Stability and responsible tuning

A controller can reduce error at first and still be unsafe. Stability asks what happens to bounded disturbances and initial errors over time. If deviations decay, the operating point is asymptotically stable; if they persist without decay it may be marginal; if they grow, it is unstable. A practical response can also be stable yet poor because it rings, overshoots a stop, or settles too slowly.

For a common second-order model, ÿ + 2ζω_n ẏ + ω_n²y = ω_n²u. Natural frequency ω_n sets a time scale, while damping ratio ζ shapes the transient. For 0 < ζ < 1, percent overshoot is approximately 100 exp[-ζπ/√(1-ζ²)]. Delay adds phase lag: sensing, filtering, computation, communication, and actuator response all mean correction arrives late. Raising loop gain then makes the controller chase old information and can turn damping into oscillation.

Responsible tuning begins with verified feedback sign, conservative output and travel limits, a repeatable test input, and logged signals. Increase proportional gain gradually until response is useful but not persistently oscillatory, then add only the action needed for a diagnosed limitation. Change one parameter at a time and return to a known-safe setting after anomalies. The build decision may be to reduce sample delay or mechanical backlash rather than “tune around” them.

A common misconception is that the highest gain before obvious oscillation is optimal. Noise, payload changes, battery voltage, and untested positions can erase that narrow margin. Another is interpreting a clipped oscillation as stable because its amplitude no longer grows; actuator saturation may merely bound an unstable linear tendency while heating hardware. Do not use aggressive self-oscillation tuning on fragile mechanisms. A clean simulation trace is evidence about its model, not proof that a camera arm or toy car will behave identically.