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(t) = 2+10(s + 2)(s + 2)(s + 4)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 10(s + 2)/((s + 2)(s + 4)). Find the steady-state error ess.

G(s) = 10(s + 2)/((s + 2)(s + 4))r(t) = 2·u(t)
ess =
2.Easy
r(t) = 2+5(s + 2)(s + 5)C(s)−

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

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

A unity-feedback loop with G(s) = K/(s(s + 2)) must follow the ramp r(t) = 2t with a steady-state error of 0.01. Find the gain K.

G(s) = K/(s(s + 2))r(t) = 2tess = 0.01
K =
4.Medium
r(t) = 0.5t+4(s + 4)s(s + 1)(s + 4)C(s)−

A ramp r(t) = 0.5t is applied to a unity-feedback loop with G(s) = 4(s + 4)/(s(s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 4(s + 4)/(s(s + 1)(s + 4))r(t) = 0.5t
ess =
5.Hard
R(s)+K = 501s(s + 5)C(s)−

A proportional controller K = 50 drives the plant G(s) = 1/(s(s + 5)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 1/(s(s + 5))K = 50
tp = s
6.Hard
R(s)+Kp​ + Ki​/s4s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 4/(s + 1)ωₙ = 8 rad/sζ = 0.8
Kp =

Answer key

  1. 1. ess = 0.5714
  2. 2. ess = 1.333
  3. 3. K = 400
  4. 4. ess = 0.125
  5. 5. tp = 0.475 s
  6. 6. Kp = 2.95

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