---
title: "Effect of pole location on stability"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:pole-location-stability"
source: https://www.bctengineer.com/q/semester_3/control_systems/pole-location-stability
site: BCT Engineering
---

# Effect of pole location on stability

Control Systems · Chapter 4: Stability

- Asked in: 2076 Chaitra, 2 marks
- Video: [Intro to Control - 7.2 Poles and Stability](https://www.youtube.com/watch?v=flLDEJYrICk) — Engineering with Prof. Kim
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/pole-location-stability
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

How does the location of poles affect the stability in control system?

## Short answer

Poles in the left half-plane (σ<0) produce stable decay; simple poles on the jω axis give sustained oscillation (marginal stability); repeated jω axis poles or right half-plane poles (σ>0) cause instability. Poles farther left decay faster; dominant poles near the jω axis govern transient response.

## Answer

The stability of a system depends only on where the poles of its closed-loop transfer function (the roots of the characteristic equation) lie in the s-plane. Each pole $s = \sigma \pm j\omega$ adds a term $e^{\sigma t}\sin(\omega t)$ to the response.

| Pole location | Response term | Stability |
|---|---|---|
| Left half plane ($\sigma<0$) | Decays to zero | Stable |
| Simple poles on $j\omega$ axis | Constant or steady oscillation | Marginally stable |
| Repeated poles on $j\omega$ axis | $t\sin\omega t$, grows | Unstable |
| Right half plane ($\sigma>0$) | Grows exponentially | Unstable |

- The further left the poles are, the faster the response decays. This means greater relative stability.
- Poles close to the $j\omega$ axis (dominant poles) decay slowly and decide the transient response.
- Imaginary part $\omega$ sets the frequency of oscillation; real poles give no oscillation.

## Check yourself

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

1. Poles in the left half of the s-plane give
   A. Decaying response and stable system
   B. Growing response
   C. Constant oscillation
   D. No response
   Answer: A. A term e^(sigma t) with sigma<0 decays to zero. (Key point: LHP poles)

2. Repeated poles on the jw axis make the system
   A. Marginally stable
   B. Unstable, with t sin(wt) terms
   C. Stable
   D. Overdamped
   Answer: B. Repeated imaginary poles produce terms growing like t sin(wt). (Key point: Repeated jw poles)

3. Moving poles further left
   A. Makes it oscillate more
   B. Makes it unstable
   C. Makes the response decay faster
   D. Has no effect
   Answer: C. The further left, the faster the decay, so greater relative stability. (Key point: Relative stability)

## Related in Control Systems

- [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
- [Effect of adding poles and zeros](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aeffect-add-pole-zero) — Control Systems, chapter 3, asked 2×
- [Effect of feedback on stability](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-stability-example) — Control Systems, chapter 1
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- [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

## Related in other subjects

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- [System function from all-pole lattice K's](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aall-pole-lattice-system-function) — Digital Signal Analysis and Processing, chapter 4