Distinguish a design change from calculation error

Ask what changed before explaining a difference

Two temperature curves differ. Did the hardware change, did the controller change, did an instrument misread the temperature, or did a calculation approximate the same equations differently? These questions need different comparisons. Start a notebook table with columns for changed input, preserved conditions, result, interpretation and unresolved alternative. A numerical difference alone does not identify its cause.

Capture standard and insulated hardware at gain 6 W/K. Both start at 20°C, use the same 100 J/K capacity, nominal ambient and 120-second duration. The hardware conductance changes from 1 to 0.5 W/K and its stipulated token cost changes from three to five. The final temperatures are about 29.998648°C and 30.767541°C; input energies are 1,887.808894 J and 1,558.581071 J. Describe this as sensitivity to the supplied hardware choice, not uncertainty in one measurement.

Change a controller parameter separately

Hold insulated hardware fixed and compare gain 2 with gain 6 W/K. The final temperature changes from about 31.480538°C to 30.767541°C. Energy changes from 1,624.624719 J to 1,558.581071 J. Record the tail-band outcome as well as the endpoint. Raising this gain improves the listed nominal comparison, but that does not establish a universal monotonic rule for every gain, delay or plant.

Conductance changes the heat-loss mechanism; gain changes how a temperature error changes the requested power. The fixed 10 W feedforward term remains in both runs. Separating these comparisons lets you say which variable was tested. Changing hardware and gain together might produce a useful candidate, but it would not isolate either effect.

Compare two calculations of one interval

The widget propagates the stipulated one-node plant exactly during each constant-input interval. A separate paper approximation, forward Euler, instead keeps the starting temperature-change rate for the whole step:

Tnext ≈ T + Δt × [P − G(T − Ta)]/C.

Use T = Ta = 20°C, P = 20 W, G = 1 W/K, C = 100 J/K and Δt = 10 s. Euler gives 20 + 10 × 20/100 = 22°C. The exact exponential gives 21.9032516393°C. Define the numerical error here as Euler minus exact: +0.0967483607 K. Euler initially overpredicts because the heat loss increases as the body warms, while this step retains the zero-loss starting rate.

This error is not an observed thermometer residual or a confidence interval. Both values come from calculations of the same stipulated model. Agreement between two solvers could support an implementation check while leaving the model’s physical assumptions entirely untested.

Do not confuse integration steps with control updates

For the same held 20 W input over 100 seconds, one Euler step gives 40°C, whereas the exact solution gives approximately 32.642411°C. That is a paper held-input example. It is not what the widget’s controller does over ten intervals: the controller can change power after each ten-second measurement. A comparison that changes both numerical step size and controller-update timing mixes two effects.

For an optional calculation, split a held-input interval into smaller Euler steps while keeping the input fixed and compare with the exact endpoint. No code is needed; a calculator and short table suffice. Numerical artifacts must not be interpreted as evidence that a physical device oscillates or becomes unstable. The thermal-capacitance reference also reminds us that representing a body by one temperature is itself conditional.

Transfer: a second approximation step

Keep the first held-input example but use a five-second Euler step. Compute its endpoint and compare it with the supplied exact endpoint 20.975411510°C. Explain what became smaller and what has not been established.

Check the shorter-step transfer

Euler gives 20 + 5 × 20/100 = 21°C. Euler minus exact is about +0.024588490 K, smaller than the ten-second single-step error. This checks a different, shorter interval for the same held input. It does not demonstrate physical accuracy, estimate a sensor's uncertainty or compare complete controller trajectories over a common duration.

If actual measurements later become available, retain them separately and define residual as observation minus prediction. Investigate calibration, boundary mismatch and spatial gradients before tuning parameters to hide every discrepancy. Return to models and residuals or continue to prototype evidence with the original failed capture intact.