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 + 2)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/((s + 2)(s + 10)). Find the position error constant Kp.

G(s) = 5/((s + 2)(s + 10))
Kp =
2.Easy
R(s)+20(s + 1)(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/((s + 1)(s + 2)). Find the position error constant Kp.

G(s) = 20/((s + 1)(s + 2))
Kp =
3.Medium
R(s)+K = 2040.5s + 1C(s)−

A process G(s) = 4/(0.5s + 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/(0.5s + 1)K = 20
ess = %
4.Medium
R(s)+K(s + 2)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 2)(s + 5)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 2)(s + 5))ess = 2 %
K =
5.Hard
R(s)+K = 251s(s + 4)C(s)−

A proportional controller K = 25 drives the plant G(s) = 1/(s(s + 4)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 4))K = 25
ts = s
6.Hard
R(s)+10 + Kd​s1s(s + 2)C(s)−

A PD controller C(s) = 10 + Kd·s drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 1/(s(s + 2))Kp = 10ζ = 0.5
Kd = s

Answer key

  1. 1. Kp = 0.25
  2. 2. Kp = 10
  3. 3. ess = 1.235 %
  4. 4. K = 490
  5. 5. ts = 2 s
  6. 6. Kd = 1.162 s

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