Answers

U5.5 Tuning

Feedback control · University · about 30 min

What this lesson is about

Finding the gain where it oscillates, and working back from there to something usable.

• The Ziegler-Nichols method
• Finding Ku on this robot
• Tuning by hand, with the chart
• Do it in the right order
• The gains are not universal

Questions 7 marks in all

1. [1 mark]The Ziegler-Nichols PID row from the lesson, for a measured Ku and Tu. What does it print?

   KU, TU = 10.0, 1.6
   kp = 0.6 * KU
   ki = 1.2 * KU / TU
   kd = 0.075 * KU * TU
   print(round(kp, 2), round(ki, 2), round(kd, 2))

   Answer:

   6.0 7.5 1.2

   Kp = 0.6 x 10 = 6, Ki = 1.2 x 10 / 1.6 = 7.5, Kd = 0.075 x 10 x 1.6 = 1.2.

2. [1 mark]Ku = 8 and Tu = 2 s. What Ki does the Ziegler-Nichols PI row give? Give two decimal places.

   Answer: 2.16 (accept within 0.005). The PI row gives Ki = 0.54 Ku / Tu = 0.54 x 8 / 2 = 2.16.

3. [1 mark]Ku = 9. What Kp does the Ziegler-Nichols P-only row give?

   Answer: 4.5 (accept within 0.001). P only uses Kp = 0.5 Ku = 4.5.

4. [1 mark]Ziegler-Nichols gains make the robot swing lively, each swing a quarter of the last. What is a common and sensible next move?

   1. AHalve Kp
   2. BDouble Ki
   3. CRemove Kd
   4. DDouble Ku and recompute
   Answer: A. Ziegler-Nichols aims for quarter amplitude decay, which most people find too aggressive. It is a starting point, and halving Kp is the usual first adjustment. On a big step, such as the lesson's quarter turn, the Ki it gives usually needs cutting as well.

5. [1 mark]The response arrives, settles short of the target, and stays short. What should change?

   1. AAdd or raise Ki
   2. BRaise Kd
   3. CLower Kp
   4. DFilter the measurement
   Answer: A. A steady offset is what the integral term exists to remove.

6. [1 mark]The response shows a slow, wallowing oscillation. What should change?

   1. ALower Ki
   2. BRaise Kp
   3. CRaise Kd
   4. DRun the loop slower
   Answer: A. Slow, large swings are the look of too much integral action. Fast oscillation at the loop rate points at Kp instead.

7. [1 mark]In what order does the lesson say to bring in the three terms?

   1. AP, then D, then I
   2. BP, then I, then D
   3. CI, then P, then D
   4. DAll three at once from Ziegler-Nichols, then fine tune
   Answer: A. Get P fast without ringing, use D to remove the ring, and add I last. Adding I early makes the offset you are trying to judge keep disappearing.

The task: find the ultimate gain

from bugbot import *
connect()

DT = 0.25

from bugbot import *
connect()

DT = 0.25

def swing(kp, ticks=32):
    """Turn a quarter turn at this gain; report how much the error swings, and how fast, in the second half."""
    target = heading() + 90
    errors = []
    for tick in range(ticks):
        error = (target - heading() + 180) % 360 - 180
        errors.append(error)
        plot("kp %s" % kp, error)
        drive(0, 0, max(-100.0, min(100.0, kp * error)))
        wait(DT)
    stop()
    wait(0.5)
    late = errors[ticks // 2:]
    mean = sum(late) / len(late)
    crossings = 0
    for a, b in zip(late, late[1:]):
        if (a - mean) * (b - mean) < 0:
            crossings += 1
    span = max(late) - min(late)
    period = len(late) * DT * 2.0 / crossings if crossings else 0.0
    return span, period

ku, tu = 0.0, 0.0
for kp in (2.0, 4.0, 6.0, 8.0, 10.0):
    span, period = swing(kp)
    print("kp", kp, "swing", round(span, 1), "period", round(period, 2))
    if span > 10.0 and ku == 0.0:
        ku, tu = kp, period
print("ku:", ku)
print("tu:", round(tu, 2))