r(t) = R sin ωt ⇒ yss(t) = R|G(iω)| sin(ωt + ∠G(iω))
|G(iω)| ≈ ?, ∠G(iω) ≈ ?
yss(t) ≈ ?
|G(iω)| = ?
Y = R|G(iω)|
Y compared with R: ?
Maximum |G(iω)|.
The height of the peak.
For the prototype second-order system,
G(s) = ωn2s2 + 2ζωns + ωn2
The frequency-domain specifications are given by
| 0 < ζ ≤ 1/√2 ≈ 0.7071 | ζ > 1/√2 | |
|---|---|---|
| Mr | 12ζ√(1 − ζ2) | 1 |
| ωr | ωn√(1 − 2ζ2) | 0 |
BW = ωn [(1 − 2ζ2) + √(4ζ4 − 4ζ2 + 2)]1/2
Optional: full derivation ↗Step input: the command jumps from 0 to 1 at t = 0. The curve shows the output y(t).
Time to first reach 50% of the final value.
Time from the first 10% crossing to the first 90% crossing.
Peak above the final value, as a fraction of the final value.
Time until the output enters and stays within ±5% of its final value.
Independent of ωn.
ζ ↑ → Mr ↓ toward 1.
Similar trend to maximum overshoot Mp.
Compare A → B, then A → C.
What changes in bandwidth, peak, rise and settling?
T(s) = Ks² + s + K
Choose K = 1, 2.5, or 4.
ωn ≥ 1.5 rad/s
ζ ≥ 0.3
| K | ωn | ζ | Both? |
|---|---|---|---|
| 1 | |||
| 2.5 | |||
| 4 |
F(s) = K G(s)
E = R − Y
Y = FE
ess = ?
ess = ?
ess = ?
Type 1: F(s) ≈ gs as s → 0, g > 0
11 + Kp
1Kv
1Ka
| Input | Find the constant | ess = ? |
|---|---|---|
| Step | Kp = lims→0 F(s) = ? | ? |
| Ramp | Kv = lims→0 sF(s) = ? | ? |
| Parabola | Ka = lims→0 s²F(s) = ? | ? |
Use the formulas. Fill each row before revealing it.
| Type | Unit step r(t) = 1 | Unit ramp r(t) = t | Unit parabola r(t) = ½t² |
|---|---|---|---|
| 0 | |||
| 1 | |||
| 2 | |||
| 3 |
Kp = lims→0 F(s)
Step: 1/(1 + Kp)
Kv = lims→0 sF(s)
Ramp: 1/Kv
Ka = lims→0 s²F(s)
Parabola: 1/Ka
Type 1: F(s) ≈ g/s as s → 0.
Rewrite F(s) into standard factors.
Only gsⁿ can provide poles at s = 0.
| n | gsⁿ | Poles at s = 0 | Type |
|---|---|---|---|
| 2 | gs² | ||
| 1 | gs | ||
| 0 | g | ||
| −1 | g/s | ||
| −2 | g/s² |
Suppose n = −1, i.e., Type 1.
F(s) = gs−1 × 1τs + 1 × ⋯ × ωn²s² + 2ζωns + ωn²
Kv = lims→0 sF(s) = ?
Stable unit-feedback system
| Input r(t) | 1 | t | ½t² |
|---|---|---|---|
| ess | ? | ? | ? |
Stable unit-feedback system
| Input r(t) | 1 | t | ½t² |
|---|---|---|---|
| ess | ? | ? | ? |