---
title: "Disturbance sensitivity and effect of gain"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:ol-cl-disturbance-gain"
source: https://www.bctengineer.com/q/semester_3/control_systems/ol-cl-disturbance-gain
site: BCT Engineering
---

# Disturbance sensitivity and effect of gain

Control Systems · Chapter 1: Control System Background

- Asked in: 2075 Chaitra, 4+4 marks
- Video: [#181 Effect of Feedback on Gain, Sensitivity, Stability, Time constant and disturbance || EC Academy](https://www.youtube.com/watch?v=_-ree7zubLA) — 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/ol-cl-disturbance-gain
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

Which of the system is more sensitive to the disturbance and prove it how? Also discuss effect of gain on response of the system.

## Short answer

Open-loop is more sensitive: disturbance output is G2D, while closed-loop output is G2D/(1+G1G2H), reduced by loop gain. Increasing gain K reduces steady-state error and speeds up response but decreases damping, potentially causing instability if too high.

## Answer

#### Which system is more sensitive to disturbance
The **open-loop system** is more sensitive to disturbance.

Let the disturbance $D(s)$ enter between two forward blocks $G_1$ (controller) and $G_2$ (plant).

- Open loop: $C_D(s) = G_2(s)D(s)$.
- Closed loop (feedback $H$), with $R = 0$ (superposition):

$$
C_D(s) = \frac{G_2(s)}{1 + G_1(s)G_2(s)H(s)}\,D(s)
$$

The disturbance output is divided by $(1+G_1G_2H)$. If $|G_1G_2H| \gg 1$, $C_D \approx \dfrac{D}{G_1H}$, which is very small for large $G_1$.

Ratio of disturbance outputs:

$$
\frac{C_{D,\,cl}}{C_{D,\,ol}} = \frac{1}{1+G_1G_2H}
$$

Since $|1+G_1G_2H| > 1$ (usually $\gg 1$), the closed-loop output is much less affected. Example: $G_1G_2H = 49$ means the disturbance effect is only 2% of that in open loop. The open loop has no way to sense the change in output, so the disturbance passes fully to the output.

#### Effect of gain on the response
Take unity feedback with $G(s) = \dfrac{K}{s(s+2)}$:

$$
\frac{C(s)}{R(s)} = \frac{K}{s^2+2s+K}, \quad \omega_n = \sqrt{K}, \quad \zeta = \frac{1}{\sqrt{K}}
$$

| $K$ | $\omega_n$ (rad/s) | $\zeta$ | $M_p$ | Ramp $e_{ss} = 2/K$ |
|---|---|---|---|---|
| 1 | 1 | 1.0 | 0% | 2.0 |
| 4 | 2 | 0.5 | 16.3% | 0.5 |
| 16 | 4 | 0.25 | 44.4% | 0.125 |

As the gain $K$ increases:
1. **Speed increases:** $\omega_n$ rises, and rise time and peak time fall.
2. **Steady-state error falls:** $e_{ss}$ for a step ($1/(1+K_p)$) or a ramp ($1/K_v$) decreases.
3. **Damping falls:** $\zeta$ decreases, so overshoot and oscillation increase.
4. **Disturbance and sensitivity effects fall** as $1/(1+GH)$.
5. **Stability may be lost:** for third- and higher-order systems, e.g. $\dfrac{K}{s(s+1)(s+2)}$, Routh's test shows instability for $K > 6$.

So the gain is chosen as a compromise between accuracy/speed and relative stability.

## Check yourself

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

1. Which system is more sensitive to disturbance?
   A. Closed loop
   B. Open loop
   C. Both equally
   D. Neither
   Answer: B. Open loop passes the whole disturbance G2 D; closed loop divides it by (1+G1G2H). (Key point: Which is more sensitive)

2. If G1G2H = 49, closed-loop disturbance effect relative to open loop is
   A. 49%
   B. 2%
   C. 49 times
   D. 50%
   Answer: B. The ratio is 1/(1+49) = 1/50 = 2%. (Key point: Ratio of disturbance outputs)

3. For K/[s(s+2)] with unity feedback, as K increases, zeta
   A. Increases
   B. Decreases (more overshoot)
   C. Stays at 1
   D. Becomes negative
   Answer: B. zeta = 1/sqrt(K) falls as K rises. (Key point: Gain effect on damping)

4. For K/[s(s+1)(s+2)] with unity feedback, the system becomes unstable for
   A. K > 2
   B. K > 3
   C. K > 6
   D. K > 12
   Answer: C. Routh's test gives s^3+3s^2+2s+K, stable for K < 6. (Key point: Stability limit)

## Related in Control Systems

- [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
- [Tracking, disturbance rejection and pole sensitivity](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atrack-reject-sensitivity-160) — Control Systems, chapter 3
- [True/false: higher forward gain reduces disturbance](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-disturbance-forward-gain) — Control Systems, chapter 1
- [K limiting disturbance effect to 2%](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adisturbance-5k-s10-6-s2) — Control Systems, chapter 3
- [Speed of response increases with gain](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aspeed-increases-with-gain) — Control Systems, chapter 1
- [Suppressing disturbance by increasing G(s) or H(s)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adisturbance-suppress-gain) — Control Systems, chapter 1

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