Root locus
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| root locus | nghiên cứu nghiệm |
| asymptote/ˈæsɪmptəʊt/ | tiếp tuyến |
| breakaway point | điểm tách |
The locus follows one angle condition
- The root locus 根轨迹 is where $\sum\angle(s-z_i) - \sum\angle(s-p_i) = 180°\cdot(2k+1)$: every sketch rule falls out of it. Branches start at open-loop poles and end at zeros or infinity.
- Rules to carry: real-axis segments with an odd count to the right; asymptote 渐近线 angles $180°(2k+1)/(n-m)$ about the centroid; the breakaway point 分离点 from $\frac{d}{ds}\bigl[\prod(s-p_i)\bigr] = 0$ (keep only on-locus roots); the $j\omega$ crossing from $1+KL_0 = 0$.
![Root locus of K/[s(s+2)(s+4)] with the ζ = 0.5 design point marked](/handout-media/university_electrical_engineering/assets/10-root-locus.png?v=1793069544)
$L = \dfrac{K}{s(s+2)(s+4)}$: centroid $-2$, asymptotes $\pm 60°, 180°$; breakaway $-0.845$ (the other root $-3.155$ is off the locus: discard); crossing at $\pm j2.83$ with $K = 48$.
{exercise:uni-ee1045-sketch}
For L = K/[s(s+2)(s+4)], the root-locus breakaway point on the real axis is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
d/ds[s³+6s²+8s] = 3s²+12s+8 = 0 gives −0.845 and −3.155, but only −0.845 lies on the real-axis segment [−2, 0]. K there is 0.845×1.155×3.155 = 3.08.
Design on a damping ray
- Constant $\zeta$ is a ray at $\arccos\zeta$ from the negative real axis. Walk the ray until the angle condition holds; the gain follows from the magnitude condition $K = \dfrac{\prod|s-p_i|}{\prod|s-z_i|}$.
- For $\zeta = 0.5$ on the running plant: the ray meets the locus at $s^{*} = -0.667 + j1.155$, and $K = |s^{*}||s^{*}+2||s^{*}+4| = 8.30$.
Check: $|s^{*}| = 1.334$ and $0.667/1.334 = 0.500$, exactly the damping asked for. $K = 8.30$ sits far below the stability limit 48, and the third pole $-4.67$ is well left of the pair, so the pair dominates.
{exercise:uni-ee1045-design}
The ζ = 0.5 ray meets the locus of K/[s(s+2)(s+4)] at −0.667 + j1.155. The gain there is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
K = |s||s+2||s+4| = 1.334 × 1.764 × 3.527 = 8.30. The angle check: 120° + 40.9° + 19.1° = 180°, and 0.667/1.334 = 0.5 exactly.
Read the picture honestly
- Circles are constant $\omega_n$; rays are constant $\zeta$. Distance from the origin is speed, angle is damping.
- The asymptotes are sketch aids: compute the actual crossing from $1 + KL_0(j\omega) = 0$ before quoting a stability limit.
{exercise:uni-ee1045-poles}
A closed-loop pole sits at −2 + j3.46. Its damping ratio is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
ζ = σ/|s| = 2/√(4+12) = 2/4 = 0.5: the cosine of the angle from the negative real axis. |s| = ω_n = 4 and the imaginary part is the ringing frequency.