|Figure6-13. Tuning a PI + controller.|
For the highest possible stiffness, KFR should be set to zero; herethe PDFF controller reduces to PDF. However, for most applications KFRshould be at least 0.4; there is a substantial loss of response forsetting KFR lower, and the stiffness is improved only marginally.
For applications that require the highest response to command,select KFR = 1 (equivalent to PI) or at least above 0.9. Setting KFR =0.65 is a good compromise for many applications.
|Figure6-14. Experiment 6C, a PI + controller.|
Experiment 6C, shown in Figure6-14 above , will be used to demonstrate the PI+ system. This issimilar to the PI controller of Experiment 6B (Figure 6-5 in Part 2)except that a command filter has been added between the sample-and-hold(“S/H”) at top left and the summing junction just under the waveformgenerator (“Wave Gen”).
This filter implements Equation 6.1 (seePart 3 ).The Live Constant KFR scales the command directly; “1 -KFR” scales the command passing through the low-pass filter. Thelow-pass break frequency is set by KI, although KI must be scaled by0.159 to convert KI (which is in rad/sec) to Hz, the scaling for filterbreak frequency.
One other minor change was required: the Live Constant KI wasconverted from a “Scale-by” constant to a standard Live Constant, withscaling accomplished by a multiplication just above the block. This wasnecessary because KI had to be provided explicitly because it is usedin two places: as the integral gain and as the break frequency for thecommand filter.
The results of tuning a P1+ system are shown in Figure 6-14 above and Figure 6-15 and Figure 6-16 below . The setting forKFR was 0.65, the compromise value. This allowed the integral gain toincrease from 100 in PI (Figure 6-5 in Part 2) to 300 while maintainingthe same overshoot.
|Figure6-15. Closed-loop Bode plot of PI + system (180 Hz bandwidth, 1.5 dBpeaking).|
The closed-loop plot shows a decline in bandwidth, from 206 Hz inthe PI system (Figure 6-7 in Part 2) to 180 Hz (Figure 6-15 above ). The open-loopplot (Figure 6-16 below ) showsa PM of 40°, a decline of 15° compared with P1.
Since KFR is outside the loop, it has no direct impact on PM.However, because it filters high-frequency components in the commandsignal, lower KFR allows higher integral gains, which, in turn, reducethe PM. However, because it filters high-frequency components in thecommand signal, lower KFR allows higher integral gains, which, in turn,reduce the PM.
|Figure6-16. Open-loop Bode plot of PI + controller (40° PM, 10.4 dB GM).|
Using the Differential Gain
The third gain that can be used for controllers is the differential, or”D,” gain. The D gain advances the phase of the loop by virtue of the90° phase lead of a derivative. Using D gain will usually allow thesystem responsiveness to increase, for example, allowing the bandwidthto nearly double in some cases.
Differential gain has shortcomings. Derivatives have high gain athigh frequencies. So while some D does help the phase margin, too muchhurts the gain margin by adding gain at the phase crossover, typicallya high frequency. This makes the D gain difficult to tune.
The designer sees overshoot improve because of increased PM, but ahigh frequency oscillation, which comes from reduced GM, becomesapparent. The high frequency problem is often hard to see in the timedomain because high-frequency ringing can be hard to distinguish fromnormal system noise.
So a control system may be accepted at installation but havemarginal stability and thus lack the robust performance expected forfactory equipment. This problem is much easier to see using Bode plotsmeasured on the working system.
Another problem with derivative gain is that derivatives aresensitive to noise. Even small amounts of noise from wiring orresolution limitations may render the D gain useless. In most cases,the D gain needs to be followed by a low-pass filter to reduce thenoise content.
The experiments in this series assume a near-noiseless system, sothe D filter is set high (2000 Hz). In many systems, especially inanalog controllers, such a value would be unrealistic.
(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).