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 + 1)(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 1)/((s + 1)(s + 2)). Find the position error constant Kp.

G(s) = 5(s + 1)/((s + 1)(s + 2))
Kp =
2.Easy
r(t) = 5+20(s + 3)(s + 1)(s + 10)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 20(s + 3)/((s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 20(s + 3)/((s + 1)(s + 10))r(t) = 5·u(t)
ess =
3.Medium
R(s)+K(s + 1)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 5)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 1)(s + 5))ess = 2 %
K =
4.Medium
r(t) = 10 + 3t+50s(s + 1)(s + 10)C(s)−

The input r(t) = 10 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50/(s(s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 50/(s(s + 1)(s + 10))r(t) = 10 + 3t
ess =
5.Hard
R(s)+K = 202s(s + 8)C(s)−

A proportional controller K = 20 drives the plant G(s) = 2/(s(s + 8)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 2/(s(s + 8))K = 20
tp = s
6.Hard
R(s)+Kp​ + Ki​/s2s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.5. Find the proportional gain Kp.

G(s) = 2/(s + 5)ωₙ = 10 rad/sζ = 0.5
Kp =

Answer key

  1. 1. Kp = 2.5
  2. 2. ess = 0.7143
  3. 3. K = 245
  4. 4. ess = 0.6
  5. 5. tp = 0.6413 s
  6. 6. Kp = 2.5

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