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(s)+2(s + 4)s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 4)/(s²(s + 1)). Find the acceleration error constant Ka.

G(s) = 2(s + 4)/(s²(s + 1))
Ka =
2.Easy
r(t) = 5+K = 41s + 1y(t)−

A proportional controller K = 4 drives the plant G(s) = 1/(s + 1) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 1)K = 4r = 5
y(∞) =
3.Medium
r(t) = 2t+5s(s + 2)(s + 4)C(s)−

A ramp r(t) = 2t is applied to a unity-feedback loop with G(s) = 5/(s(s + 2)(s + 4)). Find the steady-state error ess.

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

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 2(s + 3)/(s²(s + 4)). Find the steady-state error ess.

G(s) = 2(s + 3)/(s²(s + 4))r(t) = 2·t²/2
ess =
5.Hard
R(s)+K = 15s(s + 2)C(s)−

A proportional controller K = 1 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 = 1
tp = s
6.Hard
R(s)+K = 85s(s + 6)C(s)−

A proportional controller K = 8 drives the plant G(s) = 5/(s(s + 6)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 6))K = 8
ts = s

Answer key

  1. 1. Ka = 8
  2. 2. y(∞) = 4
  3. 3. ess = 3.2
  4. 4. ess = 1.333
  5. 5. tp = 1.571 s
  6. 6. ts = 1.333 s

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