Experiment 1
No delay: change g from 0.5 to 1.5. Predict whether overshoot means instability.
Check the prediction
The error multiplies by −0.5. It alternates sign but shrinks; overshoot alone is not instability.
Feedback and stability
Correction can become overcorrection.
Predict convergence, oscillation and instability in a discrete controller, including the effect of one update of delay.
Start with: Signed numbers and repeated updates. Error means target minus current output.
01 · Commit to a prediction
What do you expect?
It goes to 5 and then 7.5. Each correction uses the remaining error, so it approaches 10 geometrically.
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02 · Change an assumption
Predict. Change. Explain.
Before changing a control, say what should move and why. Start with the experiments below. Reset restores the starting model; it preserves your written responses.
Experiment 2
Keep g = 1.5 and add one update of delay. Explain why the same correction is now unsafe in this model.
Check the prediction
The delayed error can still request a large correction after the output has passed the target. The characteristic roots have magnitude √1.5 > 1, so errors grow.
0 to 2.5 · step 0.05
Calculated model output. The table gives the same values. Displayed values are rounded; calculations keep full precision.
Read the plotted data as a table
03 · Connect the mechanism
From the picture to the quantities.
Negative feedback uses a measurement to oppose error. The sign of the intention does not ensure stability. A correction can be too large or based on old information. The plot is a discrete, idealized integrator controlled at equal update intervals; the target is a reference line, not another simulation.
eₜ = r − xₜ
xₜ₊₁ = xₜ + g eₜ₋d
No delay: eₜ₊₁ = (1−g)eₜ
One-update delay: eₜ₊₁ = eₜ − g eₜ₋₁
xₜ₊₁ = xₜ + g eₜ₋d
No delay: eₜ₊₁ = (1−g)eₜ
One-update delay: eₜ₊₁ = eₜ − g eₜ₋₁
r = 10 is the target in arbitrary output units. x is the output in the same units. g is dimensionless, from 0 to 2.5 here. d is 0 or 1 updates. Initial x₋₁ = x₀ = 0. t counts 24 equal updates. This model has no actuator limits or physical time constant.
Worked example
For g = 0.5 without delay, errors are 10, 5, 2.5, 1.25. Thus x₂ = 7.5. For one-update delay, x₁ = 5 and x₂ = 10 because the second command still sees the initial error 10. The next command moves to 12.5. Same gain, different trajectory.
Assumptions and limits
No-delay convergence requires |1−g| < 1, giving 0 < g < 2. With one-update delay, roots of λ²−λ+g = 0 must lie inside the unit circle, giving 0 < g < 1. At g = 0 there is no correction; the boundary cases g = 2 or delayed g = 1 do not decay. These exact limits belong to this model. Saturation, disturbances, noise and nonlinear plants change the analysis.
The annotated sources distinguish established results from this lesson’s original examples.
04 · Follow the structure
Where else does this apply?
Thermal control
Old temperature measurements can keep requesting heat after enough energy has been delivered.
Boundary: Thermal inertia and continuous dynamics are absent here; these gain numbers do not tune a real heater.
Inventory replenishment
Orders respond to stock gaps while shipments are still in transit.
Boundary: Demand variation, lead-time distributions and discrete stock constraints need their own model.
05 · Retrieve without hints
Close the explanation. Try a new case.
Write an answer before opening its feedback. Later, return directly here without rereading above. Recognition, explanation and transfer are separate outcomes. No page action or answer reveal measures mastery.
Recognition and explanation
Reveal reasoning and rubric
Yes. With no delay and g = 1.5, each error is −0.5 times the previous error, so magnitudes decay.
Self-check: Separate a sign change from growth in magnitude.
Calculation
Reveal reasoning and rubric
x₁ = 12. The next error is −2, so x₂ = 12 − 2.4 = 9.6.
Self-check: Use the new error on the second update.
Novel transfer
Reveal reasoning and rubric
No. The actual recurrence is different. Model the limit and delay, examine worst cases and verify against the plant. The linear result only establishes behavior under its stated assumptions.
Self-check: Name both changed assumptions and decline an unsupported stability guarantee.
On a later day, try again and record actual evidence in the curriculum. A later unaided explanation and a fresh transfer problem give stronger evidence than immediate familiarity. No reminder is scheduled.
Sources & scope
What supports the lesson?
- Åström and Murray, Feedback Systems, Linear Systems chapter. Discrete-time systems and stability via characteristic roots.
Original teaching examples. Reference links need a connection; the lesson itself does not. Built 2026-10-11. Learner understanding is not assessed.