Worksheet

U5.3 The derivative term

Feedback control · University · about 30 min

What this lesson is about

Damping the overshoot, and why D on a noisy signal needs a filter or it is useless.

• In code
• Why D is difficult in practice
• The shape it produces

Questions 6 marks in all

1. [1 mark]The distance sensor in the lesson wobbles by about 1 cm, so two readings 0.1 s apart can differ by 4 cm with the robot all but still. What approach speed, in cm/s, does a differenced derivative report?

2. [1 mark]One tick of the PD controller from the lesson. What does it print?

   KP, KD, DT = 3.0, 1.2, 0.1
   last, error = 20.0, 14.0
   d = (error - last) / DT
   cmd = max(-70, min(70, KP * error + KD * d))
   print(round(d, 1), round(cmd, 1))

3. [1 mark]The robot is approaching the target quickly, so the error is shrinking fast. What does the D term do?

   1. AIt is negative and subtracts from the command, easing off before arrival
   2. BIt is positive and adds to the command
   3. CIt is zero until the error changes sign
   4. DIt accumulates the error over time

4. [1 mark]The setpoint jumps from 30 to 40 cm and the command spikes hugely for one tick. What is the standard fix?

   1. ADifferentiate the measurement instead of the error
   2. BLower Kp
   3. CFilter the error
   4. DClamp the integral

5. [1 mark]Which is the best treatment for derivative noise on this robot, according to the lesson?

   1. AUse the measured speed from flow() instead of differencing distance()
   2. BFilter distance() heavily before differencing
   3. CLower the loop rate so the differences are bigger
   4. DRaise Kd until the noise averages out

6. [1 mark]On the quarter turn, Kp = 4 alone overshoots by about 20 degrees. With a D term added, the response has no overshoot, a slower approach, and stops about 1.4 degrees short. What does that say about Kd?

   1. AIt is on the high side: a lot of D
   2. BThere is no D at all
   3. CIt is too small to matter
   4. DIt is making the loop wind up

The task: damp the overshoot

This robot is carrying a load, so it takes about three times as long to speed up and to slow down. That extra lag is exactly the job the D term is for. Turn to face 90 degrees and be within 3 degrees of it from three seconds onwards. Plot error and d term. On this robot P alone cannot do it, whatever the gain and however fast the loop runs: a gain low enough not to swing past arrives too late, and a gain high enough to arrive in time swings past and is still swinging at three seconds. Start from KP = 4 and find a KD that brakes it in time.

from bugbot import *
connect()

KP, KD = 4.0, 0.0
DT = 0.25
last = 90.0          # the error at the start

Plan your program here, then type it in and press Run.

Do it on the robot
www.bugbotlab.com/learn/u5-3-derivative/
The simulator checks it and tells you when it passes. Nothing to install, no account.

Challenges

1. Use -flow()[1] as the derivative instead of differencing the distance. Compare the command traces.
2. On the distance loop, turn Kd up until the robot buzzes while standing still. What is the noise doing?
3. Differentiate the measurement rather than the error, then change the setpoint mid-run and compare.