---
title: "R-H range of K for K/[(s+2)(s+4)(s²+6s+25)]"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-k-s2-s4-s2-6s"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-k-s2-s4-s2-6s
site: BCT Engineering
---

# R-H range of K for K/[(s+2)(s+4)(s²+6s+25)]

Control Systems · Chapter 4: Stability

- Asked in: 2068 Baisakh (old course), 8 marks
- Video: [#188 Problem on routh criterion to find K for stability of system || EC Academy](https://www.youtube.com/watch?v=xcssuw3YTZY) — EC Academy
- 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-s2-s4-s2-6s
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

The open loop TF of a unity feedback control system is given as G(s) = K/[(s+2)(s+4)(s²+6s+25)]. Determine the range of gain K for the system to be stable. Also determine the value of K which will cause the sustained oscillation and corresponding oscillation frequency.

## Short answer

The characteristic equation is s⁴+12s³+69s²+198s+(200+K)=0. Routh array gives stability for 0 < K < 666.25. Sustained oscillation occurs at K = 666.25 with frequency ω = 4.06 rad/s.

## Answer

Write the closed-loop characteristic equation $1+G(s)=0$ and apply the Routh criterion. Sustained oscillation occurs when the $s^1$ row becomes zero, and the auxiliary equation from the $s^2$ row gives the frequency.

#### Characteristic equation

$$
\begin{aligned}
(s+2)(s+4)(s^2+6s+25) + K &= 0 \\
(s^2+6s+8)(s^2+6s+25) + K &= 0 \\
s^4 + 12s^3 + 69s^2 + 198s + (200+K) &= 0
\end{aligned}
$$

#### Routh array

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^4$ | 1 | 69 | $200+K$ |
| $s^3$ | 12 | 198 | 0 |
| $s^2$ | $\frac{12(69)-198}{12}=52.5$ | $200+K$ | |
| $s^1$ | $\frac{52.5(198)-12(200+K)}{52.5}=\frac{7995-12K}{52.5}$ | 0 | |
| $s^0$ | $200+K$ | | |

#### Range of K

- $s^1$ row: $7995 - 12K > 0 \Rightarrow K < 666.25$
- $s^0$ row: $200 + K > 0 \Rightarrow K > -200$

$$
-200 < K < 666.25 \qquad (\text{for positive gain: } 0 < K < 666.25)
$$

#### Value of K for sustained oscillation

$$
K_{mar} = \frac{7995}{12} = 666.25
$$

#### Frequency of oscillation

Auxiliary equation from the $s^2$ row:

$$
\begin{aligned}
52.5s^2 + (200 + 666.25) &= 0 \\
s^2 &= -\frac{866.25}{52.5} = -16.5 \\
\omega &= \sqrt{16.5} = 4.062\ \text{rad/s}
\end{aligned}
$$

**Check:** with $K = 666.25$ the roots are $\pm j4.062$ and $-6 \pm j4.062$.

**Answer:** stable for $0 < K < 666.25$; sustained oscillation at $K = 666.25$ with $\omega = 4.06$ rad/s.

## Check yourself

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

1. What is the characteristic polynomial of K/[(s+2)(s+4)(s^2+6s+25)] with unity feedback?
   A. s^4+12s^3+69s^2+200s+K
   B. s^4+6s^3+69s^2+198s+(200+K)
   C. s^4+12s^3+25s^2+198s+K
   D. s^4+12s^3+69s^2+198s+(200+K)
   Answer: D. (s^2+6s+8)(s^2+6s+25) expands to s^4+12s^3+69s^2+198s+200; add K. (Key point: Characteristic equation)

2. What gives the upper limit K < 666.25?
   A. s^2 row first element positive
   B. s^0 row 200+K > 0
   C. s^1 row first element (7995-12K)/52.5 > 0
   D. s^3 row
   Answer: C. 7995-12K > 0 gives K < 666.25. (Key point: Upper limit)

3. For which K does the system oscillate continuously?
   A. K = 200
   B. K = 666.25, where the s^1 row is zero
   C. K = 7995
   D. K = 52.5
   Answer: B. The s^1 row first element vanishes at K=7995/12=666.25, giving marginal stability. (Key point: Sustained oscillation)

4. How is the oscillation frequency found and what is it?
   A. From the s^3 row; 12 rad/s
   B. From the s^1 row; 666 rad/s
   C. From s^0 row; 200 rad/s
   D. From the auxiliary equation of the s^2 row: 52.5s^2+866.25=0, so ω = 4.06 rad/s
   Answer: D. s^2 = -866.25/52.5 = -16.5, so ω = sqrt(16.5) = 4.06 rad/s. (Key point: Frequency)

## Close variants set in other exams

- [R-H range of k and oscillation frequency](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-k-s-s2-s1-2) — Control Systems, chapter 4, asked 2×

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