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)+5(s + 1)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/((s + 1)(s + 4)). Find the position error constant Kp.

G(s) = 5/((s + 1)(s + 4))
Kp =
2.Easy
R(s)+4(s + 1)s(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 1)/(s(s + 2)). Find the velocity error constant Kv.

G(s) = 4(s + 1)/(s(s + 2))
Kv =
3.Medium
r(t) = 2 + 2t+50s(s + 1)(s + 4)C(s)−

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

G(s) = 50/(s(s + 1)(s + 4))r(t) = 2 + 2t
ess =
4.Medium
r(t) = 3t+20(s + 3)s(s + 1)(s + 4)C(s)−

A ramp r(t) = 3t is applied to a unity-feedback loop with G(s) = 20(s + 3)/(s(s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 20(s + 3)/(s(s + 1)(s + 4))r(t) = 3t
ess =
5.Hard
R(s)+Kp​ + Ki​/s4s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 4 rad/s and ζ = 1. Find the proportional gain Kp.

G(s) = 4/(s + 2)ωₙ = 4 rad/sζ = 1
Kp =
6.Hard
R(s)+Kp​ + Ki​/s5s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 5/(s + 4)ωₙ = 10 rad/sζ = 0.7
Ki =

Answer key

  1. 1. Kp = 1.25
  2. 2. Kv = 2
  3. 3. ess = 0.16
  4. 4. ess = 0.2
  5. 5. Kp = 1.5
  6. 6. Ki = 20

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