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)+50(s + 1)(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50/((s + 1)(s + 2)). Find the position error constant Kp.

G(s) = 50/((s + 1)(s + 2))
Kp =
2.Easy
R(s)+5(s + 1)(s + 2)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 1)/((s + 2)(s + 5)). Find the position error constant Kp.

G(s) = 5(s + 1)/((s + 2)(s + 5))
Kp =
3.Medium
r(t) = 2·t²/2+5(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) = 5(s + 4)/(s²(s + 5)). Find the steady-state error ess.

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

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 2(s + 4)/(s²(s + 1)). Find the steady-state error ess.

G(s) = 2(s + 4)/(s²(s + 1))r(t) = 10·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 = 2 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 4/(s + 1)ωₙ = 2 rad/sζ = 0.7
Ki =
6.Hard
R(s)+K = 202s(s + 10)C(s)−

A proportional controller K = 20 drives the plant G(s) = 2/(s(s + 10)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 2/(s(s + 10))K = 20
ts = s

Answer key

  1. 1. Kp = 25
  2. 2. Kp = 0.5
  3. 3. ess = 0.5
  4. 4. ess = 1.25
  5. 5. Ki = 1
  6. 6. ts = 0.8 s

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