---
title: "Second order model from step response (peak 1.25)"
subject: "Control Systems"
chapter: 3
key: "semester_3/control_systems:second-order-from-step-125"
source: https://www.bctengineer.com/q/semester_3/control_systems/second-order-from-step-125
site: BCT Engineering
---

# Second order model from step response (peak 1.25)

Control Systems · Chapter 3: System Transfer Function and Responses

- Asked in: 2070 Chaitra (old course), 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/second-order-from-step-125
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3

## Question

The unit step response of a linear control system is shown in figure below. Find the transfer function of a second order system to model the system. [Figure: unit step response c(t) with final value 1 and peak value 1.25 occurring at t = 0.01 sec]

## Short answer

25% overshoot gives ζ = −ln(0.25)/√(π²+ln²(0.25)) = 0.404. Peak time 0.01 s gives ω_n = π/(0.01√(1−0.404²)) = 343.4 rad/s. The second‑order transfer function is T(s) = 117914/(s²+277.3s+117914).

## Answer

The response has an overshoot, so model it as an underdamped second order system with unity DC gain (final value 1):
$$
T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}
$$

#### Step 1: Read the response
- Final value $= 1$, so the DC gain is 1.
- Peak value $= 1.25$, so $M_p = \dfrac{1.25 - 1}{1} = 0.25$ (25%).
- Peak time $t_p = 0.01$ s.

#### Step 2: Damping ratio
$$
M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}
\;\Rightarrow\;
\zeta = \frac{-\ln M_p}{\sqrt{\pi^2 + (\ln M_p)^2}} = \frac{1.3863}{\sqrt{9.8696 + 1.9218}} = \frac{1.3863}{3.4339} = 0.4037
$$

#### Step 3: Natural frequency
$$
t_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}
\;\Rightarrow\;
\omega_n = \frac{\pi}{0.01\sqrt{1 - 0.4037^2}} = \frac{3.1416}{0.01(0.9149)} = 343.4\ \text{rad/s}
$$

#### Step 4: Coefficients
$$
\begin{aligned}
\omega_n^2 &= 343.4^2 = 1.179\times 10^5 \\
2\zeta\omega_n &= 2(0.4037)(343.4) = 277.3
\end{aligned}
$$

**Answer:**
$$
T(s) = \frac{C(s)}{R(s)} = \frac{117914}{s^2 + 277.3s + 117914}
$$
with $\zeta = 0.404$ and $\omega_n = 343.4$ rad/s. If the system is taken as unity feedback, the equivalent open-loop model is $G(s) = \dfrac{117914}{s(s + 277.3)}$.

## Check yourself

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

1. Final value 1 and peak 1.25 give an overshoot Mp of
   A. 1.25
   B. 0.125
   C. 0.025
   D. 0.25
   Answer: D. Mp = (peak - final)/final = 0.25 (25%). (Key point: Reading the graph)

2. Which formula gives zeta from Mp?
   A. zeta = Mp
   B. zeta = -ln Mp / sqrt(pi^2 + (ln Mp)^2)
   C. zeta = pi/ln Mp
   D. zeta = 1 - Mp
   Answer: B. Inverting Mp = exp(-pi*zeta/sqrt(1-zeta^2)) gives zeta = 0.4037. (Key point: Damping ratio)

3. With tp = 0.01 s and zeta = 0.4037, wn is
   A. 314 rad/s
   B. 100 rad/s
   C. 343.4 rad/s
   D. 277 rad/s
   Answer: C. wn = pi/(tp*sqrt(1-zeta^2)) = 3.1416/(0.01*0.9149) = 343.4 rad/s. (Key point: Natural frequency)

4. The transfer function with unity DC gain is
   A. 117914/(s^2+277.3s+117914)
   B. 277.3/(s^2+277.3s+117914)
   C. 117914/(s^2+343.4s+117914)
   D. 1/(s^2+277.3s+1)
   Answer: A. T = wn^2/(s^2+2*zeta*wn*s+wn^2) with wn^2=117914 and 2*zeta*wn = 277.3. (Key point: Final model)

## Related in Control Systems

- [Underdamped second order unit step response derivation](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Asecond-order-step-derivation) — Control Systems, chapter 3
- [Find K and P from step response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ak-p-from-step-response) — Control Systems, chapter 3
- [Unit step response of 4/[s(s+2)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aunit-step-4-s-s2) — Control Systems, chapter 3
- [Find K and T from step response (overshoot 0.254)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ak-t-from-step-0254) — Control Systems, chapter 3
- [Find M, B and K from step response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ambk-from-step-response) — Control Systems, chapter 3
- [Derive maximum overshoot of second order system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Amp-expression-derivation) — Control Systems, chapter 3

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