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Learn through experiments

A stronger correction can make control worse

How do gain and delayed sensing affect a controller?

Entering midway? Revisit Use two measurements to isolate a fault for the preceding worked example, or start here and use the explanation below.

1 · Try

Start with a concrete question

Original synthetic scenario. A virtual position controller targets 10 m. Each step applies a bounded movement, then a constant negative drift.

Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.

Go directly to the experiment · Review basic mathematics

2 · Predict

What do you expect?

Will a larger gain always reduce final error when the sensor is one step late?

3 · Experiment

Change one thing. Inspect what happens.

  1. Run gain 0.5 with current sensing and no drift; inspect error after each step.
  2. Add drift 1 and explain why a persistent offset remains.
  3. Compare gain 1 and 2 with delayed sensing; inspect overshoot and total action.
  4. Choose a controller for low error and modest effort; test it under an unfamiliar drift value.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Final position7.99805 m
Final absolute error2.00195 m
Total absolute action19.998 m

Compare the current results

Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.
06
  • Step 1 absolute error (m)6
    Step 1 absolute error (m): 6Horizontal bar from zero to 6, on the shared zero to 6 numeric axis.
  • Step 2 absolute error (m)4
    Step 2 absolute error (m): 4Horizontal bar from zero to 4, on the shared zero to 6 numeric axis.
  • Step 3 absolute error (m)3
    Step 3 absolute error (m): 3Horizontal bar from zero to 3, on the shared zero to 6 numeric axis.
  • Step 4 absolute error (m)2.5
    Step 4 absolute error (m): 2.5Horizontal bar from zero to 2.5, on the shared zero to 6 numeric axis.
  • Step 5 absolute error (m)2.25
    Step 5 absolute error (m): 2.25Horizontal bar from zero to 2.25, on the shared zero to 6 numeric axis.
  • Step 6 absolute error (m)2.125
    Step 6 absolute error (m): 2.125Horizontal bar from zero to 2.125, on the shared zero to 6 numeric axis.
  • Step 7 absolute error (m)2.0625
    Step 7 absolute error (m): 2.0625Horizontal bar from zero to 2.0625, on the shared zero to 6 numeric axis.
  • Step 8 absolute error (m)2.03125
    Step 8 absolute error (m): 2.03125Horizontal bar from zero to 2.03125, on the shared zero to 6 numeric axis.
  • Step 9 absolute error (m)2.01563
    Step 9 absolute error (m): 2.01563Horizontal bar from zero to 2.01563, on the shared zero to 6 numeric axis.
  • Step 10 absolute error (m)2.00781
    Step 10 absolute error (m): 2.00781Horizontal bar from zero to 2.00781, on the shared zero to 6 numeric axis.
  • Step 11 absolute error (m)2.00391
    Step 11 absolute error (m): 2.00391Horizontal bar from zero to 2.00391, on the shared zero to 6 numeric axis.
  • Step 12 absolute error (m)2.00195
    Step 12 absolute error (m): 2.00195Horizontal bar from zero to 2.00195, on the shared zero to 6 numeric axis.

Follow the calculation

  1. Step 1: actual 0.000, sensed 0.000, error 10.000, requested 5.000, limited action 5.000, drift −1, next 4.000.
  2. Step 2: actual 4.000, sensed 4.000, error 6.000, requested 3.000, limited action 3.000, drift −1, next 6.000.
  3. Step 3: actual 6.000, sensed 6.000, error 4.000, requested 2.000, limited action 2.000, drift −1, next 7.000.
  4. Step 4: actual 7.000, sensed 7.000, error 3.000, requested 1.500, limited action 1.500, drift −1, next 7.500.
  5. Step 5: actual 7.500, sensed 7.500, error 2.500, requested 1.250, limited action 1.250, drift −1, next 7.750.
  6. Step 6: actual 7.750, sensed 7.750, error 2.250, requested 1.125, limited action 1.125, drift −1, next 7.875.
  7. Step 7: actual 7.875, sensed 7.875, error 2.125, requested 1.063, limited action 1.063, drift −1, next 7.938.
  8. Step 8: actual 7.938, sensed 7.938, error 2.063, requested 1.031, limited action 1.031, drift −1, next 7.969.
  9. Step 9: actual 7.969, sensed 7.969, error 2.031, requested 1.016, limited action 1.016, drift −1, next 7.984.
  10. Step 10: actual 7.984, sensed 7.984, error 2.016, requested 1.008, limited action 1.008, drift −1, next 7.992.
  11. Step 11: actual 7.992, sensed 7.992, error 2.008, requested 1.004, limited action 1.004, drift −1, next 7.996.
  12. Step 12: actual 7.996, sensed 7.996, error 2.004, requested 1.002, limited action 1.002, drift −1, next 7.998.

Feedback reacts to error; gain, stale measurements, action limits and persistent drift jointly determine behavior. Larger gain can overshoot or oscillate.

Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.

4 · Understand

Explain the mechanism

What the experiment explains

Original synthetic scenario. A virtual position controller targets 10 m. Each step applies a bounded movement, then a constant negative drift.

Feedback uses the difference between target and sensed position. Proportional gain multiplies this error to propose a movement; an actuator limit clips the movement.

With position zero, gain 0.5 and drift 1, the first action is 5 and next position is 4. Next action is 3 and next position is 6. Persistent drift can leave a nonzero steady error.

Delayed sensing reacts to where the system was. A strong correction can continue after the target is crossed, causing oscillation. Error and effort should both be inspected.

Read the relationship

Action = clamp(gain × (target − sensed position), −limit, limit). Next position = current position + action − drift. Positions/actions are metres per modeled step.

Inspect the evidence

The numerical trace is recalculated from your current settings. Compare a controlled pair of trials before changing several inputs together. Inspect intermediate quantities as well as final metrics; state which assumptions make the comparison meaningful.

Boundaries of the model

  • Discrete integrator, constant drift, no inertia or sensor noise, twelve steps only.
  • This programmed proportional controller is not learned and is not a hardware safety design.

Return to the physics and engineering course and project to connect this mechanism to the broader investigation.

5 · Apply

Try a fresh case

Target 8, current/sensed position 2, gain 0.5, action limit 2 and negative drift 0.5. What is the next position?

This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.

Take the investigation further

Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.

School investigation

Manually compute four updates and label feedback, action and disturbance.

College investigation

Derive stability conditions for the unclipped zero-delay case and compare with delayed simulations.

Workplace investigation

Define acceptance limits for overshoot, settling behavior, and effort; report a failing case as well as a selected setting.

6 · Continue

Keep the evidence, choose your next step

Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.