What does a twin whose sensor reads at any distance get wrong about a loop with a range-limited lidar or radar?

research/range-twin

A twin whose sensor reads at any distance vs a range-limited one: exact step-count law (49 vs 4021 steps from 1000R), PI overshoot neither bound nor lower bound (4.1× at 1000R), exact scale invariance, and the gain fitted from logs of a given extent.

8 tests

What it found

A twin with unlimited sensor range predicts geometric convergence; a real sensor reading clip(x, −R, R) caps the approach speed at kR. The exact step-count law matches simulation in 108/108 cases; for k=0.25 the twin says 49 steps from 1000R and the real loop takes 4021 (82×), an error linear in x₀/R with slope 1/k. For a PI loop the twin's overshoot is neither a bound nor a lower bound: smaller at 2–5R, 4.1× larger at 1000R; conditional integration repairs overshoot but not settling time. Logs within range fit the gain exactly and cannot reveal R; from logs uniform on [−L, L] the fitted gain factor is 1.5q−0.5q³, q=R/L. Stylised: scalar noiseless loop, hard symmetric clip.

A figure from its own output

Figure produced by research/range-twin: steps_vs_x0 — range-twin/experiments/steps_vs_x0.svg, written by the project's own experiment code.

Paper, code and results

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.