What overshoot does a controller tuned in a fixed-step digital twin really have?

What it found

A twin's integrator adds or removes damping the plant does not have. Per-step pole moduli are exact (real e^−hc; explicit 1−hc+h²k; semi-implicit 1−hc; implicit 1/(1+hc+h²k)), and the first-order damping bias of explicit and implicit Euler changes sign at ζ=1/√2 (measured 0.742→0.711 and 0.675→0.703 as h falls 0.2→0.025). Tuned to 10% overshoot in the twin, the real plant overshoots 0.105–0.192 (semi-implicit and implicit, h=0.05–0.4) while forward Euler is conservative (0.091–0.036); at a 1% target the semi-implicit twin under-claims by up to 4.9×. The excess halves with h (first order); Richardson extrapolation of the twin’s overshoot at fixed gain recovers the real value to 0.0003–0.0012, and re-tuning with substeps costs ×4 to ×32. Stylised: one linear oscillator, textbook integrators (not any engine’s scheme), overshoot read from samples, no robot data.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
• Code
• Tests

More in Digital twins & sim-to-real

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• What does a state estimator tuned in a twin with the wrong noise levels cost in the real world?An estimator tuned in a twin with wrong noise levels: exact real MSE of the twin's Kalman gain, optimistic claim when the twin is too quiet (+121% at 10×) vs pessimistic when too noisy, no instability cliff, and a Mehra innovation retune with regret ∝ 1/n.
• What does a controller designed in a twin with the wrong actuator gain cost in the real world?A controller designed in a twin with the wrong actuator gain: exact real LQR cost, an instability band the twin does not warn about, closed-loop non-identifiability without dither, and a dithered retune with matched regret law.
• How long does a simulator without actuator limits say an unstable plant stays in control?A limit-free twin of an unstable plant with a saturating actuator overstates mean time to loss of control by 6,295× at noise sd 1 (1.5×10⁶ vs 245 steps), an exponential-in-1/s² gap (slope ≈3.7 real vs ≈12.7–13.2 twin), though both rank the deadbeat gain best.
• What does a Kalman filter tuned in an ideal-sensor twin cost on a quantizing real sensor?A Kalman filter tuned in an ideal-sensor twin, run on a quantizing sensor: twin gain costs +11% (R=1, D=4) to +64% (R=0.01), Sheppard R+D²/12 retune is within 1% of best, but its MSE claim under-predicts by up to 43% once the step exceeds the state's scale.
• How should a simulation budget be split across twin fidelities?Splitting a simulation budget across twin fidelities (multilevel Monte Carlo): smooth quantity saves 3.6×–897× (rates α=1, β=2), a failure probability only β=1 so multilevel loses at loose targets (0.4×) and reaches 7.7× at 1e-4; equal samples per level cost 156×/3.9× the optimum.

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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.