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+5(s + 1)(s + 10)C(s)−

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

G(s) = 5/((s + 1)(s + 10))r(t) = 5·u(t)
ess =
2.Easy
R(s)+20(s + 1)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 20(s + 1)/(s²(s + 4))
Ka =
3.Medium
r(t) = 5t+20s(s + 5)C(s)−

A ramp r(t) = 5t is applied to a unity-feedback loop with G(s) = 20/(s(s + 5)). Find the steady-state error ess.

G(s) = 20/(s(s + 5))r(t) = 5t
ess =
4.Medium
r(t) = 2·t²/2+4(s + 4)s²(s + 1)C(s)−

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 4(s + 4)/(s²(s + 1)). Find the steady-state error ess.

G(s) = 4(s + 4)/(s²(s + 1))r(t) = 2·t²/2
ess =
5.Hard
R(s)+K = 82s(s + 5)C(s)−

A proportional controller K = 8 drives the plant G(s) = 2/(s(s + 5)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 2/(s(s + 5))K = 8
%OS = %
6.Hard
R(s)+K = 251s(s + 5)C(s)−

A proportional controller K = 25 drives the plant G(s) = 1/(s(s + 5)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 5))K = 25
ts = s

Answer key

  1. 1. ess = 3.333
  2. 2. Ka = 5
  3. 3. ess = 1.25
  4. 4. ess = 0.125
  5. 5. %OS = 8.084 %
  6. 6. ts = 1.6 s

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