Answers

U5.6 Saturation and windup

Feedback control · University · about 25 min

What this lesson is about

What a controller should do when the actuator has nothing left to give.

• Where this robot saturates
• The symptom
• The fix
• Where else this bites

Questions 6 marks in all

1. [1 mark]A PI controller with conditional integration, run on four errors. What does it print?

   KP, KI, DT = 1.5, 0.7, 0.1
   integral = 0.0
   for error in (30, 40, 30, 10):
       want = KP * error + KI * integral
       cmd = max(-55, min(55, want))
       if abs(want - cmd) < 0.01:
           integral += error * DT
       print(round(cmd, 2), round(integral, 2))

   Answer:

   45.0 3.0
   55 3.0
   47.1 6.0
   19.2 7.0

   Tick 1 wants 45, not saturated, so the integral becomes 3. Tick 2 wants 62.1, clipped to 55, so the integral is frozen at 3. Ticks 3 and 4 are unsaturated and resume integrating to 6 and 7.

2. [1 mark]drive() is asked for 250 and then for 6. What does the robot get each time?

   1. A100, then nothing
   2. B250, then 6
   3. C100, then 6
   4. D100, then 15
   Answer: A. The drive clips at 100 at the top and does nothing below about 15 percent. Both are saturation, one at each end.

3. [1 mark]What is the name of the anti-windup method that stops accumulating the integral while the actuator is saturated?

   Answer: conditional integration. Conditional integration freezes the integral while the command is pinned, and resumes it the moment the controller has authority again.

4. [1 mark]With back-calculation, integral -= (want - cmd) / Kt. If want = 62.1, cmd = 55 and Kt = 2, by how much is the integral reduced? Give two decimal places.

   Answer: 3.55 (accept within 0.005). (62.1 - 55) / 2 = 7.1 / 2 = 3.55. The integral is unwound in proportion to how much was clipped.

5. [1 mark]A robot switches from manual driving to a PI controller and jumps. What is the fix?

   1. AInitialise the integral so the controller's first output matches what was already being sent
   2. BReset the integral to zero at the switch
   3. CRaise Kp during the switch
   4. DClamp the command to 15 percent
   Answer: A. That is bumpless transfer. Any system with modes needs it, because an integral left from before the switch produces a step in the command.

6. [1 mark]A diagonal drive command saturates the lateral axis but not the forward one. What happens?

   1. AThe robot travels in the wrong direction, because one component is clipped and the other is not
   2. BThe robot slows down but keeps its direction
   3. CNothing, because each axis is controlled separately
   4. DThe robot stops
   Answer: A. The lateral axis is slower, so clipping it alone changes the ratio of the two components, and with it the direction.

The task: anti-windup

from bugbot import *
connect()

KP, KI = 1.5, 0.7
TARGET = 12.0
DT = 0.1
integral = 0.0

from bugbot import *
connect()

KP, KI = 1.5, 0.7
TARGET = 12.0           # closer to the blocker than the robot can get
DT = 0.1
integral = 0.0

for tick in range(500):
    error = distance() - TARGET
    want = KP * error + KI * integral
    cmd = max(-55.0, min(55.0, want))
    # only add to the integral while the actuator still has something to give
    if abs(want - cmd) < 0.01:
        integral += error * DT
    plot("error", error)
    plot("i term", KI * integral)
    drive(cmd, 0, 0)
    wait(DT)
stop()