What does a twin with no actuator deadband promise a proportional loop, and what does the real one do?

research/deadband-twin

An actuator deadband the twin lacks: real loop rests at exactly d/k while the twin goes to 0, tolerance ε needs k ≥ d/ε, a fitted linear gain is 2Q(d/s) (excitation-dependent, small-signal gain 0), and bisection compensation leaves exact (d−d̂)/k or a (d̂−d)/(2−k) chatter (8 probes: 0.784 → 0.0005).

8 tests

What it found

With a deadband d the real servo loop rests at exactly |x| = d/k (0.500 at d=0.2, k=0.4) while the twin converges to zero, so the twin certifies tolerances the real loop never meets and needs k ≥ d/ε, which fails beyond k = 1. A linear twin fitted from Gaussian excitation of std s recovers 2Q(d/s) (Stein’s lemma; 0.0457 vs 0.0455 at s=0.5), an excitation-dependent gain, while the small-signal gain is 0. Bisection needs about 8 real probes to cut the stall 0.784 → 0.0005, with exact residuals (d−d̂)/k under-compensated and a period-2 chatter (d̂−d)/(2−k) over-compensated. Stylised: scalar integrator, noiseless, k ≤ 1.

A figure from its own output

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

Paper, code and results

More in Digital twins & sim-to-real

Does measurement noise, or dither, hide an actuator deadband the twin lacks?Follow-up to deadband-twin: measurement noise flips the linear twin's error from 27× optimistic (s=0.01) to conservative (0.37× at s=0.05), dither helps only without noise, and mild under-compensation beats the exact width under noise (single-seed simulations).
How large a manoeuvre can a twin without an actuator rate limit be trusted for?A twin without an actuator rate limit is exact only for x₀ ≤ r/k; beyond it the real loop overshoots (0.90 of x₀ at x₀/r=10⁴) and settles in O(amplitude) steps, with an empirical overshoot threshold 2n(2n−1) for k=1/n.
Does a twin with a rigid drivetrain miss the hunting cycle caused by gear backlash?A rigid-drivetrain twin says the PI loop converges but gear backlash makes the real loop hunt at 1.3–2.9× the gap (exactly ∝ h); describing-function prediction 7–10% high, and bistable starts can hide it.
How wrong is the stopping distance of a twin that models friction as purely viscous?A viscous-only twin never stops (6.5–17.7 s to reach 1e-3…1e-9 vs 1.95 s real) and its fitted gain b + c·E[v]/E[v²] depends on the logged speed range; fitted on [1,3] it overestimates stopping distance ×1.8–17 below and underestimates above v*=7.3, no single viscous gain beats 0.87 worst error over [0.05,3], affine fit recovers exactly.
Does a twin with correct sliding friction still miss stick-slip hunting from static friction?A twin with the right sliding friction says the PID loop converges, and with no stiction it does; a breakaway excess Fs−Fc makes the real loop stick-slip hunt at ≈0.5–0.9 ΔF/kp (independent of Fc), and short runs can mistake a long stick for convergence.
How much bandwidth can a controller tuned in a twin with an instantaneous actuator really have?A twin with an instantaneous actuator is stable at any gain; the real loop with lag τ is stable iff τ<kd/kp=2ζ/ωn, hunts at the twin's own ωn, decays no faster than 1/(3τ) for any gains, and the real-optimal bandwidth is ≈0.26–0.46 of the limit.

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.