The fifth controller discussed in this series of articles is PID+: aPID controller modified with the command filter (Figure 6-24 below ). As with PI+, thegoal for PID+ is to allow higher integral gains for improved DCstiffness. Again, the input filter cancels peaking caused by highintegral gains; as with PI+, the command response suffers as thestiffness improves.
|Figure6-24. Experiment 6E, a PID+ controller|
Tuning a PID+ controller is the same as tuning a PID controllerexcept the value of KFR must be selected before tuning the integralgain (similar to PI+). The process is shown in Figure 6-25 below .
|Figure6-25. Tuning a PID+ controller.|
The results of the tuning process of Figure 6-25 above are shown inFigures 6-24 and Figure 6-26 andFigure 6-27 below . The integral gain increasedto 300, up from 120 in the PID controller.
The closed-loop Bode plot shows the bandwidth fell to 282 Hz, downfrom 359 Hz in the PID controller. However, the PID+ controller hasonly 140° phase lag, which is superior to the 170° phase lag ofthe closed-loop PID controller.
|Figure6-26. Closed-loop Bode plot of a PID+ controller (282-Hz bandwidth, 0.4dB peaking).|
Comparing the PID+ and PI+ controllers, introduction of D gainallows the PID+ controller to have higher bandwidth (282 Hz compared to180 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 controllerbut with 45° PM, 10° less than the PID controller. This isexpected; as with the PI+ controller, the command filter allows thecontroller to work with a lower PM.
|Figure6-27. Open-loop Bode plot of a PID+ controller (45° PM, 7.9 dB GM).|
The sixth controller covered in this series is a PD controller. Herethe P controller is augmented with a D term to allow the higherproportional gain. The controller is shown in Figure 6-28 below . It is identicalto the PID controller with a zero I gain.
|Figure6-28. Experiment 6F, a PD controller.|
Tuning a PD controller (Figure 6-29below ) is the same as tuning a PID controller, but assume KI iszero. The effects of noise are the same as those experienced with thePID controller.
|Figure6-29. 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 issquare. The introduction of the D gain allowed the P gain to be raisedfrom 1.2 to 1.7.
This allows much higher bandwidth (353 Hz for the PD controllercompared to 186 Hz for the P controller), although the phase lag atthat bandwidth is much higher (162° for the PD controller comparedto 110° for the P controller). As with the PID controller, the PDcontroller is fast but more susceptible to stability problems.
|Figure6-30. Closed-loop Bode plot of a PD controller (353 Hz bandwidth, 0 dBpeaking).|
|Figure6-31. 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 Pcontroller). The PD controller is useful in the cases where the fastestresponse is required.
Choosing A Controller
The results of tuning each of the six controllers in this series aretabulated in Table 6-3 below .Each has its strengths and weaknesses. The simple P controller providesperformance suitable for many applications.
|Table6-3. 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 butreduces bandwidth.
The D term provides higher responsiveness but erodes gain margin andadds phase shift, which is a disadvantage if this loop is to beenclosed in an outer loop.
|Figure6-32. Selecting the controller.|
The chart in Figure 6-32 above provides a procedure for selecting a controller. First determinewhether the application needs a D gain; if not, avoid it, because itadds complexity, increases noise susceptibility, and steals gainmargin.
Next, make sure the application can support D gains; systems thatare 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, usethe standard form (PI or PID); if maximum DC stiffness is required, addthe input filter by using PI+ or PID+ control.
To read Part 1, go to “Moving beyond PID“
To read Part 2, go to “Howto tune a Proportional Controller.”
To read Part 3, go to “How to tune a PIcontroller“
To read Part 4, go to “Tuning a Pl+Controller.“
To read Part 5, go to “Tuning A PIDcontroller.“
(Editor's Note: Experiments 6A-6F
All the examples in this series ofarticles were run on Visual Mode1Q. Each of the sixexperiments, 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 experimentingwith an analog controller, set the sample time to 0.0001 second, whichis so much faster than the power converter that the power converterdominates the system, causing it to behave like an analog controller.
The default gains reproduce the resultsshown in this series, but you can go further. Change the powerconverter bandwidth and investigate the effect on the differentcontrollers.
Assume noise is a problem, reducethe low-pass filter on the D gain (fD), and observe how this reducesthe benefit available from the derivative-based controllers (PID, PID+,and PD). Adjust the power converter bandwidth and the sample time, andobserve the results. )
This series of articles was excerptedfrom ControlSystem Design Guide by George Ellis with the permission of thepublisher – Elsevier/Academic Books – and can be purchased online whichretains all copyrights.
George Ellis is senior scientistat Danaher Motion. He hasdesigned and applied motion control systems for over 20 years and haswritten for Machine Control Magazine, Control Engineering, MotionSystems Design, Power Control and Intelligent Motion, EDN Magazine. Inaddition to Control System Design Guide, he is also the author ofObservers in Control Systems (Academic Press).