Steady-state error & PID control worksheet with answers

Free printable Steady-state error & PID control 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
r(t) = 10+K = 32s + 1y(t)−

A proportional controller K = 3 drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 1)K = 3r = 10
y(∞) =
2.Easy
r(t) = 10+K = 25s + 10y(t)−

A proportional controller K = 2 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 10)K = 2r = 10
y(∞) =
3.Medium
r(t) = 2·t²/2+4(s + 2)s²(s + 10)C(s)−

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 4(s + 2)/(s²(s + 10)). Find the steady-state error ess.

G(s) = 4(s + 2)/(s²(s + 10))r(t) = 2·t²/2
ess =
4.Medium
R(s)+K(s + 2)(s + 4)C(s)−

A unity-feedback loop has G(s) = K/((s + 2)(s + 4)). Find the gain K that gives a steady-state error of 25 % for a step input.

G(s) = K/((s + 2)(s + 4))ess = 25 %
K =
5.Hard
R(s)+K = 25s(s + 2)C(s)−

A proportional controller K = 2 drives the plant G(s) = 5/(s(s + 2)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 5/(s(s + 2))K = 2
tp = s
6.Hard
R(s)+K = 2010s(s + 4)C(s)−

A proportional controller K = 20 drives the plant G(s) = 10/(s(s + 4)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 10/(s(s + 4))K = 20
ts = s

Answer key

  1. 1. y(∞) = 8.571
  2. 2. y(∞) = 5
  3. 3. ess = 2.5
  4. 4. K = 24
  5. 5. tp = 1.047 s
  6. 6. ts = 2 s

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