Root locus of K/[s(s+1)(s²+4s+5)] with ξ = 0.5

$$G(s) = \frac{K}{s(s+1)(s^2+4s+5)}, \qquad H(s) = 1$$

Characteristic equation: $s^4 + 5s^3 + 9s^2 + 5s + K = 0$.

1. Poles, zeros, branches

Poles: $0,\ -1,\ -2 \pm j1$ ($n = 4$). No zeros ($m = 0$). All four branches go to infinity.

2. Real-axis locus

Only between $0$ and $-1$.

3. Asymptotes

$$\sigma_A = \frac{0 - 1 - 2 - 2}{4} = -1.25, \qquad \theta_A = \pm 45^\circ,\ \pm 135^\circ$$

4. Breakaway point

$$\frac{dK}{ds} = -(4s^3 + 15s^2 + 18s + 5) = 0$$

Real root: $s = -0.393$. Gain there: $K = 0.393 \times 0.607 \times 3.583 = 0.855$. The other roots, $-1.68 \pm j0.60$, are not on the locus.

5. Angle of departure from $-2 + j1$

From $0$: $153.43^\circ$; from $-1$: $135^\circ$; from $-2-j1$: $90^\circ$.

$$\phi_d = 180^\circ - 378.43^\circ = -198.43^\circ \equiv +161.57^\circ$$

6. $j\omega$ crossing (Routh)

$K_{mar} = 8$. Auxiliary equation: $8s^2 + 8 = 0 \Rightarrow s = \pm j1$.

Sketch

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                 -2       -1        0
x pole   * breakaway (-0.393)   X jw crossing (+-j1)
# zeta = 0.5 poles (-0.288 +- j0.5, K = 2.04)

• Branches from $0$ and $-1$ break away at $-0.393$, curve toward the ±45° asymptotes, and cross the $j\omega$-axis at $\pm j1$.
• Branches from $-2 \pm j1$ leave at $\pm 161.6^\circ$ and follow the ±135° asymptotes.

7. Dominant closed-loop poles for ξ = 0.5

Constant-ξ line: $\theta = \cos^{-1}0.5 = 60^\circ$ from the negative real axis, i.e. $s = \omega_n(-0.5 + j0.866)$. Search along it for $\angle G = -180^\circ$. The solution is at $\omega_n = 0.577$, giving $s = -0.288 + j0.500$.

$$\angle G = -(120.00 + 35.07 - 16.30 + 41.22)^\circ = -180.0^\circ \;\checkmark$$

Gain from the magnitude criterion:

$$K = 0.577 \times 0.870 \times 1.783 \times 2.276 = 2.04$$

With $K = 2.04$, the other two closed-loop poles are at $-2.21 \pm j1.11$. They are about 7.7 times farther from the $j\omega$-axis than the dominant pair, so the pair at $-0.288 \pm j0.5$ is truly dominant.

Answer: The dominant closed-loop poles for ξ = 0.5 are s = −0.288 ± j0.500 ($\omega_n = 0.577$ rad/s), at K ≈ 2.04. The system is stable for 0 < K < 8.