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) = 5+50(s + 1)(s + 5)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 50/((s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 50/((s + 1)(s + 5))r(t) = 5·u(t)
ess =
2.Easy
R(s)+5s(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/(s(s + 4)). Find the velocity error constant Kv.

G(s) = 5/(s(s + 4))
Kv =
3.Medium
r(t) = 5t+10(s + 1)s(s + 1)C(s)−

A ramp r(t) = 5t is applied to a unity-feedback loop with G(s) = 10(s + 1)/(s(s + 1)). Find the steady-state error ess.

G(s) = 10(s + 1)/(s(s + 1))r(t) = 5t
ess =
4.Medium
R(s)+K(s + 1)(s + 4)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 4)). Find the gain K that gives a steady-state error of 20 % for a step input.

G(s) = K/((s + 1)(s + 4))ess = 20 %
K =
5.Hard
r(t) = 5t+1 + 4/s5(s + 5)(s + 10)C(s)−

A PI controller C(s) = 1 + 4/s drives the plant G(s) = 5/((s + 5)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 1 + 4/sG(s) = 5/((s + 5)(s + 10))r(t) = 5t
ess =
6.Hard
R(s)+K = 405s(s + 4)C(s)−

A proportional controller K = 40 drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 5/(s(s + 4))K = 40
tp = s

Answer key

  1. 1. ess = 0.4545
  2. 2. Kv = 1.25
  3. 3. ess = 0.5
  4. 4. K = 16
  5. 5. ess = 12.5
  6. 6. tp = 0.2244 s

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