Root locus of k(s+2)/[s(s+1)(s²+8s+64)]

$$G(s)H(s) = \frac{k(s+2)}{s(s+1)(s^2+8s+64)}$$

Characteristic equation: $s^4 + 9s^3 + 72s^2 + (64+k)s + 2k = 0$.

1. Poles, zeros, branches

• Poles: $0, -1, -4 \pm j6.93$ ($n = 4$). Zero: $-2$ ($m = 1$).
• There are 4 branches: one ends at $-2$ and three go to infinity.

2. Real-axis locus

$[-1, 0]$ and $(-\infty, -2]$.

3. Asymptotes

$$\begin{aligned} \sigma_A &= \frac{(0 - 1 - 4 - 4) - (-2)}{3} = -2.33 \\ \theta_A &= 60^\circ,\ 180^\circ,\ 300^\circ \end{aligned}$$

4. Breakaway / break-in points

$$\frac{dk}{ds} = 0 \Rightarrow 3s^4 + 26s^3 + 126s^2 + 288s + 128 = 0$$

Real roots: $s = -0.572$ ($k = 10.24$), which is a breakaway between 0 and −1, and $s = -3.51$ ($k = 281.5$), which is a break-in on $(-\infty, -2]$. The other two roots are complex and are not on the locus.

5. Angle of departure from $-4 + j6.93$

Angles: from zero $-2$: $106.1^\circ$; from pole $0$: $120^\circ$; from $-1$: $113.4^\circ$; from $-4 - j6.93$: $90^\circ$.

$$\phi_d = 180^\circ + 106.1^\circ - (120^\circ + 113.4^\circ + 90^\circ) = -37.3^\circ$$

6. Imaginary-axis crossing (Routh)

Setting the $s^1$ row to zero: $k^2 - 358k - 37376 = 0 \Rightarrow k = 442.5$. Auxiliary equation: $\dfrac{584 - 442.5}{9}s^2 + 885 = 0 \Rightarrow \omega = 7.50$ rad/s.

Shape of the locus

• The branches from $0$ and $-1$ meet at $-0.572$ and break away into the complex plane. They loop to the left and re-enter the real axis at the break-in point $-3.51$. From there, one branch goes right to the zero at $-2$ and the other goes left to $-\infty$.
• The complex poles leave at $\mp 37.3^\circ$ (moving right), cross the $j\omega$-axis at $\pm j7.50$, and follow the ±60° asymptotes.

                                      |     ....
                                      |  ....
                                     .X...
                       x..............|
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                         ...........  |
........................*.....o---x...x-----------
                         ...........  |
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                       x..............|
                                     .X...
                                      |  ....
                                      |     ....
       -8             -4     -2       0
x pole   o zero (-2)   * break-in (-3.51)
X jw crossing (+-j7.5, k = 442.5)

Stability regions

Answer: Marginal stability at k = 442.5. The frequency of oscillation there is ω = 7.50 rad/s.