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)+C(s)G(s)C(s)−

The controller C(s) = 2(s + 5)/s drives the plant G(s) = 2/(s(s + 3)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 2(s + 5)/sG(s) = 2/(s(s + 3))
N =
2.Easy
r(t) = 2+50(s + 1)(s + 5)C(s)−

A step of size 2 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) = 2·u(t)
ess =
3.Medium
r(t) = 0.5·t²/2+20(s + 3)s²(s + 10)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 3)/(s²(s + 10)). Find the steady-state error ess.

G(s) = 20(s + 3)/(s²(s + 10))r(t) = 0.5·t²/2
ess =
4.Medium
r(t) = 2 + 4t+5(s + 3)s(s + 4)(s + 10)C(s)−

The input r(t) = 2 + 4t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5(s + 3)/(s(s + 4)(s + 10)). Find the steady-state error ess.

G(s) = 5(s + 3)/(s(s + 4)(s + 10))r(t) = 2 + 4t
ess =
5.Hard
R(s)+K = 251s(s + 8)C(s)−

A proportional controller K = 25 drives the plant G(s) = 1/(s(s + 8)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 1/(s(s + 8))K = 25
%OS = %
6.Hard
R(s)+K = 401s(s + 6)C(s)−

A proportional controller K = 40 drives the plant G(s) = 1/(s(s + 6)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 1/(s(s + 6))K = 40
%OS = %

Answer key

  1. 1. N = 2
  2. 2. ess = 0.1818
  3. 3. ess = 0.08333
  4. 4. ess = 10.67
  5. 5. %OS = 1.516 %
  6. 6. %OS = 18.4 %

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