Measure Again Before Choosing the Next Move

Feedback Uses the Result of the Last Action

BOTIE

A steam engine's speed changes when its load changes. A governor responds to that speed and adjusts the steam supply, so the next action depends on what the machine is actually doing. This is feedback: measure the result and use it to change the input.

Maxwell's 1868 paper On Governors examined why such regulators could settle or develop oscillations. Feeding a measurement back into a machine creates a changing loop, so a correction that points in the right direction can still be poorly timed or too strong. The loop's behaviour needs analysis of its own.

Our robot uses a much smaller proportional rule. A larger position error asks for a larger correction, and a new measurement informs the next move. A learned detector may provide that measurement; its static accuracy alone does not establish the stability of the resulting controller.

Sources for this history

Botie’s First Move Was Almost Right

A model predicts where a part should be placed. The motor, which is the actuator applying the command, moves the part, but friction leaves it short of the target. Sending the original move again can overshoot. We need to measure the remaining error before choosing the next command.

A proportional controller turns that remaining error into a command. Its gain controls how strongly it reacts. The physical system being controlled is called the plant. Its response and the time between updates determine what that correction actually does.

An ideal integrator without inertia, noise, delay or actuator limits; no physical device is controlled.

Step 1 of 1

A simple position update

The teaching plant treats command as velocity.

The command is 2 m/s and the next position is 8 m.
Quantity	Value
Target	10 m
Measured position	6 m
Gain	0.5 per second
Step	1 second

Target
Value
10 m
Measured position
Value
6 m
Gain
Value
0.5 per second
Step
Value
1 second

The command is 2 m/s and the next position is 8 m.

Compare target with the latest measurement

The next state comes from the measured error and the plant’s stated update.

Correct the remaining position error

Botie’s actuator stops short and needs a measured correction.

BOTIE

Botie’s actuator stops short and needs a measured correction.

Choose an example

Target position

Measured position

Proportional gain

Timestep

Next position under the teaching plant

Follow the operations

1. Measured position
```
6
```
2. Target position
```
10
```
3. Subtract
```
4
```
4. Proportional gain
```
0.5
```
5. Multiply
```
2
```
6. Timestep
```
1
```
7. Multiply
```
2
```
8. Add
```
8
```
9. Result
```
8
```

An ideal integrator without inertia, noise, delay or actuator limits; no physical device is controlled.

1. Compare target with the latest measurement

Error is target minus measured position. A positive error requests motion in the positive direction. Gain converts position error into the command’s units.

The controller uses feedback, so the next action depends on what happened after the previous action. A fixed open-loop sequence does not observe those deviations.

Measurement delay makes the controller react to older errors. The previous lesson’s timing assumptions now affect the whole loop.

2. Follow one closed-loop step

Our idealised plant adds command times timestep to position. Real actuators have inertia, limits and other dynamics.

error = 10 - 6 = 4
command = 0.5 × 4 = 2
next position = 6 + 2 × 1 = 8

The next error is smaller. Reusing the original error would ignore the correction we just achieved.

Observe the error multiplier

The same plant can shrink or amplify its error depending on the gain and interval.

Python

for gain in [.5,1.5,2.5]:
    print(1-gain*1)

Output

0.5
-0.5
-1.5

The sign determines whether the error changes direction. Its magnitude determines growth for this particular ideal plant.

Check yourself

Which error drives the next action?

Question 1 of 2

What is the next position?

Use the information below.

Target 10, position 6, gain .5, dt 1.

3. A stronger gain can overshoot

For this ideal scalar plant with no delay, the next error is the current error multiplied by one minus gain times timestep. Its magnitude shrinks only when that multiplier has magnitude below one.

e_next = (1 - gain × dt) × e
convergence for this plant: 0 < gain × dt < 2

A product above one alternates the error sign; a product above two amplifies it. These conditions belong to this simplified plant, not every robot.

4. Keep control evaluation separate from prediction accuracy

Inspect settling, overshoot and command limits over complete trajectories. Add sensor noise and delay in simulation before considering hardware. A small average position error can hide a dangerous peak.

A learned policy can replace or augment a control rule, but it still needs closed-loop evaluation under the relevant physical constraints.

The workshop and Python rollout simulate an ideal integrator. They do not provide hardware tuning recommendations.