---
title: "R-H root distribution of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-s6-2-8-12-20"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-s6-2-8-12-20
site: BCT Engineering
---

# R-H root distribution of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16

Control Systems · Chapter 4: Stability

- Asked in: 2071 Shrawan, 6 marks
- Video: [routh hurwitz criterion special cases | when all elements of a row are zero](https://www.youtube.com/watch?v=BqH0mB9GGAQ) — Smart Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rh-s6-2-8-12-20
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

Using R-H criteria, tell how many roots of polynomial are in right half s-plane, in left half s-plane and on jω axis. s⁶ + 2s⁵ + 8s⁴ + 12s³ + 20s² + 16s + 16 = 0

## Short answer

Routh array shows no sign changes, so zero roots in RHP. An all-zero s³ row gives an auxiliary equation whose solution yields four jω-axis roots (±j1.414, ±j2). Factoring out leaves s²+2s+2 giving two LHP roots at −1±j1. Thus, RHP=0, LHP=2, jω axis=4.

## Answer

**Result:** RHP = 0, LHP = 2, $j\omega$ axis = 4. The system is **marginally stable**.

By the Routh–Hurwitz criterion, the number of sign changes in the first column equals the number of RHP roots. A zero row shows roots symmetric about the origin, which are found from the auxiliary equation.

$$
s^6 + 2s^5 + 8s^4 + 12s^3 + 20s^2 + 16s + 16 = 0
$$

#### Routh array

| Row | Col 1 | Col 2 | Col 3 | Col 4 |
|---|---|---|---|---|
| $s^6$ | 1 | 8 | 20 | 16 |
| $s^5$ | 2 | 12 | 16 | 0 |
| $s^4$ | $\frac{2(8)-1(12)}{2}=2$ | $\frac{2(20)-1(16)}{2}=12$ | 16 | |
| $s^3$ | $\frac{2(12)-2(12)}{2}=0$ | $\frac{2(16)-2(16)}{2}=0$ | | |

The $s^3$ row is all zeros, so there are roots placed symmetrically about the origin. Auxiliary equation from the $s^4$ row:

$$
A(s) = 2s^4 + 12s^2 + 16, \qquad \frac{dA}{ds} = 8s^3 + 24s
$$

| Row | Col 1 | Col 2 |
|---|---|---|
| $s^3$ | 8 | 24 |
| $s^2$ | $\frac{8(12)-2(24)}{8}=6$ | $\frac{8(16)-0}{8}=16$ |
| $s^1$ | $\frac{6(24)-8(16)}{6}=2.67$ | 0 |
| $s^0$ | 16 | |

First column: $1,\ 2,\ 2,\ 8,\ 6,\ 2.67,\ 16$. **No sign change**, so no root is in the RHP.

#### Roots of the auxiliary equation

$$
\begin{aligned}
2s^4+12s^2+16 &= 2(s^2+2)(s^2+4) = 0 \\
s &= \pm j1.414,\ \pm j2
\end{aligned}
$$

These four roots lie on the $j\omega$ axis and are **not repeated**. The other factor is

$$
\frac{s^6+2s^5+8s^4+12s^3+20s^2+16s+16}{(s^2+2)(s^2+4)} = s^2+2s+2 \;\Rightarrow\; s = -1\pm j1
$$

| Region | Number | Roots |
|---|---|---|
| Right half plane | 0 | none |
| $j\omega$ axis | 4 | $\pm j1.414$, $\pm j2$ |
| Left half plane | 2 | $-1\pm j1$ |

The $j\omega$-axis roots are simple, so the system is **marginally stable**: it oscillates at 1.414 rad/s and 2 rad/s.

## Check yourself

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

1. The s^3 row is all zero, so we
   A. Declare instability
   B. Divide by s
   C. Ignore it
   D. Differentiate the auxiliary equation from the s^4 row
   Answer: D. A(s)=2s^4+12s^2+16, dA/ds = 8s^3+24s supplies the new row. (Key point: Zero row)

2. The first column 1, 2, 2, 8, 6, 2.67, 16 has
   A. No sign change, so no RHP roots
   B. One sign change
   C. Two sign changes
   D. Three sign changes
   Answer: A. All elements are positive. (Key point: Sign changes)

3. The auxiliary equation roots are
   A. +-1.414 and +-2
   B. +-j1.414 and +-j2
   C. +-j2 repeated
   D. -1+-j1
   Answer: B. 2(s^2+2)(s^2+4)=0. (Key point: Imaginary roots)

4. The system is marginally stable because
   A. All roots are in the LHP
   B. There is a sign change
   C. The jw-axis roots are simple and none are in the RHP
   D. The roots are repeated
   Answer: C. Simple imaginary roots give sustained oscillation; the other roots -1+-j1 are in the LHP. (Key point: Verdict)

## Close variants set in other exams

- [R-H root distribution of s⁵+3s⁴+5s³+4s²+s+3](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-s5-3-5-4-1) — Control Systems, chapter 4, asked 4×
- [R-H root distribution of s⁵+4s⁴+2s³+8s²+s+4](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-s5-4-2-8-1) — Control Systems, chapter 4, asked 4×

## Related in Control Systems

- [Relative and absolute stability; R-H of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-relative-absolute-s6) — Control Systems, chapter 4, asked 2×