PI controllers have two zones: high and low. The high zone is served byKp and the low by K. As Figure 6-6below shows, the process for setting the proportional gain isthe same as it was in the P controller, described earlier in Part 2.
Figure6-6. Tuning a PI controller
After the higher zone is complete, KI can be tuned. Here it israised for 15% overshoot to a square wave. Again, a square wave is anunreasonably harsh command to follow perfectly; a modest amount ofovershoot to a square wave is tolerable in most applications.
As Figure 6.6 in Part 2 andFigures 6-7 and 6-8 below show ,the PI controller is similar to the P controller, but with slightlypoorer stability measures. The integral gain is high enough to cause a15% overshoot to a step. The bandwidth has gone up a bit (from 186 Hzto 206 Hz), but the peaking is about 1.3 dB. The PM has fallen 9°,and the Gain Margin (GM) is nearly unchanged, just down 0.4 dB to 11.7dB.
|Figure6-7. Closed-loop Bode plot for a PI controller (206-Hz bandwidth, 1.3dBof peaking).|
|Figure6-8. Open-loop plot of PI controller (56°, PM 11.7dB GM).|
Analog PI Control
A simple analog circuit can be used to implement PI control. As shownin the schematic of Figure 6-9 below ,a series resistor and capacitor are connected across the feedback pathof an op-amp to form the proportional (RL) and integral (CL) gains.
Clamping diodes clamp the op-amp and prevent the capacitor fromcharging much beyond the saturation level. A small leakage path due tothe diodes is shown as a resistor. The input-scaling resistors areassumed here to be equal (RC = RF).
|Figure 6-9. Schematic for analogPI controller.|
The control block diagram for Figure 6-9 is shown in Figure 6-10 below . Note that thegains in this figure are constructed to parallel those of the generalPI controller in Figure 6-5 in Part2 .Tuning the analog controller is similar to tuning the generalcontroller.
Short (remove) the capacitor to convert the system to a Pcontroller, and determine the appropriate setting of RL, as was donefor KP. Then adjust CL for 15% overshoot. The analog controller willbehave much like the digital controller.
|Figure6-10. Block diagram of analog PI controller.|
One compromise that must be made for analog PI control is thatop-amps cannot form true integrators. The diodes and capacitor willhave some leakage, and, unlike a true integrator, the op-amp haslimited gains at low frequency.
Often, the PI controller is modeled as a lag network, with a largeresistor across the op-amp feedback path, as shown in Figure 6-9 . This “leaky” integratoris sometimes called a lag circuit.
In some cases a discrete resistor is used to cause leakageintentionally. This is useful to keep the integral from charging whenthe control system is disabled. Although the presence of the resistordoes have some effect on the control system, it is usually small enoughand at low enough frequency not to be of much concern.
As the name indicates, PI+ control is an enhancement to PI. Because ofthe overshoot, the integral gain in PI controllers is limited inmagnitude. PI+ control uses a low-pass filter on the command signal toremove overshoot.
In this way, the integral gain can be raised to higher values. PI+is useful in applications where the rejection of DC disturbances isparamount, for example, in a motion controller driving a high-frictionmechanism such as a worm gear. The primary shortcoming of PI+ is thatthe command filter also reduces the controller's command response.
|Figure6-11. Block diagram for PI + control|
The PI+ controller is shown in Figure6-11 above . The system is the PI controller of Figure 6-5 in Part 2 with a commandfilter added. The degree to which a PI+ controller filters the commandsignal is determined by the gain KFR.
As can be seen in Figure 6-11 above, when KFR is 1, all filtering is removed and the controller isidentical to a PI controller. Filtering is most severe when KFR iszero. As can also be seen in Figure 6-11, when KFR is zero, command isfiltered by K1/(s + KI), which is a single-pole low-pass filter at thefrequency KI (in rad/sec).
This case will allow the highest integral gain but also will mostseverely limit the controller command response. Typically, KFR = 0 willallow an increase of almost three times in the integral gain but willreduce the bandwidth by about one-half when compared with KFR = 1 (PIcontrol).
Finding the optimal value of KFR depends on the application, but avalue of 0.65 has been found to work in many applications. This valuetypically allows the integral gain to more than double while reducingthe bandwidth by only 15%-20%.
One question about PI+ that naturally arises is why to select KI asthe frequency of the command low-pass filter? Why not set thatfrequency either higher or lower? The reason is that this frequency isexcellent at canceling the peaking caused by the integral gain.
One way to look at PI+ control is that it uses the command filter toattenuate the peaking caused by PI. The peaking caused by KI can becanceled by the attenuation of a low-pass filter with a break of KI.
|Figure6-12. Alternative implementation for PI + control, a PDFF controller.|
Comparing Pl+ and PDFF
Pl+ is often referred to asPDFF (pseudo-derivativefeedback with feed-forward)bythe author and others such as D.Y.Ohm. This method isshown in Figure 6-12 above. Although the equivalence between Figures 6-11 and 6-12 above is notobvious, upon inspection construction of the control law for Figure6-11 is:
And of the control law for Figure 6-12 is:
With some algebra, Equation 6.1 reduces to Equation 6.2.
PDFF is an extension of a control method developed by R.M. Phelan inAutomatic Control Systems (Cornell University Press), calledPDF, which is equivalent to PDFF with KFR set to 0. PDFF is analternative way to implement PI+; it is useful in digital systemsbecause there are no multiplications before the integral.
Multiplication, when not carefully constructed, causes numericalnoise. That noise prior to the integrator may cause drift in thecontrol loop as the round-off error accumulates in the integrator. PDFFhas a single operation, a subtraction, which is usually noiseless,before the integration and thus easily avoids such noise.
(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. )
Thisseries of articles was excerpted from 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).