The basics of control system design: Part 4 - Tuning a Pl+ ControllerTuning a PI+ controller (Figure 6-13 below) is similar to tuning a PI controller (see Part 3) except that you must choose the amount of filtering (KFR) before tuning the integral gain.
|Figure 6-13. Tuning a PI + controller.|
For the highest possible stiffness, KFR should be set to zero; here the PDFF controller reduces to PDF. However, for most applications KFR should be at least 0.4; there is a substantial loss of response for setting 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.
|Figure 6-14. Experiment 6C, a PI + controller.|
Experiment 6C, shown in Figure 6-14 above, will be used to demonstrate the PI+ system. This is similar 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 waveform generator ("Wave Gen").
This filter implements Equation 6.1 (see Part 3). The Live Constant KFR scales the command directly; "1 - KFR" scales the command passing through the low-pass filter. The low-pass break frequency is set by KI, although KI must be scaled by 0.159 to convert KI (which is in rad/sec) to Hz, the scaling for filter break frequency.
One other minor change was required: the Live Constant KI was converted from a "Scale-by" constant to a standard Live Constant, with scaling accomplished by a multiplication just above the block. This was necessary because KI had to be provided explicitly because it is used in two places: as the integral gain and as the break frequency for the command 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 for KFR was 0.65, the compromise value. This allowed the integral gain to increase from 100 in PI (Figure 6-5 in Part 2) to 300 while maintaining the same overshoot.
|Figure 6-15. Closed-loop Bode plot of PI + system (180 Hz bandwidth, 1.5 dB peaking).|
The closed-loop plot shows a decline in bandwidth, from 206 Hz in the PI system (Figure 6-7 in Part 2) to 180 Hz (Figure 6-15 above). The open-loop plot (Figure 6-16 below) shows a 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 command signal, lower KFR allows higher integral gains, which, in turn, reduce the PM. However, because it filters high-frequency components in the command signal, lower KFR allows higher integral gains, which, in turn, reduce the PM.
|Figure 6-16. 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 the 90° phase lead of a derivative. Using D gain will usually allow the system responsiveness to increase, for example, allowing the bandwidth to nearly double in some cases.
Differential gain has shortcomings. Derivatives have high gain at high frequencies. So while some D does help the phase margin, too much hurts the gain margin by adding gain at the phase crossover, typically a high frequency. This makes the D gain difficult to tune.
The designer sees overshoot improve because of increased PM, but a high frequency oscillation, which comes from reduced GM, becomes apparent. The high frequency problem is often hard to see in the time domain because high-frequency ringing can be hard to distinguish from normal system noise.
So a control system may be accepted at installation but have marginal stability and thus lack the robust performance expected for factory equipment. This problem is much easier to see using Bode plots measured on the working system.
Another problem with derivative gain is that derivatives are sensitive to noise. Even small amounts of noise from wiring or resolution limitations may render the D gain useless. In most cases, the D gain needs to be followed by a low-pass filter to reduce the noise content.
The experiments in this series assume a near-noiseless system, so the D filter is set high (2000 Hz). In many systems, especially in analog controllers, such a value would be unrealistic.
(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. )
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).