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

# R-H stability of 2/(2s⁵+3s⁴+2s³+s²+2s)

Control Systems · Chapter 4: Stability

- Asked in: 2076 Asoj, 4 marks
- Video: [Special Case of Routh Array II : Numerical 2](https://www.youtube.com/watch?v=3E8MevPWNRA) — Learning Electronics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rh-2-2s5-3s4
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

Check stability for the system with open loop transfer function G(s)H(s) = 2/(2s⁵ + 3s⁴ + 2s³ + s² + 2s) using R-H criterion.

## Short answer

Characteristic equation 2s^5+3s^4+2s^3+s^2+2s+2=0. Routh array first column: 2,3,1.33,−0.5,6,2. Two sign changes indicate 2 poles in RHP; the closed-loop system is unstable.

## Answer

**Result:** 2 sign changes, so the closed-loop system has 2 roots in the RHP and is **unstable**.

The closed-loop stability is decided by the characteristic equation $1+G(s)H(s)=0$.

$$
\begin{aligned}
2s^5+3s^4+2s^3+s^2+2s + 2 &= 0
\end{aligned}
$$

#### Routh array

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^5$ | 2 | 2 | 2 |
| $s^4$ | 3 | 1 | 2 |
| $s^3$ | $\frac{3(2)-2(1)}{3}=\frac{4}{3}$ | $\frac{3(2)-2(2)}{3}=\frac{2}{3}$ | 0 |
| $s^2$ | $\frac{\frac43(1)-3\cdot\frac23}{\frac43}=-0.5$ | $\frac{\frac43(2)-0}{\frac43}=2$ | |
| $s^1$ | $\frac{-0.5\cdot\frac23-\frac43(2)}{-0.5}=6$ | 0 | |
| $s^0$ | 2 | | |

First column: $2,\ 3,\ 1.33,\ -0.5,\ 6,\ 2$.

There are **2 sign changes** ($1.33 \to -0.5$ and $-0.5 \to 6$), so 2 closed-loop poles lie in the right half s-plane.

**Answer:** the closed-loop system is **unstable**, with 2 poles in the RHP and 3 in the LHP. The roots are approximately $0.51\pm j0.75$, $-0.76\pm j0.80$ and $-1$.

## Check yourself

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

1. For G(s)H(s)=2/(2s^5+3s^4+2s^3+s^2+2s), the characteristic equation is
   A. 2s^5+3s^4+2s^3+s^2+2s=0
   B. 2s^5+3s^4+2s^3+s^2+2s-2=0
   C. s^5+3s^4+2s^3+s^2+2s+2=0
   D. 2s^5+3s^4+2s^3+s^2+2s+2=0
   Answer: D. 1+GH=0, adding the numerator 2 to the denominator. (Key point: Characteristic equation)

2. The s^3 row first element is
   A. 4/3
   B. -0.5
   C. 3
   D. 2/3
   Answer: A. (3*2 - 2*1)/3 = 4/3. (Key point: Array element)

3. The first column 2, 3, 1.33, -0.5, 6, 2 has
   A. 0 sign changes
   B. 2 sign changes
   C. 1 sign change
   D. 3 sign changes
   Answer: B. 1.33 to -0.5 and -0.5 to 6. (Key point: Sign changes)

4. The system is
   A. Stable
   B. Marginally stable
   C. Unstable with 2 poles in the RHP
   D. Unstable with 3 poles in the RHP
   Answer: C. Two sign changes mean two RHP poles. (Key point: Conclusion)

## Related in Control Systems

- [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×
- [Nyquist stability of 1/[s(1+2s)(1+s)] and gain margin](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-1-s-1-2s-1) — Control Systems, chapter 6, asked 3×
- [Nyquist stability of (4s+1)/[s²(s+1)(2s+1)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-4s1-s2-s1-2s1) — Control Systems, chapter 6, asked 3×
- [Nyquist stability of 2/[s(s+1)(2s+1)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-2-s-s1-2s1) — Control Systems, chapter 6, asked 3×
- [Bode plot of 48(1+s)/[s²(1+3s)(1+0.5s)(2+0.2s)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-48-1s-s2-1-3s) — Control Systems, chapter 6
- [State space advantages; state model of (2s²+3s+1)/(s³+5s²+6s+7)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-adv-tf-2s2-3s-1) — Control Systems, chapter 8

## Related in other subjects

- [Excitation system with stabilizing transformer: transfer function](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-stabilizing-transformer-model) — Power Plant Equipment, chapter 3, asked 13×
- [Function of excitation system and its transfer function](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-function-tf) — Power Plant Equipment, chapter 3, asked 2×
- [Equal area criterion for transient stability](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aequal-area-criterion-concept) — Power System Analysis II, chapter 6, asked 4×
- [T/F: stability criterion implies thermal criterion](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Atf-stability-implies-thermal) — Transmission and Distribution System Design, chapter 4, asked 2×