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)+20s(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s(s + 2)). Find the velocity error constant Kv.

G(s) = 20/(s(s + 2))
Kv =
2.Easy
r(t) = 5+50(s + 2)(s + 4)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 50/((s + 2)(s + 4)). Find the steady-state error ess.

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

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 2)/(s²(s + 4)). Find the steady-state error ess.

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

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 1t with a steady-state error of 0.02. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 1tess = 0.02
K =
5.Hard
R(s)+K = 401s(s + 4)C(s)−

A proportional controller K = 40 drives the plant G(s) = 1/(s(s + 4)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 1/(s(s + 4))K = 40
%OS = %
6.Hard
R(s)+Kp​ + Ki​/s10s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.8. Find the integral gain Ki.

G(s) = 10/(s + 2)ωₙ = 3 rad/sζ = 0.8
Ki =

Answer key

  1. 1. Kv = 10
  2. 2. ess = 0.6897
  3. 3. ess = 0.4
  4. 4. K = 50
  5. 5. %OS = 35.09 %
  6. 6. Ki = 0.9

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