Root locus of s³+9s²+Ks+K = 0

Converting to root-locus form

$$s^3 + 9s^2 + K(s + 1) = 0 \;\Rightarrow\; 1 + \frac{K(s+1)}{s^2(s+9)} = 0$$

Equivalent open-loop transfer function: $G(s)H(s) = \dfrac{K(s+1)}{s^2(s+9)}$.

Root locus construction

1. Poles: $0, 0, -9$. Zero: $-1$. With $n = 3$ and $m = 1$, two branches go to infinity.
2. Real-axis locus: $[-9, -1]$.
3. Asymptotes:

$$\sigma_A = \frac{(0 + 0 - 9) - (-1)}{2} = -4, \qquad \theta_A = \pm 90^\circ$$

4. Departure from the double pole at 0: $2\phi = 180^\circ$, so the branches leave vertically at $\pm 90^\circ$ and curve into the LHP.
5. Breakaway point:

$$K = -\frac{s^3 + 9s^2}{s + 1}, \qquad \frac{dK}{ds} = 0 \Rightarrow 2s^3 + 12s^2 + 18s = 0$$ $$2s(s + 3)^2 = 0 \Rightarrow s = 0,\ -3,\ -3$$

$s = -3$ is a triple root point:

$$K = -\frac{-27 + 81}{-2} = 27$$

Check: $(s + 3)^3 = s^3 + 9s^2 + 27s + 27$, which matches the characteristic equation with $K = 27$. ✓ 6. Branch angles at $s = -3$: three branches meet, so arriving and leaving branches are spaced $60^\circ$ apart. The two complex branches from the origin and the real branch from $-9$ arrive (from directions $\pm 60^\circ$ and $180^\circ$). They leave at $0^\circ$ (to the zero at $-1$) and $\pm 120^\circ$ (toward the asymptotes $\sigma = -4$). 7. $j\omega$ crossing (Routh):

No sign change for $K > 0$, so there is no $j\omega$ crossing.

                           .                 |
                           .                 |
                           .                 |
                           ..                |
                            .                |
                            .                |
                            ..       .....   |
                             .. ......   ....|
----x..........................*........o---.x----
                             .. ......   ....|
                            ..       .....   |
                            .                |
                            .                |
                           ..                |
                           .                 |
                           .                 |
                           .                 |
   -9                     -4  -3             0
x pole (0, 0, -9)  o zero (-1)  * triple point (-3, K=27)

Comment on stability

• For $0 < K < 27$: one real pole lies between $-9$ and $-3$, plus a complex pair moving from the origin toward $-3$. The system is stable and underdamped.
• At $K = 27$: there is a triple pole at $s = -3$. The system is stable with a critically damped type response.
• For $K > 27$: one real pole lies between $-3$ and $-1$, and a complex pair moves toward $\sigma = -4$. The system is stable and underdamped.
• The first column of the Routh array is positive for all $K > 0$.

Answer: The locus has a triple meeting point at s = −3 (K = 27), asymptotes at σ = −4 (±90°), and stays entirely in the LHP. The system is stable for all K > 0.