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 + 3)(s + 1)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 3)/((s + 1)(s + 10)). Find the position error constant Kp.

G(s) = 20(s + 3)/((s + 1)(s + 10))
Kp =
2.Easy
r(t) = 2+20(s + 4)(s + 10)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 20/((s + 4)(s + 10)). Find the steady-state error ess.

G(s) = 20/((s + 4)(s + 10))r(t) = 2·u(t)
ess =
3.Medium
r(t) = 2·t²/2+20(s + 4)s²(s + 5)C(s)−

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 4)/(s²(s + 5)). Find the steady-state error ess.

G(s) = 20(s + 4)/(s²(s + 5))r(t) = 2·t²/2
ess =
4.Medium
r(t) = 0.5·t²/2+5(s + 1)s²(s + 10)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 1)/(s²(s + 10)). Find the steady-state error ess.

G(s) = 5(s + 1)/(s²(s + 10))r(t) = 0.5·t²/2
ess =
5.Hard
R(s)+Kp​ + Ki​/s2s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 2 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 2/(s + 2)ωₙ = 2 rad/sζ = 0.8
Kp =
6.Hard
R(s)+K2s(s + 10)C(s)−

A proportional controller K drives the plant G(s) = 2/(s(s + 10)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.4.

G(s) = 2/(s(s + 10))ζ = 0.4
K =

Answer key

  1. 1. Kp = 6
  2. 2. ess = 1.333
  3. 3. ess = 0.125
  4. 4. ess = 1
  5. 5. Kp = 0.6
  6. 6. K = 78.12

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