---
title: "Effect of feedback on stability"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:feedback-stability-example"
source: https://www.bctengineer.com/q/semester_3/control_systems/feedback-stability-example
site: BCT Engineering
---

# Effect of feedback on stability

Control Systems · Chapter 1: Control System Background

- Asked in: 2080 Baisakh, 3 marks
- Video: [Part 3 of 5 : Effect of Feedback on Stability of Control System](https://www.youtube.com/watch?v=0V3usJJueAg) — 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/feedback-stability-example
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

How does feedback affect the stability of a control system? Explain with suitable example.

## Short answer

Feedback can stabilize or destabilize a system. For example, an unstable plant 1/(s-1) becomes stable with gain K>1, while a third-order system K/[s(s+1)(s+2)] becomes unstable for K>6.

## Answer

Feedback changes the location of the closed-loop poles (roots of $1 + G(s)H(s) = 0$), so it can make a system **more stable or less stable**.

- **Stabilizing effect:** an unstable plant $G(s) = \dfrac{1}{s-1}$ (pole at $+1$) with proportional feedback gain $K$ gives $\dfrac{K}{s-1+K}$; the pole moves to $s = 1-K$, which is stable for $K > 1$.
- **Destabilizing effect:** take $G(s) = \dfrac{K}{s(s+1)(s+2)}$ with unity feedback. The open-loop poles $0, -1, -2$ do not move with $K$, but the closed-loop characteristic equation is

$$
s^3 + 3s^2 + 2s + K = 0
$$

Routh array: row $s^1$ term $= \dfrac{3\times2 - K}{3}$. The system is stable only for $0 < K < 6$; at $K = 6$ it oscillates at $\omega = \sqrt{2}$ rad/s and for $K > 6$ it is unstable. So too much feedback gain can destabilize a system.

## Check yourself

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

1. Feedback affects stability because it changes
   A. only the zeros of the plant
   B. the closed-loop poles, the roots of 1 + G(s)H(s) = 0
   C. the open-loop poles
   D. only the steady-state value
   Answer: B. Closed-loop poles are roots of the characteristic equation 1+GH=0. (Key point: Pole location)

2. Plant 1/(s-1) with proportional feedback K has closed-loop pole at
   A. s = 1 + K, stable for K > 1
   B. s = K - 1, stable for K > 1
   C. s = 1 - K, stable for K > 1
   D. s = 1 - K, stable for K < 1
   Answer: C. Closed loop is K/(s-1+K), pole at 1-K. (Key point: Stabilizing example)

3. K/[s(s+1)(s+2)] with unity feedback is stable for
   A. all K > 0
   B. K > 6 only
   C. 0 < K < 2
   D. 0 < K < 6, oscillating at sqrt(2) rad/s when K = 6
   Answer: D. Routh: s^3+3s^2+2s+K; the s^1 term (6-K)/3 must be positive. (Key point: Destabilizing example)

## Related in Control Systems

- [Stability of 56/[s(s⁴+7s³+6s²+42s+8)] feedback system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Astability-56-s5) — Control Systems, chapter 4
- [Feedback effect in DC motor speed control](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-dc-motor-speed) — Control Systems, chapter 1
- [Control system and why feedback makes response faster](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-feedback-faster) — Control Systems, chapter 1
- [Characterizing control system by speed, accuracy, stability](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aspeed-accuracy-stability) — Control Systems, chapter 1
- [Derivative feedback controller: block diagram and effect](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aderivative-feedback-bd-effect) — Control Systems, chapter 7, asked 2×
- [Effect of feedback on gain and speed of response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-gain-speed) — Control Systems, chapter 1

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