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)+2(s + 2)s(s + 10)C(s)−

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

G(s) = 2(s + 2)/(s(s + 10))
Kv =
2.Easy
R(s)+20(s + 2)s²(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 2)/(s²(s + 2)). Find the acceleration error constant Ka.

G(s) = 20(s + 2)/(s²(s + 2))
Ka =
3.Medium
r(t) = 4t+4(s + 3)s(s + 2)(s + 4)C(s)−

A ramp r(t) = 4t is applied to a unity-feedback loop with G(s) = 4(s + 3)/(s(s + 2)(s + 4)). Find the steady-state error ess.

G(s) = 4(s + 3)/(s(s + 2)(s + 4))r(t) = 4t
ess =
4.Medium
R(s)+K(s + 2)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 2)(s + 10)). Find the gain K that gives a steady-state error of 20 % for a step input.

G(s) = K/((s + 2)(s + 10))ess = 20 %
K =
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 = 0.8. Find the integral gain Ki.

G(s) = 4/(s + 2)ωₙ = 4 rad/sζ = 0.8
Ki =
6.Hard
R(s)+K = 161s(s + 6)C(s)−

A proportional controller K = 16 drives the plant G(s) = 1/(s(s + 6)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 1/(s(s + 6))K = 16
tp = s

Answer key

  1. 1. Kv = 0.4
  2. 2. Ka = 20
  3. 3. ess = 2.667
  4. 4. K = 80
  5. 5. Ki = 4
  6. 6. tp = 1.187 s

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