Root locus of K(s+9)/[s(s²+4s+11)]

$$G(s) = \frac{K(s+9)}{s(s^2+4s+11)}$$

Characteristic equation: $s^3 + 4s^2 + (11+K)s + 9K = 0$.

Root locus data

1. Poles: $0,\ -2 \pm j2.646$. Zero: $-9$. With $n = 3$ and $m = 1$, two branches go to infinity.
2. Real-axis locus: $[-9, 0]$.
3. Asymptotes:

$$\sigma_A = \frac{(0 - 2 - 2) - (-9)}{2} = +2.5, \qquad \theta_A = \pm 90^\circ$$

The asymptotes lie in the right half-plane, so the complex branches must cross the $j\omega$-axis. 4. Breakaway: $dK/ds = 0 \Rightarrow 2s^3 + 31s^2 + 72s + 99 = 0$. Roots: $s = -13.03$ (gives $K < 0$; not on the locus) and $-1.24 \pm j1.51$ (complex). So there is no breakaway point. The pole at $0$ moves along the real axis to the zero at $-9$. 5. Angle of departure from $-2 + j2.646$: from zero $-9$: $20.71^\circ$; from pole $0$: $127.09^\circ$; from the conjugate pole: $90^\circ$.

$$\phi_d = 180^\circ + 20.71^\circ - (127.09^\circ + 90^\circ) = -16.38^\circ$$

6. $j\omega$ crossing (Routh):

Sketch

                                   | ..
                                   |..
                                   ..
                                  .X
                                ...|
                            x....  |
                                   |
                                   |
----o----..........................x--------------
                                   |
                                   |
                            x....  |
                                ...|
                                  .X
                                   ..
                                   |..
                                   | ..
   -9               -4             0       2.5
x pole   o zero (-9)
X jw crossing (+-j4.45, K = 8.8)

(i) Range of K for stability

All first-column entries must be positive:

• $9K > 0 \Rightarrow K > 0$
• $44 - 5K > 0 \Rightarrow K < 8.8$

Answer (i): 0 < K < 8.8

(ii) Undamped natural frequency of oscillation

At $K = 8.8$ the auxiliary equation is

$$4s^2 + 9(8.8) = 0 \;\Rightarrow\; s^2 = -19.8 \;\Rightarrow\; s = \pm j4.45$$

Answer (ii): ω = 4.45 rad/s (sustained oscillation when K = 8.8).