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)102 s4

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 10. At t = 2 s it reads 4. Find the time constant τ.

y(∞) = 10y(2) = 4
τ = s
2.Easy
t (s)y(t)y(∞) = ?

A first-order system G(s) = 15/(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) = 15/(s + 5)U = 10
y(∞) =
3.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 47.5/(s² + 15s + 95) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 47.5/(s² + 15s + 95)
σ =
4.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 20/(s² + 4s + 40) and receives a unit step input. Find the damping ratio ζ.

G(s) = 20/(s² + 4s + 40)
ζ =
5.Hard
t (s)y(t)15 %

A second-order system without zeros must have a step response with 5 % overshoot. Find the damping ratio ζ.

%OS = 5 %
ζ =
6.Hard
R(s)+145s(s + 8)G(s)C(s)−

A unity negative feedback loop has G(s) = 145/(s(s + 8)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 145/(s(s + 8))
ts = s

Answer key

  1. 1. τ = 3.915 s
  2. 2. y(∞) = 30
  3. 3. σ = −7.5
  4. 4. ζ = 0.3162
  5. 5. ζ = 0.6901
  6. 6. ts = 1 s

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