---
title: "R-H range of K for s⁴+2s³+(4+K)s²+9s+25"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-s4-2-4k-9-25"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-s4-2-4k-9-25
site: BCT Engineering
---

# R-H range of K for s⁴+2s³+(4+K)s²+9s+25

Control Systems · Chapter 4: Stability

- Asked in: 2075 Chaitra, 6 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-s4-2-4k-9-25
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

Using R-H criteria find the range of K for system having characteristic equation shown below, to be stable. s⁴ + 2s³ + (4+K)s² + 9s + 25 = 0

## Short answer

Form the Routh array and enforce all first-column entries positive. The s² row gives K > 0.5, and the s¹ row requires 18K > 109, so K > 109/18 ≈ 6.056. Thus, the system is stable for K > 6.056 (no upper limit).

## Answer

Form the Routh array with $K$ and make every first-column element positive.

$$
s^4 + 2s^3 + (4+K)s^2 + 9s + 25 = 0
$$

#### Routh array

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^4$ | 1 | $4+K$ | 25 |
| $s^3$ | 2 | 9 | 0 |
| $s^2$ | $\frac{2(4+K)-9}{2}=\frac{2K-1}{2}$ | 25 | |
| $s^1$ | $\frac{\frac{2K-1}{2}(9)-2(25)}{\frac{2K-1}{2}}=\frac{18K-109}{2K-1}$ | 0 | |
| $s^0$ | 25 | | |

#### Conditions

1. $s^2$ row: $\frac{2K-1}{2}>0 \Rightarrow K>0.5$
2. $s^1$ row: with $2K-1>0$, we need $18K-109>0 \Rightarrow K > \frac{109}{18} = 6.056$
3. $s^0$ row: $25>0$ always true.

Condition 2 is stricter, so:

$$
K > 6.056
$$

#### Marginal value

At $K = 109/18 = 6.056$, the $s^1$ row is zero. The auxiliary equation $\frac{50}{9}s^2 + 25 = 0$ gives $s = \pm j2.12$, so the system would oscillate at 2.12 rad/s.

**Answer:** the system is stable for $K > 6.056$ (i.e. $K>109/18$). There is no upper limit.

## Check yourself

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

1. The s^2 row first element is
   A. (2K+1)/2
   B. K/2
   C. (4+K)/2
   D. (2K-1)/2
   Answer: D. [2(4+K)-9]/2 = (2K-1)/2. (Key point: s^2 row)

2. This element requires
   A. K > 0.5
   B. K < 0.5
   C. K > 6.056
   D. K > 0
   Answer: A. (2K-1)/2 > 0 gives K > 0.5. (Key point: Condition from s^2)

3. The s^1 element (18K-109)/(2K-1) positive gives
   A. K < 6.056
   B. K > 109/18 = 6.056
   C. K > 0.5 only
   D. K > 25
   Answer: B. With 2K-1>0 we need 18K-109>0. (Key point: Stricter condition)

4. The final stability range is
   A. 0.5 < K < 6.056
   B. 0 < K < 6.056
   C. K > 6.056 with no upper limit
   D. K < 0.5
   Answer: C. The stricter condition K>6.056 is the answer; there is no upper bound. (Key point: Final range)

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