---
title: "Closed loop disturbance rejection and command tracking"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:cl-disturbance-tracking"
source: https://www.bctengineer.com/q/semester_3/control_systems/cl-disturbance-tracking
site: BCT Engineering
---

# Closed loop disturbance rejection and command tracking

Control Systems · Chapter 1: Control System Background

- Asked in: 2072 Chaitra, 4 marks
- Video: [Part 5 of 5 : Effect of Feedback on  Disturbance/Noise of Control System](https://www.youtube.com/watch?v=sN7kpigwxS8) — 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/cl-disturbance-tracking
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

Discuss how a closed loop system has better disturbance rejection and command input tracking capabilities in comparison to an open loop system.

## Short answer

In closed loop, disturbance effect is divided by (1+G1G2H), heavily suppressed with high loop gain, while open-loop passes it fully. Tracking error is 1/(1+K) for step input, so output follows reference more accurately and faster than in an uncalibrated open-loop system.

## Answer

A closed-loop system measures the output and corrects any error, so it both **rejects disturbances** and **tracks the command input** better than an open-loop system.

#### Disturbance rejection
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 $G_1$.

In open loop the whole disturbance $G_2D$ reaches the output, because nothing senses it.

#### Command input tracking
Tracking error for unity feedback: $E(s) = \dfrac{R(s)}{1+G(s)}$, and $e_{ss} = \lim_{s\to0}sE(s)$.

- Step input: $e_{ss} = \dfrac{1}{1+K_p}$; with an integrator in $G$ (type 1), $e_{ss} = 0$.
- In open loop, the output equals the reference only if the plant gain is exactly calibrated; any change in the plant gives an error, since the sensitivity is 1. In closed loop the sensitivity is $1/(1+GH)$.

The closed-loop time constant $\tau/(1+K)$ is also smaller, so the output follows a changing reference more quickly.

**Example:** in a motor speed system with loop gain 99, a 10% change in motor gain changes the speed by only about 0.1% in closed loop, compared with 10% in open loop, and a load disturbance has only 1% of its open-loop effect.

## Check yourself

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

1. In open loop, how much of the disturbance reaches the output?
   A. None
   B. All of G2 D
   C. G2 D/(1+GH)
   D. Half
   Answer: B. Nothing senses the disturbance, so G2 D passes fully. (Key point: Open-loop disturbance)

2. For a type-1 unity-feedback system and a step input, e_ss is
   A. 1
   B. 1/(1+Kp)=0
   C. Infinity
   D. Kp
   Answer: B. An integrator makes Kp infinite, so e_ss = 0. (Key point: Step tracking error)

3. For unity feedback, the error is E(s) =
   A. R/(1+G)
   B. R(1+G)
   C. G R
   D. R/G
   Answer: A. E = R/(1+G), with e_ss = lim s E(s). (Key point: Tracking error formula)

4. With loop gain 99, a 10% change in motor gain changes closed-loop speed by about
   A. 10%
   B. 1%
   C. 0.1%
   D. 99%
   Answer: C. 10%/(1+99) = 0.1%. (Key point: Motor example)

## Related in Control Systems

- [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
- [Why closed loop rejects disturbances better](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adefine-cs-disturbance-rejection) — Control Systems, chapter 1
- [Traffic light system of Kathmandu: open or closed loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atraffic-light-justify) — Control Systems, chapter 1
- [Why closed loop response is faster than open loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Acl-faster-than-ol) — Control Systems, chapter 1, asked 4×
- [Closed vs open loop step response analytically](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Acl-vs-ol-step-response) — Control Systems, chapter 1, asked 2×
- [Open and closed loop systems with block diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aol-cl-bd-role) — Control Systems, chapter 1, asked 2×

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