Root locus with poles 0, −6, −2 ± j3

Open-loop poles at $0, -6, -2 \pm j3$ and no zeros give

$$G(s)H(s) = \frac{K}{s(s+6)(s^2+4s+13)}$$

Characteristic equation: $s^4 + 10s^3 + 37s^2 + 78s + K = 0$.

1. Branches

$n = 4$, $m = 0$: 4 branches, all ending at infinity.

2. Real-axis locus

Between $0$ and $-6$ (one real pole to the right is an odd count). There is no locus left of −6.

3. Asymptotes

$$\begin{aligned} \sigma_A &= \frac{0 - 6 - 2 - 2}{4} = -2.5 \\ \theta_A &= \pm 45^\circ,\ \pm 135^\circ \end{aligned}$$

4. Breakaway point

$K = -(s^4 + 10s^3 + 37s^2 + 78s)$

$$\frac{dK}{ds} = -(4s^3 + 30s^2 + 74s + 78) = 0$$

Roots: $s = -4.20$ and $s = -1.65 \pm j1.39$. Only $s = -4.20$ is on the real-axis locus. Gain there:

$$K = 4.20 \times 1.80 \times 13.84 \approx 104.6$$

5. Angle of departure from $-2 + j3$

Angles to $-2+j3$: from $0$: $123.69^\circ$; from $-6$: $36.87^\circ$; from $-2-j3$: $90^\circ$.

$$\phi_d = 180^\circ - (123.69^\circ + 36.87^\circ + 90^\circ) = -70.56^\circ$$

At $-2-j3$, it is $+70.56^\circ$.

6. Imaginary-axis crossing

$K_{mar} = 227.76$. Auxiliary equation: $29.2s^2 + 227.76 = 0 \Rightarrow \omega = \sqrt{7.8} = 2.79$ rad/s.

Sketch

                                       |
    ...                                |      ....
      ....                             |   ....
         ....                x.        .....
            ...               .........X
              ...                      |
                ...                    |
                  ..                   |
----------x........*...................x----------
                  ..                   |
                ...                    |
              ...                      |
            ...               .........X
         ....                x.        .....
      ....                             |   ....
    ...                                |      ....
                                       |
         -6      -4.2       -2         0
x pole   * breakaway (-4.2)
X jw crossing (+-j2.79, K = 227.8)

The branches from $0$ and $-6$ meet at $-4.2$ and break away at ±90°, then bend to the ±135° asymptotes. The complex poles leave at $\mp 70.6^\circ$, curve right, cross the $j\omega$-axis at $\pm j2.79$ and follow the ±45° asymptotes.

Answer: Breakaway at s = −4.20 (K ≈ 104.6). The departure angle is ∓70.6°. The system is stable for 0 < K < 227.8 and oscillates at 2.79 rad/s when K = 227.8.