A continuous-time twin certifies PD at every gain: which gains survive a zero-order-hold sampled loop?

research/sample-twin

A continuous-time twin certifies PD at every gain; the real zero-order-hold loop is stable iff kp·T/2 < kd < 2/T (ωn·T < min(4ζ, 1/ζ)), overshoot is 5× the twin's at ωn·T = 1.2, raising ωn past ωn·T ≈ 1 slows settling, and deadbeat (2 samples) is the real speed limit.

8 tests

What it found

Double integrator under PD control computed at period T and held. The twin (T→0) is stable for every gain; the exact step matrix gives the real stable region k_pT/2 < k_d < 2/T, i.e. ω_nT < min(4ζ, 1/ζ), matched to bisection and an independent RK4 held-input run (ζ=0.7 fails at ω_nT=1.4286). The twin is accurate for ω_nT ≲ 0.2 but at 1.2 the real overshoot is 0.234 against its 0.046, and at fixed T raising ω_n past ω_nT≈1 slows settling (0.4 → 5.2) then diverges. Deadbeat gains (k_pT²=1, k_dT=3/2) settle in 2 samples, the real speed limit. Stylised: noiseless, no computation delay.

A figure from its own output

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

Paper, code and results

More in Digital twins & sim-to-real

A linear-drag twin of a quadratic-drag vehicle: what does it get wrong about braking, terminal speed and the safe speed under a stopping limit?A linear-drag twin of a quadratic-drag vehicle: coast-down time and terminal speed are off by fit-speed ratios, least-squares fit gives k = 3c·vmax/4, and the safe speed under a stopping limit is mis-stated by −21% to +41% with sign set by the fit speed.
A kinematic-bicycle twin ignores tyre slip: how wrong is its turning radius, and does calibrating that reveal where a twin-tuned steering loop goes unstable?A kinematic-bicycle twin of a car ignores tyre slip: steady-turn radius is off by exactly 1 + Kv²/L (twin 4–63% too small, half at the characteristic speed), calibrating K fixes that but both twins certify steering gains stable at every speed while 11 of 30 real gain pairs lose stability below 60 m/s.
A grid twin over-states every route by a heading-dependent factor: does a finer grid fix it, and can it rank two routes the wrong way round?A grid twin over-states every route by a heading-dependent factor that cell size does not shrink (8-connected: mean 8(√2−1)/π = 1.0548, worst 1.0824 at 22.5°), calibrating the mean leaves −5.2% to +2.6%, and the twin ranks two routes (lengths 1 and 1.03) the wrong way round 19% of the time; rotating the map flips a fixed pair for 31% of orientations.
A global-shutter twin of a rolling-shutter camera ignores target motion: how biased is a twin-trained orientation estimate, and what does the same frame reveal?A global-shutter twin of a rolling-shutter camera ignores target motion: a moving rectangle is sheared and stretched by exact closed forms, a twin-trained orientation estimate is biased by half the edge tilt (1.79° at 1000 px/s), and one frame of a known-size target identifies pose and velocity (0.027° and 16 px/s at 0.25 px noise). Stylised.
A snapshot twin of a spinning lidar ignores that the platform moves during the sweep: how biased is the map, what does one scan reveal, and what do yaw and acceleration do?A snapshot-scan twin of a spinning lidar ignores motion during the sweep: length biased by exactly vτ/2 (0.5 m at 10 m/s), parallel walls converge by vτ/2π·(1/d₁+1/d₂), one scan gives speed to 0.13 m/s, yaw scales bearings by −κβ/(1+κ), acceleration biases odometry by a·t_f without accumulating.
A satellite-positioning twin with white range noise certifies an error radius the real receiver does not meet: how wrong is it once multipath is correlated, and does averaging fix it?A white-noise satellite-positioning twin vs a receiver with correlated multipath: exact HDOP √(4/n) and bias map 2b/n, averaging loses the 1/√T gain (95% radius covers 33% at T=10, 5% at T=100), and a residual integrity test tuned on the twin false-alarms 21% at equal multipath. Stylised.

All 164 Digital twins & sim-to-real projects →

Other research topics

Proper scoring & elicitation28 Effort & peer prediction9 Verification games & slashing19 Detection & tolerances14 Markets as routers13 Prediction-market design9 Paying contributors8 Decentralised training systems23 Digital twins & sim-to-real164 LLM chip design loops40 Robot manufacturing loops53 Ambient AI assistant46 Self-replicating manufacturing34 Simulation-first design64 Engineered clothing45 Task-specific AI accelerators40 Algorithmic chip-design loops42 Open-source purpose-built robots44 The software-engineering loop43 Adversarial human imitation41 Open-source residential construction41 Autonomous multi-angle capture34 Sensing, imaging and 3D31 Energy supply and dense energy storage39 Robot societies among people31 Agentic co-design of low-cost robots27 Laundry-hanging robots25 Machine learning on qualitative data23 Robots on the building site27 Civilizations & group psychology1 History & human sciences1

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.