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)40.8 s?

A first-order system G(s) = 2/(0.8s + 1) receives a step input of size 2 at t = 0, starting from rest. Find the output y(0.8) at t = 0.8 s.

G(s) = 2/(0.8s + 1)U = 2t = 0.8 s
y(0.8) =
2.Easy
t (s)y(t)1005 s50

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 50. Find the time constant τ.

y(∞) = 100y(5) = 50
τ = s
3.Medium
t (s)y(t)0.5ts​ = ?

A system has the transfer function G(s) = 20.5/(s² + 2s + 41) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 20.5/(s² + 2s + 41)
ts = s
4.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 34/(s² + 9s + 68) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 34/(s² + 9s + 68)
σ =
5.Hard
t (s)y(t)130 %

A second-order system without zeros must have a step response with 30 % overshoot. Find the damping ratio ζ.

%OS = 30 %
ζ =
6.Hard
t (s)y(t)1tp​ = 2 s20 %

The step response of a second-order system without zeros shows 20 % overshoot, and the first peak occurs at t = 2 s. Find the natural frequency ωn.

%OS = 20 %tp = 2 s
ωn = rad/s

Answer key

  1. 1. y(0.8) = 2.528
  2. 2. τ = 7.213 s
  3. 3. ts = 4 s
  4. 4. σ = −4.5
  5. 5. ζ = 0.3579
  6. 6. ωn = 1.765 rad/s

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