---
title: "R-H range of K for K/[s(s²+s+1)(s+2)]"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-k-s-s2-s1-s2"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s-s2-s1-s2
site: BCT Engineering
---

# R-H range of K for K/[s(s²+s+1)(s+2)]

Control Systems · Chapter 4: Stability

- Asked in: 2072 Chaitra, 4 marks
- Video: [Numerical 2 : Marginal value of K and Frequency of sustained oscillations](https://www.youtube.com/watch?v=OyDSqsS0eVc) — 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-k-s-s2-s1-s2
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

Find the range of 'K' for stable operation using R-H criteria. [Figure: R(s) → summing point (+, −) → K → 1/[s(s²+s+1)(s+2)] → C(s); unity negative feedback]

## Short answer

Characteristic equation: s⁴ + 3s³ + 3s² + 2s + K =0. Routh array requires K>0 and 2−9K/7>0, so K<14/9≈1.556. The stable range is 0 < K < 14/9.

## Answer

For unity feedback, the characteristic equation is $1+G(s)=0$, with $G(s)=\frac{K}{s(s^2+s+1)(s+2)}$.

#### Characteristic equation

$$
\begin{aligned}
s(s^2+s+1)(s+2) + K &= 0 \\
s^4 + 3s^3 + 3s^2 + 2s + K &= 0
\end{aligned}
$$

#### Routh array

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^4$ | 1 | 3 | $K$ |
| $s^3$ | 3 | 2 | 0 |
| $s^2$ | $\frac{3(3)-1(2)}{3}=\frac{7}{3}$ | $K$ | |
| $s^1$ | $\frac{\frac73(2)-3K}{\frac73}=2-\frac{9K}{7}$ | 0 | |
| $s^0$ | $K$ | | |

#### Conditions

- $s^1$ row: $2 - \frac{9K}{7} > 0 \Rightarrow K < \frac{14}{9} = 1.556$
- $s^0$ row: $K > 0$

$$
0 < K < 1.556
$$

At $K = 14/9$ the system is marginally stable. The auxiliary equation $\frac73 s^2 + \frac{14}{9} = 0$ gives $\omega = \sqrt{2/3} = 0.816$ rad/s.

**Answer:** stable for $0 < K < \frac{14}{9}$ (= 1.556).

## Check yourself

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

1. The characteristic equation is
   A. s^4+3s^3+3s^2+K s+2=0
   B. s^3+3s^2+3s+2+K=0
   C. s^4+2s^3+3s^2+3s+K=0
   D. s^4+3s^3+3s^2+2s+K=0
   Answer: D. s(s^2+s+1)(s+2) expands to s^4+3s^3+3s^2+2s. (Key point: Characteristic equation)

2. The s^2 row first element is
   A. 7/3
   B. 3
   C. 2
   D. 9/7
   Answer: A. (3*3 - 1*2)/3 = 7/3. (Key point: s^2 element)

3. The s^1 element is
   A. 2 + 9K/7
   B. 2 - 9K/7
   C. K - 2
   D. 9K/7
   Answer: B. [(7/3)(2) - 3K]/(7/3) = 2 - 9K/7. (Key point: s^1 element)

4. The stable range of K is
   A. K > 14/9
   B. 0 < K < 9/14
   C. 0 < K < 14/9
   D. K < 0
   Answer: C. 2-9K/7>0 gives K<14/9; K>0 from s^0. (Key point: Range)

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