Step response worksheet with answers

Free printable Step response worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (s)y(t)y(∞) = ?

A first-order system G(s) = 20/(s + 10) receives a step input of size 1 at t = 0, starting from rest. Find the final value y(∞) of the output.

G(s) = 20/(s + 10)U = 1
y(∞) =
2.Easy
t (s)y(t)1005 s60

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 100. At t = 5 s it reads 60. Find the time constant τ.

y(∞) = 100y(5) = 60
τ = s
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 20/(s² + 3s + 20) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 20/(s² + 3s + 20)
σ =
4.Medium
t (s)y(t)3

A system has the transfer function G(s) = 252/(s² + 7s + 84) and receives a unit step input. Find the damping ratio ζ.

G(s) = 252/(s² + 7s + 84)
ζ =
5.Hard
R(s)+150s(s + 10)G(s)C(s)−

A unity negative feedback loop has G(s) = 150/(s(s + 10)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 150/(s(s + 10))
ts = s
6.Hard
t (s)y(t)1tp​ = 0.25 s2 %

The step response of a second-order system without zeros shows 2 % overshoot, and the first peak occurs at t = 0.25 s. Find the natural frequency ωn.

%OS = 2 %tp = 0.25 s
ωn = rad/s

Answer key

  1. 1. y(∞) = 2
  2. 2. τ = 5.457 s
  3. 3. σ = −1.5
  4. 4. ζ = 0.3819
  5. 5. ts = 0.8 s
  6. 6. ωn = 20.07 rad/s

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