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 + 2)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/((s + 2)(s + 4)). Find the position error constant Kp.

G(s) = 20/((s + 2)(s + 4))
Kp =
2.Easy
R(s)+2(s + 5)s²(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 5)/(s²(s + 5)). Find the acceleration error constant Ka.

G(s) = 2(s + 5)/(s²(s + 5))
Ka =
3.Medium
r(t) = 5·t²/2+20(s + 5)s²(s + 1)C(s)−

The parabolic input r(t) = 5·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 5)/(s²(s + 1)). Find the steady-state error ess.

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

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 4(s + 4)/(s²(s + 5)). Find the steady-state error ess.

G(s) = 4(s + 4)/(s²(s + 5))r(t) = 0.5·t²/2
ess =
5.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 = 3 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 4/(s + 1)ωₙ = 3 rad/sζ = 0.8
Kp =
6.Hard
r(t) = 2t+1 + 5/s4(s + 1)(s + 10)C(s)−

A PI controller C(s) = 1 + 5/s drives the plant G(s) = 4/((s + 1)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 2t.

C(s) = 1 + 5/sG(s) = 4/((s + 1)(s + 10))r(t) = 2t
ess =

Answer key

  1. 1. Kp = 2.5
  2. 2. Ka = 2
  3. 3. ess = 0.05
  4. 4. ess = 0.1563
  5. 5. Kp = 0.95
  6. 6. ess = 1

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