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)2.50.2 s?

A first-order system G(s) = 0.5/(0.2s + 1) receives a step input of size 5 at t = 0, starting from rest. Find the output y(0.2) at t = 0.2 s.

G(s) = 0.5/(0.2s + 1)U = 5t = 0.2 s
y(0.2) =
2.Easy
t (s)y(t)82 s3.2

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 8. At t = 2 s it reads 3.2. Find the time constant τ.

y(∞) = 8y(2) = 3.2
τ = s
3.Medium
t (s)y(t)1ts​ = ?

A system has the transfer function G(s) = 92/(s² + 13s + 92) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 92/(s² + 13s + 92)
ts = s
4.Medium
t (s)y(t)0.5tp​ = ?

A system has the transfer function G(s) = 26/(s² + 6s + 52) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 26/(s² + 6s + 52)
tp = s
5.Hard
t (s)y(t)1tp​ = 0.5 s30 %

The step response of a second-order system without zeros shows 30 % overshoot, and the first peak occurs at t = 0.5 s. Find the natural frequency ωn.

%OS = 30 %tp = 0.5 s
ωn = rad/s
6.Hard
t (s)y(t)1tp​ = 1.5 s30 %

The step response of a second-order system without zeros shows 30 % overshoot, and the first peak occurs at t = 1.5 s. Find the natural frequency ωn.

%OS = 30 %tp = 1.5 s
ωn = rad/s

Answer key

  1. 1. y(0.2) = 1.58
  2. 2. τ = 3.915 s
  3. 3. ts = 0.6154 s
  4. 4. tp = 0.4791 s
  5. 5. ωn = 6.729 rad/s
  6. 6. ωn = 2.243 rad/s

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