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Module 2 · Lesson 4 · 11 min video · 30 min practice

Identifying a PT1 model from measurements

Estimate K and T from imperfect step-response data and test whether the model is credible.

  • Estimate K from the input and output changes.
  • Estimate T with the 63.2% crossing.
  • Diagnose noise, delay, and model mismatch from residuals.
Unofficial TUM ACE alignment: ACE §4: Identifikation aus Sprungantworten

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8–15 minute visual explanation. The interactive experiment below is usable now.

Prediction

A measured response begins at 20°C and settles at 26°C after a 2 kW input step. What is the process gain?

Decide whether to divide absolute values or changes from the operating point.

Identify changes, not coordinates

For a step from u0u_0 to u1u_1 and output from y0y_0 to y∞y_\infty, estimate

K^=y∞−y0u1−u0.\hat K=\frac{y_\infty-y_0}{u_1-u_0}.

Then find the time at which the output has traversed 63.2% of its total change. Measured from the input-step time, that crossing estimates TT for a delay-free PT1 model.

Concept check

For the temperature experiment above, what is K?

Reactive experiment: corrupt the measurement

As noise and delay grow, a single crossing becomes less trustworthy. Identification is not merely reading two numbers; it is deciding whether the model structure explains the data.

Worked example

If the response changes from 10 to 18 after an input change from 1 to 3, then K=(18−10)/(3−1)=4K=(18-10)/(3-1)=4. The 63.2% target is 10+0.632(8)=15.0610+0.632(8)=15.06. The elapsed time from the step to that crossing estimates TT.

Derivation

Why baseline subtraction matters

A model in deviation variables predicts Δy(t)=K Δu (1-exp(-t/T)). Neither the original input nor output offset belongs to the dynamic gain. Fitting absolute coordinates silently mixes the operating point into K.

Unofficial TUM ACE exam prep

Document the construction

Mark pre-step and final values, the 63.2% level, input-step time, and crossing time on the response. If visible delay exists, identify it separately instead of calling the full crossing time T.

Guided exercise

Increase noise until the first crossing jumps early. Propose one robust alternative: smooth the data, average several experiments, or fit all samples by least squares. State the bias your choice could introduce.

Hint

A method that uses the entire curve is usually less sensitive to one noisy sample than a single threshold crossing.

Summary

  • Estimate gain from changes around the operating point.
  • The 63.2% rule estimates TT only for a delay-free PT1 model.
  • Always inspect residual structure and measurement quality before trusting parameters.

Retrieve it later

Describe a complete PT1 identification experiment, including one diagnostic for model mismatch.

Answer from memory tomorrow, then return to check your reasoning.