---
title: "Control system and why feedback makes response faster"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:what-is-cs-feedback-faster"
source: https://www.bctengineer.com/q/semester_3/control_systems/what-is-cs-feedback-faster
site: BCT Engineering
---

# Control system and why feedback makes response faster

Control Systems · Chapter 1: Control System Background

- Asked in: 2076 Chaitra, 2+6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/what-is-cs-feedback-faster
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

What is control system? "When feedback is added on a control system, its response is faster." Illustrate this statement mathematically.

## Short answer

A control system regulates output to a desired value. Adding negative feedback reduces the time constant: for a first-order plant K/(τs+1), closed-loop time constant becomes τ/(1+K), so the response is (1+K) times faster, e.g., τ=1s, K=9 gives τ_cl=0.1s.

## Answer

#### Control system
A **control system** is an arrangement of components connected to command, direct or regulate a system so that the output reaches the desired value. When the output is measured and fed back to correct the input, it is a feedback (closed-loop) system.

#### Feedback makes the response faster
Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

**Open loop:** for a unit step input

$$
c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau
$$

**Closed loop (unity feedback):**

$$
\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{K}{\tau s + 1 + K} = \frac{\dfrac{K}{1+K}}{\dfrac{\tau}{1+K}s + 1}
$$

$$
c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}
$$

Comparing:

| Quantity | Open loop | Closed loop |
|---|---|---|
| Time constant | $\tau$ | $\tau/(1+K)$ |
| Settling time (2%) | $4\tau$ | $4\tau/(1+K)$ |
| Bandwidth | $1/\tau$ rad/s | $(1+K)/\tau$ rad/s |
| DC gain | $K$ | $K/(1+K)$ |

The time constant is reduced by $(1+K)$, so the output reaches its final value $(1+K)$ times sooner.

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.

```text
 c(t)
  |        .----------- closed loop (fast)
  |      /    ..------- open loop (slow)
  |     / .'
  |    /.'
  |   /'
  +-------------------------> t
```

(The curves are scaled to the same final value; in practice the reduced closed-loop gain is made up by a preamplifier.)

For a second-order system $K/[s(s+a)]$ with unity feedback, $\omega_n = \sqrt{K}$ also rises with gain, so rise time and peak time fall. Hence adding feedback makes the response faster.

## Check yourself

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

1. With unity feedback and G=K/(tau s+1), the step response is
   A. c(t) = K(1 - e^(-t/tau))
   B. c(t) = K/(1+K) (1 - e^(-t/tau))
   C. c(t) = K/(1+K) (1 - e^(-t(1+K)/tau))
   D. c(t) = (1+K)(1 - e^(-t/tau))
   Answer: C. The exponent uses the closed-loop time constant tau/(1+K). (Key point: Closed-loop step response)

2. 2% settling time changes from 4 tau to
   A. 4 tau(1+K)
   B. 4 tau K
   C. 4 tau
   D. 4 tau/(1+K)
   Answer: D. Settling time is 4 times the closed-loop time constant. (Key point: Settling time)

3. For K/[s(s+a)] with unity feedback, raising K raises
   A. wn = sqrt(K), so rise and peak times fall
   B. wn, so rise time increases
   C. only the steady-state error
   D. nothing
   Answer: A. Higher wn means a faster response. (Key point: Second order)

## Related in Control Systems

- [True/false: feedback makes response faster](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-feedback-faster) — 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×
- [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
- [Derivative feedback in second order time response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aderivative-feedback-second-order) — Control Systems, chapter 7

## Related in other subjects

- [Short note: Feedback control system](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Afeedback-control-note) — Project Engineering, chapter 4, asked 2×