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)+10(s + 4)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10/((s + 4)(s + 5)). Find the position error constant Kp.

G(s) = 10/((s + 4)(s + 5))
Kp =
2.Easy
R(s)+2(s + 4)s²(s + 4)C(s)−

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

G(s) = 2(s + 4)/(s²(s + 4))
Ka =
3.Medium
r(t) = 1 + 1t+20(s + 4)s(s + 1)C(s)−

The input r(t) = 1 + 1t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20(s + 4)/(s(s + 1)). Find the steady-state error ess.

G(s) = 20(s + 4)/(s(s + 1))r(t) = 1 + 1t
ess =
4.Medium
r(t) = 2 + 2t+50s(s + 1)(s + 4)C(s)−

The input r(t) = 2 + 2t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50/(s(s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 50/(s(s + 1)(s + 4))r(t) = 2 + 2t
ess =
5.Hard
R(s)+K2s(s + 6)C(s)−

A proportional controller K drives the plant G(s) = 2/(s(s + 6)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 2/(s(s + 6))ζ = 0.707
K =
6.Hard
R(s)+Kp​ + Ki​/s1s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 1/(s + 2)ωₙ = 3 rad/sζ = 0.7
Kp =

Answer key

  1. 1. Kp = 0.5
  2. 2. Ka = 2
  3. 3. ess = 0.0125
  4. 4. ess = 0.16
  5. 5. K = 9.003
  6. 6. Kp = 2.2

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