Stability and steady-state error
A response can be bounded yet inaccurate, or accurate in theory but unreachable with real actuator limits.
First, picture it
An upright pole is unstable without active balancing: a small tilt grows. A motor speed loop may be stable but sit below target under load. Stability and accuracy answer different questions.
What the model says
For a minimal continuous-time LTI model, asymptotic stability requires every closed-loop pole to have negative real part. The final-value theorem can estimate the steady value only when its pole conditions hold; applying it to an unstable loop gives nonsense. Integral action increases low-frequency loop gain and can remove constant-error under suitable stability and actuator conditions, but saturation can prevent that ideal result.
e_ss=lim(t→∞)e(t)=lim(s→0)sE(s) [only if final-value conditions hold]Check closed-loop stability before using the final-value theorem.
A pole's real part sets growth or decay; its imaginary part sets oscillation frequency. The boundary itself needs separate analysis.
Make it concrete
Apply a constant load to the motor. P-only control may keep a persistent speed error; adding I can reduce it, unless the demanded voltage exceeds the limit.
Test load rejection'Bounded in this 12-second run' is not a mathematical proof of stability for all time and inputs.
Check your understanding+
Can a stable motor speed loop still miss the target?
Yes. It can settle with nonzero error, especially under load or saturation.