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

# Effect of feedback on dynamic response

Control Systems · Chapter 1: Control System Background

- Asked in: 2071 Shrawan, 6 marks
- Video: [Effects of Feedback](https://www.youtube.com/watch?v=BJFyyqpdNg4) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/feedback-dynamic-response
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

Discuss how the dynamic responses of control system are affected by feedback.

## Answer

Feedback changes the **dynamic (transient) response** of a system: it changes the time constant, bandwidth, damping and stability.

#### 1. Time constant and speed
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):**

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

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

Feedback reduces the time constant by $(1+K)$, so the response is faster and the bandwidth increases from $1/\tau$ to $(1+K)/\tau$.

#### 2. Damping and overshoot (second order)
With $G(s) = \dfrac{K}{s(\tau s+1)}$ and unity feedback:

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

Raising the feedback loop gain $K$ increases $\omega_n$ (faster rise) but reduces $\zeta$, so overshoot $M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}$ increases. Adding **rate (tachometer) feedback** $K_ts$ changes damping to

$$
\zeta' = \frac{1 + KK_t}{2\sqrt{K\tau}}
$$

which reduces overshoot without changing $\omega_n$.

#### 3. Stability
Feedback moves the poles. It can stabilize an unstable plant (e.g. $1/(s-1)$ with gain $K > 1$ gives the stable pole $s = 1 - K$) or destabilize a stable one when gain is too high (e.g. $K/[s(s+1)(s+2)]$ is unstable for $K > 6$).

#### 4. Reduced effect of disturbances and parameter changes on the transient
Both are reduced by $(1+GH)$, so the transient shape stays nearly the same even when plant parameters drift.

## Check yourself

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

1. A plant K/(tau s + 1) is given unity feedback. The new time constant is
   A. tau/(1+K)
   B. tau(1+K)
   C. tau/K
   D. tau
   Answer: A. The closed loop is [K/(1+K)] / [(tau/(1+K)) s + 1], so the time constant falls by (1+K). (Key point: Time constant with feedback)

2. For K/[s(tau s+1)] with unity feedback, increasing K causes
   A. higher damping ratio and less overshoot
   B. higher natural frequency but lower damping ratio, so more overshoot
   C. lower natural frequency and slower rise
   D. no change in damping because K only scales the output
   Answer: B. wn = sqrt(K/tau) rises but zeta = 1/(2 sqrt(K tau)) falls, so overshoot increases. (Key point: Gain vs damping)

3. What does adding rate (tachometer) feedback Kt s do?
   A. Increases wn and overshoot together
   B. Removes the integrator and makes the system type 0
   C. Increases damping to (1+K Kt)/(2 sqrt(K tau)) and reduces overshoot without changing wn
   D. Reduces damping but speeds the response
   Answer: C. Rate feedback adds to the s-term of the characteristic equation, raising zeta while wn stays sqrt(K/tau). (Key point: Rate feedback)

4. Feedback on the plant 1/(s-1) with gain K gives the pole s = 1 - K. For which K is it stable?
   A. K < 1
   B. K > 6
   C. Any K, since the pole never moves
   D. K > 1
   Answer: D. The pole 1 - K lies in the left half-plane only when K > 1. (Key point: Stability and feedback)

## Related in Control Systems

- [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
- [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
- [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
- [True/false: feedback makes response faster](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-feedback-faster) — 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
- [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×

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