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)+5(s + 3)s(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 3)/(s(s + 5)). Find the velocity error constant Kv.

G(s) = 5(s + 3)/(s(s + 5))
Kv =
2.Easy
R(s)+20(s + 1)(s + 1)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/((s + 1)(s + 5)). Find the position error constant Kp.

G(s) = 20(s + 1)/((s + 1)(s + 5))
Kp =
3.Medium
R(s)+K = 2041s + 1C(s)−

A process G(s) = 4/(1s + 1) is controlled by a proportional gain K = 20 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

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

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

G(s) = 5(s + 5)/(s²(s + 4))r(t) = 0.5·t²/2
ess =
5.Hard
R(s)+K4s(s + 5)C(s)−

A proportional controller K drives the plant G(s) = 4/(s(s + 5)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 4/(s(s + 5))ζ = 0.8
K =
6.Hard
R(s)+K = 255s(s + 10)C(s)−

A proportional controller K = 25 drives the plant G(s) = 5/(s(s + 10)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 10))K = 25
ts = s

Answer key

  1. 1. Kv = 3
  2. 2. Kp = 4
  3. 3. ess = 1.235 %
  4. 4. ess = 0.08
  5. 5. K = 2.441
  6. 6. ts = 0.8 s

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