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) = 1+10(s + 2)(s + 2)(s + 5)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 10(s + 2)/((s + 2)(s + 5)). Find the steady-state error ess.

G(s) = 10(s + 2)/((s + 2)(s + 5))r(t) = 1·u(t)
ess =
2.Easy
R(s)+4(s + 1)s(s + 1)(s + 4)C(s)−

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

G(s) = 4(s + 1)/(s(s + 1)(s + 4))
Kv =
3.Medium
r(t) = 10·t²/2+5(s + 1)s²(s + 10)C(s)−

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 1)/(s²(s + 10)). Find the steady-state error ess.

G(s) = 5(s + 1)/(s²(s + 10))r(t) = 10·t²/2
ess =
4.Medium
r(t) = 1 + 0.5t+50(s + 2)s(s + 4)C(s)−

The input r(t) = 1 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50(s + 2)/(s(s + 4)). Find the steady-state error ess.

G(s) = 50(s + 2)/(s(s + 4))r(t) = 1 + 0.5t
ess =
5.Hard
R(s)+K = 1610s(s + 6)C(s)−

A proportional controller K = 16 drives the plant G(s) = 10/(s(s + 6)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 10/(s(s + 6))K = 16
ts = s
6.Hard
R(s)+K = 501s(s + 6)C(s)−

A proportional controller K = 50 drives the plant G(s) = 1/(s(s + 6)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 1/(s(s + 6))K = 50
tp = s

Answer key

  1. 1. ess = 0.3333
  2. 2. Kv = 1
  3. 3. ess = 20
  4. 4. ess = 0.02
  5. 5. ts = 1.333 s
  6. 6. tp = 0.4906 s

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