---
title: "Root locus of k(s+2)/[s(s+1)(s²+8s+64)]"
subject: "Control Systems"
chapter: 5
key: "semester_3/control_systems:rl-k-s2-s-s1-s2"
source: https://www.bctengineer.com/q/semester_3/control_systems/rl-k-s2-s-s1-s2
site: BCT Engineering
---

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

Control Systems · Chapter 5: Root Locus Technique

- Asked in: 2076 Asoj, 10 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rl-k-s2-s-s1-s2
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5

## Question

Sketch Root locus plot for the system open loop transfer function G(s)H(s) = k(s+2)/[s(s+1)(s²+8s+64)]. Discuss the region for stability, instability and marginal stability. What is frequency of oscillation at the point of marginal stability?

## Short answer

G(s)H(s)=k(s+2)/[s(s+1)(s²+8s+64)]. Real-axis breakaway at -0.572, break-in at -3.51. Stable for 0<k<442.5. Marginally stable at k=442.5 with sustained oscillation frequency 7.50 rad/s.

## Answer

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

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)

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^4$ | 1 | 72 | $2k$ |
| $s^3$ | 9 | $64+k$ | |
| $s^2$ | $(584-k)/9$ | $2k$ | |
| $s^1$ | $\dfrac{(584-k)(64+k) - 162k}{584-k}$ | | |
| $s^0$ | $2k$ | | |

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.

```text
                                      |     ....
                                      |  ....
                                     .X...
                       x..............|
                                      |
                                      |
                                      |
                                      |
                         ...........  |
........................*.....o---x...x-----------
                         ...........  |
                                      |
                                      |
                                      |
                                      |
                       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

| Range of $k$ | Nature |
|---|---|
| $0 < k < 442.5$ | Stable (all poles in LHP) |
| $k = 442.5$ | Marginally stable, poles at $\pm j7.50$ |
| $k > 442.5$ | Unstable (two poles in RHP) |

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

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With 4 poles and 1 zero, how many branches go to infinity?
   A. 4
   B. 3
   C. 1
   D. 2
   Answer: B. One branch ends at the zero, so n - m = 3 branches go to infinity. (Key point: Number of branches and ends)

2. What is the asymptote centroid?
   A. -2.33
   B. -3.0
   C. -1.5
   D. -4.0
   Answer: A. sigma_A = (0 - 1 - 4 - 4 - (-2))/3 = -7/3 = -2.33. (Key point: Asymptote centroid)

3. How are the crossing gain and frequency found?
   A. Routh array: set the s^1 row to zero and use the s^2 auxiliary equation
   B. Set dK/ds = 0
   C. Use the angle of departure
   D. Set K = 1
   Answer: A. Zero s^1 row gives k = 442.5 and the auxiliary equation gives w = 7.50 rad/s. (Key point: Imaginary-axis crossing)

4. What is the stability result?
   A. Stable for k > 442.5
   B. Unstable for 0 < k < 442.5
   C. Stable for 0 < k < 442.5; marginal at 442.5; unstable above
   D. Stable for all k
   Answer: C. Poles cross into the RHP at k = 442.5, oscillating at 7.50 rad/s. (Key point: Stability regions)

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