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)80.5 s5.6

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 8. At t = 0.5 s it reads 5.6. Find the time constant τ.

y(∞) = 8y(0.5) = 5.6
τ = s
2.Easy
t (s)y(t)8t = ?99 %

A first-order system G(s) = 4/(4s + 1) receives a step input of size 2 at t = 0, starting from rest. How long does the output take to reach 99 % of its final value? Find t.

G(s) = 4/(4s + 1)U = 299 %
t = s
3.Medium
t (s)y(t)2

A system has the transfer function G(s) = 146/(s² + 10s + 73) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 146/(s² + 10s + 73)
σ =
4.Medium
t (s)y(t)1tp​ = ?

A system has the transfer function G(s) = 57/(s² + 9s + 57) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 57/(s² + 9s + 57)
tp = s
5.Hard
R(s)+85s(s + 8)G(s)C(s)−

A unity negative feedback loop has G(s) = 85/(s(s + 8)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 85/(s(s + 8))
ts = s
6.Hard
t (s)y(t)1tp​ = 0.1 s20 %

The step response of a second-order system without zeros shows 20 % overshoot, and the first peak occurs at t = 0.1 s. Find the natural frequency ωn.

%OS = 20 %tp = 0.1 s
ωn = rad/s

Answer key

  1. 1. τ = 0.4153 s
  2. 2. t = 18.42 s
  3. 3. σ = −5
  4. 4. tp = 0.5182 s
  5. 5. ts = 1 s
  6. 6. ωn = 35.3 rad/s

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