Identifiability

Learning goals¶

• Separate noise, poor excitation, and structural non-identifiability

• Design a simple two-parameter loss-surface or sensitivity diagnostic

• Distinguish physical vs equivalent fitted parameters

Reading¶

• Identifiability and experiment design Ljung, 1999

• Physical versus equivalent parameters

Three failure modes¶

Physical vs equivalent parameters¶

A fitted coefficient is physical when it matches a device/control constant under a consistent model map and informative data. It is equivalent (effective) when it mainly absorbs mismatch, weak excitation, or an inconsistent residual — even if trajectory error looks small.

Low NRMSE does not certify physical meaning.

Worked example¶

For $(H,D)$ on one load step, plot $\mathcal{L}(H,D)$. If the valley is elongated along $D$, damping is weakly informed by that experiment — redesign excitation before trusting $\hat{D}$.

Lab¶

• Diagnostics required in Lab 3

• Deeper measurement design: Measurements and excitation

Further reading¶

• Lab 3 success criteria; physical versus equivalent parameters in the identification part

• Maths background (external, link only): low-rank approximation / PCA in the EECS 245 notes Rampure, 2026 (CC BY-NC-ND 4.0) — useful intuition for flat directions in a loss surface

Common misconceptions¶

• “Unique minimizer of training loss ⇒ unique physical $\theta$”

• Bound-clamped estimates reported as if interior physical optima

• Ignoring that a second disturbance may invalidate an equivalent fit