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) = 10/(s + 5) receives a step input of size 10 at t = 0, starting from rest. Find the final value y(∞) of the output.

G(s) = 10/(s + 5)U = 10
y(∞) =
2.Easy
t (s)y(t)510 s4.5

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 5. At t = 10 s it reads 4.5. Find the time constant τ.

y(∞) = 5y(10) = 4.5
τ = s
3.Medium
t (s)y(t)2ts​ = ?

A system has the transfer function G(s) = 50/(s² + 6s + 25) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 50/(s² + 6s + 25)
ts = s
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 69/(s² + 13s + 69) and receives a unit step input. Find the damping ratio ζ.

G(s) = 69/(s² + 13s + 69)
ζ =
5.Hard
k = 200 N/mc = 24 N·s/mm = 8 kgF = 25 Nx(t)

A mass m = 8 kg hangs on a spring k = 200 N/m with a viscous damper c = 24 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 25 N is suddenly applied. Find the largest displacement xmax in mm.

m = 8 kgk = 200 N/mc = 24 N·s/mF = 25 N
xmax = mm
6.Hard
R(s)+200s(s + 20)G(s)C(s)−

A unity negative feedback loop has G(s) = 200/(s(s + 20)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 200/(s(s + 20))
ts = s

Answer key

  1. 1. y(∞) = 20
  2. 2. τ = 4.343 s
  3. 3. ts = 1.333 s
  4. 4. ζ = 0.7825
  5. 5. xmax = 171.5 mm
  6. 6. ts = 0.4 s

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