.

If none is required, avoid the integral gain. If some is needed, use the standard form (PI or PID); if maximum DC stiffness is required, add the input filter by using PI+ or PID+ control.

To read Part 1, go to "Moving beyond PID"
To read Part 2, go to "How to tune a Proportional Controller."
To read Part 3, go to "How to tune a PI controller"
To read Part 4, go to "Tuning a Pl+ Controller."
To read Part 5, go to "Tuning A PID controller."

(Editor's Note: Experiments 6A-6F
All the examples in this series of articles were run on Visual Mode1Q. Each of the six experiments, 6A-6F, models one of the six methods, P, PI, PI+, PID, PID+, and PD, respectively.

These are models of digital systems, with sample frequency defaulting to 2 kHz. If you prefer experimenting with an analog controller, set the sample time to 0.0001 second, which is so much faster than the power converter that the power converter dominates the system, causing it to behave like an analog controller.

The default gains reproduce the results shown in this series, but you can go further. Change the power converter bandwidth and investigate the effect on the different controllers.

Assume noise is a problem, reduce the low-pass filter on the D gain (fD), and observe how this reduces the benefit available from the derivative-based controllers (PID, PID+, and PD). Adjust the power converter bandwidth and the sample time, and observe the results. )

This series of articles was excerpted from Control System Design Guide by George Ellis with the permission of the publisher - Elsevier/Academic Books - and can be purchased online which retains all copyrights.

George Ellis is senior scientist at Danaher Motion. He has designed and applied motion control systems for over 20 years and has written for Machine Control Magazine, Control Engineering, Motion Systems Design, Power Control and Intelligent Motion, EDN Magazine. In addition to Control System Design Guide, he is also the author of Observers in Control Systems (Academic Press).