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)+4(s + 1)s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 1)/(s(s + 10)). Find the velocity error constant Kv.

G(s) = 4(s + 1)/(s(s + 10))
Kv =
2.Easy
R(s)+20(s + 1)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/(s²(s + 10)). Find the acceleration error constant Ka.

G(s) = 20(s + 1)/(s²(s + 10))
Ka =
3.Medium
r(t) = 10t+Ks(s + 1)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 10t with a steady-state error of 0.02. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 10tess = 0.02
K =
4.Medium
r(t) = 10 + 4t+50(s + 2)s(s + 1)C(s)−

The input r(t) = 10 + 4t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50(s + 2)/(s(s + 1)). Find the steady-state error ess.

G(s) = 50(s + 2)/(s(s + 1))r(t) = 10 + 4t
ess =
5.Hard
R(s)+K = 201s(s + 4)C(s)−

A proportional controller K = 20 drives the plant G(s) = 1/(s(s + 4)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 1/(s(s + 4))K = 20
%OS = %
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 peak time tp of the step response.

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

Answer key

  1. 1. Kv = 0.4
  2. 2. Ka = 2
  3. 3. K = 500
  4. 4. ess = 0.04
  5. 5. %OS = 20.79 %
  6. 6. tp = 0.5642 s

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