Acceleration Error Constant⚠ unverified
Physics / Controls · Compute the steady-state error from the acceleration error constant
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| ka | ka | — | 1.0 | Acceleration error constant (Ka) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | ess | — | Steady-state acceleration error. Returns infinity when ka is zero |
The science & history
Understanding the Parameters
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$K_a$ — larger $K_a$ → smaller parabolic tracking error. Type 2+ systems can have infinite $K_a$ (zero $e_{ss}$ to a parabola).
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$e_{ss}$ — $1/K_a$ for unit $t^{2}/2$ reference; scale for other amplitudes.
Derivation (Approaching a Proof)
Final-value theorem on $E(s) = R(s)/(1+G(s))$ with $R(s) = 1/s^{3}$ (Laplace of $t^{2}/2$) yields $e_{ss} = 1/K_a$ when $K_a$ is finite and nonzero. Definition: $K_a \equiv \lim_{s\to 0} s^{2} G(s)$ for unity feedback.
History
Position / velocity / acceleration error constants $K_p$, $K_v$, $K_a$ are classical servo figures of merit (type number classification).
Related Concepts: Velocity Error Constant, Pid Output, Phase Margin
Notes: Registry calculator acceleration-error-constant (unverified). Computes $e_{ss}=1/K_a$,
not $K_a$ from $G(s)$.