A system with zero built-in damping — like a ball rolling on a frictionless beam — can’t be stabilised by proportional control alone, no matter the gain.
Proportional-only feedback on a pure double integrator is only marginally stable in theory, and real-world lag (a servo, a sensor) tips it into genuine instability. Derivative action supplies the damping the physical system never had.
Any real system has some damping, so pure marginal stability is an idealisation — a ball on a real beam has rolling resistance and will eventually settle. The rule is directionally right and quantitatively optimistic: the damping present is usually far too small to rely on.
Estimate with the rule, then check it against the calculator that models it properly.
Open Ball and Beam Control Simulator →A system with zero built-in damping — like a ball rolling on a frictionless beam — can’t be stabilised by proportional control alone, no matter the gain. Proportional-only feedback on a pure double integrator is only marginally stable in theory, and real-world lag (a servo, a sensor) tips it into genuine instability.