---
title: "Relative and absolute stability; R-H of s⁵+s⁴+2s³+2s²+3s+5"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-relative-absolute-s5"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-relative-absolute-s5
site: BCT Engineering
---

# Relative and absolute stability; R-H of s⁵+s⁴+2s³+2s²+3s+5

Control Systems · Chapter 4: Stability

- Asked in: 2081 Bhadra, 3+5 marks
- Video: [Example on Routh Array Stable System](https://www.youtube.com/watch?v=BApzhX2WBe0) — TutorialsPoint
- 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-s5
- 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: s⁵ + s⁴ + 2s³ + 2s² + 3s + 5 = 0 using R-H criteria.

## Short answer

Absolute stability checks if all poles are in the left half-plane; relative stability measures how far they are from instability. The polynomial has two sign changes, so two roots are in the right half-plane and the system is unstable.

## 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 $s^5 + s^4 + 2s^3 + 2s^2 + 3s + 5 = 0$

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^5$ | 1 | 2 | 3 |
| $s^4$ | 1 | 2 | 5 |
| $s^3$ | $\frac{1(2)-1(2)}{1}=0 \to \varepsilon$ | $\frac{1(3)-1(5)}{1}=-2$ | |
| $s^2$ | $\frac{2\varepsilon+2}{\varepsilon}$ | 5 | |
| $s^1$ | $-2-\frac{5\varepsilon^2}{2\varepsilon+2}$ | 0 | |
| $s^0$ | 5 | | |

Only the first element of the $s^3$ row is zero (the row is not all zero), so it is replaced by a small positive number $\varepsilon$. Let $\varepsilon \to 0^+$:

| Row | First-column sign |
|---|---|
| $s^5$ | $+1$ |
| $s^4$ | $+1$ |
| $s^3$ | $+\varepsilon$ |
| $s^2$ | $\frac{2}{\varepsilon} \to +\infty$ |
| $s^1$ | $\to -2$ |
| $s^0$ | $+5$ |

There are **2 sign changes** ($+\infty \to -2$ and $-2 \to +5$).

**Answer:** two roots lie in the right half s-plane, so the system is **unstable**. The other three roots are in the LHP. The actual roots are $\approx 0.72\pm j1.17$, $-0.60\pm j1.34$ and $-1.24$.

## Check yourself

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

1. What does absolute stability mean?
   A. Whether the system is stable or not, i.e. all closed-loop poles are in the left half plane
   B. How fast the response settles
   C. The gain margin of a stable system
   D. The damping ratio only
   Answer: A. 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 is a yes/no test using Routh
   B. It measures how far the poles are from the jω axis, e.g. through GM, PM or damping ratio
   C. It counts RHP roots only
   D. It applies only to unstable systems
   Answer: B. 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. In the s^3 row the first element is 0 but the row is not all zero. What is done?
   A. Stop and declare the system marginally stable
   B. Use the auxiliary equation
   C. Multiply the polynomial by s
   D. Replace the zero by a small positive epsilon and continue
   Answer: D. A lone zero in the first column is replaced by a small epsilon, and the array is completed with epsilon tending to 0+. (Key point: Zero in first column)

4. The first column signs are +, +, +eps, +2/eps, -2, +5. What is the result?
   A. Stable, no sign changes
   B. 1 sign change, 1 RHP root
   C. 2 roots on the jω axis
   D. 2 sign changes, so 2 RHP roots and unstable
   Answer: D. Two sign changes (large positive to -2, and -2 to +5) mean 2 RHP roots; the system is unstable. (Key point: Conclusion)

## Close variants set in other exams

- [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×

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