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)+20(s + 2)s(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 2)/(s(s + 1)). Find the velocity error constant Kv.

G(s) = 20(s + 2)/(s(s + 1))
Kv =
2.Easy
R(s)+20(s + 2)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/((s + 2)(s + 10)). Find the position error constant Kp.

G(s) = 20/((s + 2)(s + 10))
Kp =
3.Medium
R(s)+K(s + 1)(s + 4)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 4)). Find the gain K that gives a steady-state error of 4 % for a step input.

G(s) = K/((s + 1)(s + 4))ess = 4 %
K =
4.Medium
r(t) = 1 + 1t+4s(s + 1)(s + 5)C(s)−

The input r(t) = 1 + 1t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4/(s(s + 1)(s + 5)). Find the steady-state error ess.

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

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 4/(s + 5)ωₙ = 8 rad/sζ = 0.6
Kp =
6.Hard
r(t) = 2t+5 + 4/s20(s + 1)(s + 2)C(s)−

A PI controller C(s) = 5 + 4/s drives the plant G(s) = 20/((s + 1)(s + 2)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 2t.

C(s) = 5 + 4/sG(s) = 20/((s + 1)(s + 2))r(t) = 2t
ess =

Answer key

  1. 1. Kv = 40
  2. 2. Kp = 1
  3. 3. K = 96
  4. 4. ess = 1.25
  5. 5. Kp = 1.15
  6. 6. ess = 0.05

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