Compare gain without assuming bigger is better

Gain is a response rule, not a quality score

The proportional gain Kp converts sensed temperature error into a power contribution. It has units W/K: a one-kelvin error contributes 2, 6 or 12 W in the three allowed settings. A larger gain asks the actuator to respond more strongly to the same error, but neither energy use nor requirement compliance follows from that number alone.

Begin with the nominal scenario and write a prediction before comparing gains 2 and 12 W/K. Keep the standard hardware, 20°C start, 20°C ambient, 30°C target and 120-second duration fixed. Compare the entire temperature trace and all four tail samples, not just which final number rounds to 30°C.

Clipping can make different commands identical

At the initial sensed temperature of 20°C, the error is 10 K. With the fixed 10 W feedforward term, gains 2, 6 and 12 request 30, 70 and 130 W respectively. All three exceed the nominal 20 W ceiling, so all apply exactly 20 W in the first interval and reach the same next temperature, about 21.903°C.

This explains why multiplying gain by six does not multiply the initial heating rate by six. The requested-power column and applied-power plot answer different questions. Once the temperature error shrinks, one controller may leave saturation before another, causing their trajectories to separate. Inspect the receipts to find that transition rather than assuming the gain acts without limits.

At sensed 29°C the requests are 12, 16 and 22 W. The first two are delivered without upper clipping; the last is clipped to 20 W. At sensed 31°C the requests are 8, 4 and −2 W; the last is clipped to zero. Zero applied power does not create active cooling: the warm body loses heat to its surroundings through the stated conductance.

Compare nominal performance on several indicators

Nominal gain 2 ends near 29.766220°C, with peak equal to that final value and input energy 1,821.035488 J. Gain 6 ends near 29.998648°C, using 1,887.808894 J. Gain 12 ends near 29.999949°C, but its peak is approximately 30.068294°C, and it uses 1,893.370525 J.

All satisfy the supplied nominal tail band, peak, energy and token criteria. A learner who prioritizes final closeness might prefer a higher gain; one who prioritizes lower input within the permitted band might choose differently. Write the priority explicitly. A close endpoint does not prove less overshoot, less energy, or satisfactory behavior under a changed sensor condition.

The trace lasts only 120 seconds. It does not prove asymptotic stability, behavior for every initial temperature, or tolerance of arbitrary delay. Nor does it estimate a distribution of physical failures. The readings and constants are original deterministic model assumptions.

Increasing gain cannot calibrate the sensor

Now compare the same gains in the +2°C offset scenario. Gain 2 ends near 28.491828°C; gain 12 near 28.153858°C. Both fail the lower tail requirement. Stronger response to a biased reading pushes the true state toward a different operating point; the controller cannot identify the bias merely from its own error signal.

For a paper equilibrium calculation with no clipping or delay, outward transfer is T − 20 watts. The offset reading is T + 2, so the request is 10 + Kp(28 − T). Equating them gives T = 30 − 2Kp/(1 + Kp). At gain 2 the equilibrium is about 28.667°C; at gain 12 it is about 28.154°C. The finite endpoint need not equal equilibrium. This formula applies only where the resulting command is not clipped.

Transfer: change gain at one stated reading

For a hypothetical sensed value of 28.5°C, calculate requests at gains 2 and 12, then their nominal applied powers. Predict whether the ratio of applied powers equals the ratio of gains.

Check the clipped-gain transfer

Error is 1.5 K. Requests are 10 + 2 × 1.5 =13 W and10 + 12 × 1.5 =28 W. Applied powers are 13 and 20 W. Their ratio is 20/13, about 1.54, not the gain ratio 6. Both the fixed feedforward contribution and clipping prevent that proportional shortcut.

Use NIST’s measurement discussion to distinguish calibration evidence from additional controller strength. Return to sensitivity and error for model-comparison language, then investigate sensor delay. A design claim should name the tested gain and scenario instead of declaring that the largest gain is best.