System identification

• Estimate gain, offset, and time constant from input-output data.
• Validate a model using residuals and a separate experiment before using it for design.

• experimental-method
• signals-systems

System identification works backward from observed inputs and outputs to a model. Rather than assuming a fan, heater, or motor matches a nominal equation, excite it safely, record what happened, choose a model class, estimate parameters, and test predictions. The goal is not the most complicated model; it is a model accurate enough for a stated decision within a stated operating region.

For a first-order model τdy/dt + y = Ku + b, steady tests at several inputs can estimate gain K and offset b from a line y_ss = Ku + b. A step response can estimate τ from the time to reach 63.2% of the change after accounting for offset and any dead time. Parameter fitting minimizes an error such as sum of squared residuals Σ(y_k - ŷ_k)², but a low fit error alone is not validation.

Excitation must reveal the dynamics without leaving safe operation. An input that barely changes will be dominated by noise; a single large step may trigger saturation or a different regime. Use several bounded levels, timestamps from the same clock when possible, and record battery voltage, payload, and environment. Reserve a separate dataset or input pattern for validation. A useful build decision might be whether one speed model covers a toy car both unloaded and carrying a camera; if residuals depend on payload, it likely does not.

A common misconception is that correlation establishes the plant model. Input and output can move together because of shared trends, feedback, or a hidden variable. Another failure is fitting and judging on the same trace, which rewards overfitting. Inspect residuals versus time and input: structure means information remains unexplained. Never extend the model beyond tested voltage, speed, temperature, or load without marking that extrapolation. Identification estimates behavior; it does not certify a mechanism or make unsafe testing acceptable.

Problem

A low-voltage motor reaches 210 rpm at 2.0 V and 450 rpm at 4.0 V under the same load. Assume y_ss = Ku + b. Estimate K and b, then predict 3.0 V speed.

Steps

1. Gain K = (450 - 210) rpm ÷ (4.0 - 2.0) V = 120 rpm/V.
2. Offset b = 210 rpm - (120 rpm/V × 2.0 V) = -30 rpm.
3. At 3.0 V, predicted speed is 120 × 3.0 - 30 = 330 rpm.

Result

The fitted local model is y = 120u - 30 rpm and predicts 330 rpm at 3.0 V.

Sanity check

The 3.0 V prediction lies halfway between the two measured speeds, as linear interpolation requires; the negative intercept is not a claim that reverse speed occurs at 0 V.

A model fits its training step almost perfectly but predicts a second step poorly. What is the best conclusion?

A. The model has not validated for that condition B. The second measurement must be discarded C. Perfect training fit proves the physics D. Increase numerical precision only

Your answer: ______________________________

Reasoning:

Goal

Identify and validate a local static-plus-first-order model for a low-energy system.

Equipment

• Low-voltage motor with speed sensor or small fan with airflow proxy
• Current-limited supply
• Microcontroller logger
• Stopwatch

Procedure

1. Choose three bounded input levels and a separate validation level; state the safe operating range and expected trend.
2. Record actual steady outputs at the fitting levels, then apply one modest step and estimate gain, offset, and τ.
3. Predict the reserved validation trace before running it; compare residuals and settling time.
4. Deliberate fault: add a small payload or partially obstruct only the bench-scale airflow path; diagnose systematic residual change as an operating-condition mismatch.

What to observe

• Expected: the fitted model predicts best near tested conditions and degrades when load changes.
• Measure: actual input, output, timestamps, supply voltage, load, parameter estimates, and residuals; never fabricate smooth values.

Transfer challenge

Plan identification data needed before selecting feedforward commands for a camera pan axis at two payloads.

Cautions

• Use a guarded low-voltage bench device; do not open or probe an energized household fan or vacuum cleaner.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: