Account for the age of a feedback reading

A correct reading can describe the wrong moment

A sensor can report an accurate past temperature while providing misleading information about what the heater should do now. In this investigation, the delayed scenario uses the state from two samples earlier. Samples are ten seconds apart, so the delay is 20 seconds. The plant still updates every ten seconds; this is not a larger integration step.

Write a prediction for gain 6 W/K before changing from nominal to delayed sensing. Keep hardware, ambient, target, actuator limit and run duration fixed. Predict whether continued heating after the true temperature reaches target could occur without any sensor offset.

Read the requested and effective observation times

At a command time of 70 seconds, the delayed controller requests the state at 50 seconds. Its receipt records both the true current temperature and the older sensor value. At early times, the requested time can be negative: at time zero it is −20 seconds, and at ten seconds it is −10 seconds. The model uses the stipulated initial 20°C history for those requests, with effective stored sample time zero. It does not invent measured observations before the run.

At 20 seconds, the requested and effective observation time is zero. At 30 seconds, both refer to ten seconds. Check these rows and distinguish a timestamp from the numerical temperature reported. Equal values at several early commands do not mean several independent measurements were made.

Follow one delayed command through the plant

For delayed gain 6, the true temperature at 70 seconds is about 30.068294°C, already above target. Yet the 50-second temperature is only 27.869387°C. The sensed error is therefore 30 − 27.869387 ≈ 2.130613 K. Requested power is 10 + 6 × 2.130613 ≈ 22.783679 W, clipped to 20 W.

The heater continues at maximum input for the next ten seconds, reaching about 31.013421°C at 80 seconds. It is responding to old information, not intentionally choosing the current overshoot. At 80 seconds it uses the 60-second value, about 29.023767°C, and requests approximately 15.857396 W. The reduction arrives after the plant has already become warmer.

This receipt-level explanation connects information, command and energy. A graph alone might tempt you to call the curve “oscillatory” without showing why. The exact exponential plant step eliminates one possible confusion: the observed delayed behavior is not produced by an Euler integration approximation in this model.

Check the whole requirement window

Capture nominal and delayed gain 6 on the same axes. The nominal four tail samples stay near 30°C. The delayed values at 90, 100, 110 and 120 seconds are approximately 31.474386, 31.295085, 30.593203 and 29.694914°C. The final value is inside 29–31°C, but the first two samples violate the upper bound. A final-only inspection would miss the failure.

The delayed peak is below the separate 32°C limit. Its energy, about 1,905.208 J, is below 2,000 J. Keep those successful criteria alongside the failed tail band. A failure of one requirement cannot be cancelled by averaging unrelated indicators.

At lower gain 2, the delayed tail is approximately 30.341073, 30.359363, 30.293904 and 30.201020°C. That comparison suggests a useful redesign within this case. It does not establish a universal delay tolerance or guarantee that every lower gain meets every response requirement.

Transfer: identify the sample, not a guessed temperature

On paper, suppose a different sampler makes decisions every five seconds and still uses the state two samples earlier. At time 35 seconds, what time does it observe? Would that preserve this widget’s 20-second delay? No new temperature trajectory is supplied.

Check the sample-age transfer

Two five-second intervals give a 10-second delay, so the command at 35 seconds uses the 25-second state. It does not preserve a 20-second delay; that would require four five-second intervals. This arithmetic does not predict a temperature because the changed sampling/control sequence needs its own model calculation. It is a paper extension, not a selectable widget setting.

NASA’s verification guidance supports retaining test conditions and criterion evidence. Our delayed case is a synthetic exercise, not a physical device certification. Return to the control loop if sample selection is unclear, or continue to actuator capacity to diagnose a different reason for missing the target.