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+40(s + 2)(s + 1)(s + 2)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 40(s + 2)/((s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 40(s + 2)/((s + 1)(s + 2))r(t) = 5·u(t)
ess =
2.Easy
R(s)+10(s + 1)s(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s(s + 2)). What is the system type N?

G(s) = 10(s + 1)/(s(s + 2))
N =
3.Medium
r(t) = 0.5·t²/2+2(s + 3)s²(s + 1)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 2(s + 3)/(s²(s + 1)). Find the steady-state error ess.

G(s) = 2(s + 3)/(s²(s + 1))r(t) = 0.5·t²/2
ess =
4.Medium
r(t) = 2 + 2t+4s(s + 4)(s + 5)C(s)−

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

G(s) = 4/(s(s + 4)(s + 5))r(t) = 2 + 2t
ess =
5.Hard
R(s)+K = 162s(s + 8)C(s)−

A proportional controller K = 16 drives the plant G(s) = 2/(s(s + 8)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 2/(s(s + 8))K = 16
ts = s
6.Hard
R(s)+K = 55s(s + 8)C(s)−

A proportional controller K = 5 drives the plant G(s) = 5/(s(s + 8)) in a unity-feedback loop. Find the peak time tp of the step response.

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

Answer key

  1. 1. ess = 0.122
  2. 2. N = 1
  3. 3. ess = 0.08333
  4. 4. ess = 10
  5. 5. ts = 1 s
  6. 6. tp = 1.047 s

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