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)+4s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4/(s(s + 10)). Find the velocity error constant Kv.

G(s) = 4/(s(s + 10))
Kv =
2.Easy
r(t) = 2+50(s + 1)(s + 4)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 50/((s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 50/((s + 1)(s + 4))r(t) = 2·u(t)
ess =
3.Medium
r(t) = 5t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 5t with a steady-state error of 0.5. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 5tess = 0.5
K =
4.Medium
r(t) = 10 + 0.5t+4s(s + 10)C(s)−

The input r(t) = 10 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4/(s(s + 10)). Find the steady-state error ess.

G(s) = 4/(s(s + 10))r(t) = 10 + 0.5t
ess =
5.Hard
R(s)+Kp​ + Ki​/s1s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 1/(s + 1)ωₙ = 3 rad/sζ = 0.6
Kp =
6.Hard
r(t) = 1t+5 + 1/s20(s + 5)(s + 10)C(s)−

A PI controller C(s) = 5 + 1/s drives the plant G(s) = 20/((s + 5)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 1t.

C(s) = 5 + 1/sG(s) = 20/((s + 5)(s + 10))r(t) = 1t
ess =

Answer key

  1. 1. Kv = 0.4
  2. 2. ess = 0.1481
  3. 3. K = 50
  4. 4. ess = 1.25
  5. 5. Kp = 2.6
  6. 6. ess = 2.5

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