A twin with an unlimited actuator says a PI step response ignores step size: what does the real loop with a command limit do?

What it found

Integrator plant under PI control, real actuator limited to |u|≤U with constant load d; the twin has neither. The twin is exact iff k_px₀≤U (zero deviation up to x₀=0.714), so gentle step tests certify it. Beyond that the integrator winds up to exactly x₀²/(2U): overshoot rises from the twin’s 21% to 74%, 93%, 97% and the settling time from 4.9 to 17, 165, 944 (x₀/U=5, 20, 50); a load d≥U is never rejected and settling blows up as d→U. “Faster is better” fails: the twin’s settling time falls 32× over ω_n=0.25–8 while the real one is best near ω_n=1 and rises to 77; conditional integration recovers most of it. Companion to the lag and slew twins. Stylised: one integrator, static limit, noiseless.

A figure from its own output

Paper, code and results

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

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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.