System identification
Fit a useful dynamic model from designed experiments and validate it on data not used for fitting.
- Estimate gain, offset, and time constant from input-output data.
- Validate a model using residuals and a separate experiment before using it for design.
Related study — optional context
Understand
The working idea
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.
Follow the reasoning
Worked example
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.
- 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.
The fitted local model is y = 120u - 30 rpm and predicts 330 rpm at 3.0 V.
Sanity checkThe 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.
Try it yourself
Independent check
Move to the bench
Identify and validate a local static-plus-first-order model for a low-energy system.
- Low-voltage motor with speed sensor or small fan with airflow proxy
- Current-limited supply
- Microcontroller logger
- Stopwatch
Experiment
- Choose three bounded input levels and a separate validation level; state the safe operating range and expected trend.
- Record actual steady outputs at the fitting levels, then apply one modest step and estimate gain, offset, and τ.
- Predict the reserved validation trace before running it; compare residuals and settling time.
- Deliberate fault: add a small payload or partially obstruct only the bench-scale airflow path; diagnose systematic residual change as an operating-condition mismatch.
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.
Plan identification data needed before selecting feedforward commands for a camera pan axis at two payloads.
- Use a guarded low-voltage bench device; do not open or probe an energized household fan or vacuum cleaner.
Keep evidence
Bench note
Record what you expected, what you actually measured or observed, and one difference worth investigating. This is self-reported evidence, not a certification.