A twin whose wind gusts are matched to the logged variance each step: how wrong is the position variance of a PD-held vehicle, and does matching the spectrum instead fix it?

research/gust-twin

A twin whose wind gusts are iid per step with the logged variance: real position variance is s²(a+k_d)/(k_d k_p(a²+a k_d+k_p)), the twin's is 141× smaller at a=1, dt=0.01 and scales as dt, its 1e-3 margin is exceeded 78% of the time, and a spectrum-matched white twin is always conservative.

12 tests

What it found

Real gust: Ornstein–Uhlenbeck with rate a and variance s²; exact real variance s²(a+k_d)/(k_dk_p(a²+ak_d+k_p)). A twin drawing iid gusts per step has variance s²dt/(2k_dk_p): 141× too small at a=1, dt=0.01 (270× at a=0.05), a standard deviation scaling as √dt, and a 10⁻³ margin that the real system exceeds 78% of the time. A spectrum-matched white twin is always conservative, real/twin = a(a+k_d)/(a²+ak_d+k_p) < 1 (0.71 at a=1). Fitting the OU rate from logged wind recovers the real variance (within 1% at 10⁵ samples). Stylised: linear loop, Gaussian gusts, no wind data.

A figure from its own output

Figure produced by research/gust-twin: std_vs_dt — gust-twin/experiments/std_vs_dt.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.