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)+4(s + 2)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 2)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 4(s + 2)/(s²(s + 4))
Ka =
2.Easy
R(s)+10(s + 1)s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s²(s + 1)). Find the acceleration error constant Ka.

G(s) = 10(s + 1)/(s²(s + 1))
Ka =
3.Medium
R(s)+K(s + 1)(s + 2)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 2)). Find the gain K that gives a steady-state error of 10 % for a step input.

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

A ramp r(t) = 2t is applied to a unity-feedback loop with G(s) = 4/(s(s + 5)). Find the steady-state error ess.

G(s) = 4/(s(s + 5))r(t) = 2t
ess =
5.Hard
R(s)+Kp​ + Ki​/s10s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 10/(s + 4)ωₙ = 5 rad/sζ = 0.7
Ki =
6.Hard
R(s)+Kp​ + Ki​/s5s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 4 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 5/(s + 5)ωₙ = 4 rad/sζ = 0.8
Kp =

Answer key

  1. 1. Ka = 2
  2. 2. Ka = 10
  3. 3. K = 18
  4. 4. ess = 2.5
  5. 5. Ki = 2.5
  6. 6. Kp = 0.28

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