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

# Relative and absolute stability; R-H of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16

Control Systems · Chapter 4: Stability

- Asked in: 2078 Bhadra, 2+6 marks
- Video: [Routh-Hurwitz Criterion, Beyond Stability](https://www.youtube.com/watch?v=wGC5C_7Yy-s) — Brian Douglas
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rh-relative-absolute-s6
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

What is relative and absolute stability? Check the stability of system with characteristic equation: s⁶ + 2s⁵ + 8s⁴ + 12s³ + 20s² + 16s + 16 = 0 using R-H criteria.

## Short answer

Absolute stability means all poles in the left half-plane; relative stability quantifies distance from jω-axis. Four simple poles lie on the jω-axis and two in the left half-plane, so the system is marginally stable.

## Answer

#### Absolute and relative stability

- **Absolute stability** answers yes or no: is the system stable or not? A system is absolutely stable if all closed-loop poles lie in the left half of the s-plane, so every bounded input gives a bounded output and the natural response decays to zero. The Routh–Hurwitz criterion checks absolute stability.
- **Relative stability** measures *how* stable a stable system is, i.e. how far its poles are from the $j\omega$ axis (instability boundary). Poles further left decay faster and give less oscillation. It is measured by the real part of the dominant poles (settling time), damping ratio, or in the frequency domain by **gain margin** and **phase margin**. In Routh's method, it is found by shifting the axis ($s = z - \sigma$) and testing again.

| Absolute stability | Relative stability |
|---|---|
| Stable or unstable (qualitative) | Degree of stability (quantitative) |
| Checks sign of real parts of poles | Checks distance of poles from $j\omega$ axis |
| Routh–Hurwitz, pole location | Shifted Routh, GM, PM, damping ratio |

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

#### 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 | $-1\pm j1$ |

#### Conclusion

There is no root in the RHP, and the four roots on the $j\omega$ axis are simple (non-repeated). The system is **marginally stable** (not absolutely stable): its response contains sustained oscillations at $\omega = 1.414$ rad/s and $\omega = 2$ rad/s.

## Check yourself

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

1. What does absolute stability mean?
   A. How fast the response settles
   B. The gain margin of a stable system
   C. Whether the system is stable or not, i.e. all closed-loop poles are in the left half plane
   D. The damping ratio only
   Answer: C. Absolute stability is a yes/no property: all poles in the LHP, so the natural response decays to zero. (Key point: Absolute stability)

2. Which description fits relative stability?
   A. It measures how far the poles are from the jω axis, e.g. through GM, PM or damping ratio
   B. It is a yes/no test using Routh
   C. It counts RHP roots only
   D. It applies only to unstable systems
   Answer: A. Relative stability is the degree of stability, measured by pole distance from the jω axis, gain margin and phase margin. (Key point: Relative stability)

3. The s^3 row of s^6+2s^5+8s^4+12s^3+20s^2+16s+16 is all zero. What does that indicate?
   A. All roots in the RHP
   B. The system is stable
   C. Roots symmetric about the origin, found from the s^4 auxiliary equation
   D. The polynomial has a negative coefficient
   Answer: C. An all-zero row means symmetric roots; the auxiliary equation 2s^4+12s^2+16=0 gives them. (Key point: Zero row)

4. What are the roots of 2s^4+12s^2+16=0?
   A. ±j1.414 and ±j2
   B. ±1.414 and ±2
   C. -1±j1
   D. ±j4 and ±j2
   Answer: A. 2(s^2+2)(s^2+4)=0 gives s=±j1.414, ±j2. (Key point: Auxiliary roots)

5. What is the verdict for the system?
   A. Marginally stable, with simple jω roots and no RHP roots
   B. Unstable, 2 RHP roots
   C. Absolutely stable
   D. Unstable, repeated jω roots
   Answer: A. No sign change and four simple jω roots mean sustained oscillation at 1.414 and 2 rad/s: marginally stable. (Key point: Conclusion)

## Close variants set in other exams

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

## Related in Control Systems

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