---
title: "Overshoot and peak time for 5/[s(s+4)]"
subject: "Control Systems"
chapter: 3
key: "semester_3/control_systems:mp-tp-5-s-s4"
source: https://www.bctengineer.com/q/semester_3/control_systems/mp-tp-5-s-s4
site: BCT Engineering
---

# Overshoot and peak time for 5/[s(s+4)]

Control Systems · Chapter 3: System Transfer Function and Responses

- Asked in: 2066 Bhadra (old course), 8 marks
- Video: [Problem to calculate rise time, peak time , peak overshoot , Max Peak Overshoot and settling time |](https://www.youtube.com/watch?v=pKBcv8YPEa4) — Mathspedia
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/mp-tp-5-s-s4
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3

## Question

Evaluate the percentage overshoot and peak time for the unity feedback system with open loop transfer function G(s) = 5/[s(s+4)].

## Short answer

Closed-loop transfer function is 5/(s²+4s+5) with ωn=2.236 rad/s and ζ=0.894. Percentage overshoot = 0.19%, peak time = 3.14 s.

## Answer

Find the closed-loop transfer function, read off $\omega_n$ and $\zeta$, then use the standard second order formulas.

#### Closed-loop transfer function
$$
\frac{C(s)}{R(s)} = \frac{G}{1+G} = \frac{5}{s^2 + 4s + 5}
$$

#### Parameters
$$
\begin{aligned}
\omega_n^2 &= 5 \;\Rightarrow\; \omega_n = \sqrt{5} = 2.236\ \text{rad/s} \\
2\zeta\omega_n &= 4 \;\Rightarrow\; \zeta = \frac{4}{2(2.236)} = 0.894
\end{aligned}
$$
Since $\zeta < 1$, the response is underdamped.
$$
\omega_d = \omega_n\sqrt{1 - \zeta^2} = 2.236\sqrt{1 - 0.8} = 2.236(0.4472) = 1\ \text{rad/s}
$$
(The closed-loop poles are $s = -2 \pm j1$.)

#### Percentage overshoot
$$
\%M_p = 100\,e^{-\pi\zeta/\sqrt{1-\zeta^2}} = 100\,e^{-\pi(0.894)/0.447} = 100\,e^{-2\pi} = 0.187\%
$$

#### Peak time
$$
t_p = \frac{\pi}{\omega_d} = \frac{\pi}{1} = 3.14\ \text{s}
$$

**Answer:** $\%M_p \approx 0.19\%$ and $t_p = 3.14$ s. The system is heavily damped, so the overshoot is very small.

## Check yourself

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

1. For G=5/[s(s+4)] with unity feedback, C/R is
   A. 5/(s^2+4s)
   B. 5/(s^2+4s+5)
   C. 5/(s^2+5s+4)
   D. 1/(s^2+4s+5)
   Answer: B. G/(1+G) = 5/(s^2+4s+5). (Key point: Closed loop)

2. wn^2 = 5 and 2*zeta*wn = 4 give
   A. zeta = 0.447
   B. zeta = 0.894
   C. zeta = 1.79
   D. zeta = 2
   Answer: B. zeta = 4/(2*2.236) = 0.894, underdamped. (Key point: Zeta)

3. wd equals
   A. 2.236 rad/s
   B. 2 rad/s
   C. 1 rad/s
   D. 4 rad/s
   Answer: C. wd = wn*sqrt(1-zeta^2) = 2.236*0.447 = 1; the poles are -2 +/- j1. (Key point: Damped frequency)

4. Percentage overshoot and peak time are
   A. 18.7%, 3.14 s
   B. 0.19%, 3.14 s
   C. 0.19%, 1.57 s
   D. 1.9%, 3.14 s
   Answer: B. Mp = e^(-2*pi) = 0.19%, tp = pi/wd = 3.14 s. (Key point: Results)

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