Trace a sensor–controller–actuator loop

Follow information and energy through different paths

A heater does not receive a temperature target directly. A controller first compares a reading with a target, calculates a command and sends that command to an actuator. The actuator supplies energy to the body, whose temperature changes. A sensor then reports a value for the next decision. Draw arrows around this loop and label each with a quantity and unit: temperature in °C, error in K, requested and applied power in W, and input energy in J.

In this investigation the standard enclosure is fixed: heat capacity 100 J/K, conductance to ambient 1 W/K, initial and ambient temperatures 20°C, and target 30°C. The controls select gain 2, 6 or 12 W/K and one of four public scenarios. Twelve ten-second intervals produce a 120-second run. Hardware and zero sensor correction are fixed here, unlike the hardware comparison in the previous package.

Separate the command’s two contributions

Requested power is 10 + Kp × (30 − sensed temperature) watts. Kp is the proportional gain in W/K. The 10 W contribution is called feedforward: it supplies the nominal heat loss expected at target for this standard plant. The proportional contribution responds to a measured discrepancy. It is not an integral term that accumulates error over time.

For a sensed temperature of 27°C at gain 6 W/K, error is 30 − 27 = 3 K. The proportional contribution is 18 W and the total request is 28 W. The nominal actuator can supply at most 20 W, so applied power is 20 W. A command is information about desired action; it does not make unavailable power appear.

At target, sensed error is zero and the requested power is still 10 W. That input balances G × (30 − 20) = 10 W of outward transfer in the nominal standard plant. Zero error therefore need not mean zero heater power. Conversely, a negative request is clipped to zero: there is no active cooling actuator in this model.

Inspect the event order

Select nominal gain 6 and inspect the first receipt. The sensed value is 20°C, error 10 K, request 70 W and applied power 20 W. That input is held for ten seconds. The next true temperature is about 21.903°C, after which the next decision uses the appropriate sensor value. The software does not continuously recompute the command during that interval.

The plant accounts for input, stored-energy change and ambient transfer. Its exact exponential step solves a stipulated one-temperature model. The thermal-capacitance reference explains that such spatial uniformity is conditional; the widget does not establish it for actual hardware. Return to the energy balance if the distinction between a command and stored heat is unclear.

A sensor can alter the decision without changing the plant

Capture nominal gain 6, then change only to the +2°C sensor-offset scenario. The sensor reports two degrees above the selected true temperature. No delay is added and the plant itself has not changed. The controller sees a smaller positive error and reduces heating too early. The final true temperature is about 28.285°C, while its biased indication would be about 30.285°C.

A sensor reading above target is not proof that the true body is above target. Trace one offset receipt from true state to raw indication, error, requested power and next state. Keep the sensor scenario separate from the limited-heater scenario, which changes the actuator capacity rather than the information it receives.

Transfer: a paper calibration correction

Suppose separate synthetic reference checks at 20, 30 and 40°C indicate 22, 32 and 42°C. On paper, propose an additive correction. For a true temperature of 27°C, calculate the gain-6 request before and after that correction, and the nominal applied power in each case.

Check the sensor-correction transfer

The supplied checks are consistent with subtracting 2°C. Without correction the indication is 29°C, so the request is 10 + 6 × 1 = 16 W and all 16 W is applied. Corrected indication is 27°C, giving a 28 W request clipped to 20 W. This is a paper correction demonstration: the lab fixes correction at zero. Exact synthetic references do not establish real traceability or correctness outside the checked range.

NIST’s traceability policy distinguishes a measurement’s documented reference relationship from fitness for a particular use. Keep those limits with your calibration claim. Continue to gain comparisons to test whether a larger response to error actually improves the required behavior.