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)+4(s + 3)s(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 3)/(s(s + 5)). Find the velocity error constant Kv.

G(s) = 4(s + 3)/(s(s + 5))
Kv =
2.Easy
R(s)+4(s + 1)s(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 1)/(s(s + 5)). Find the velocity error constant Kv.

G(s) = 4(s + 1)/(s(s + 5))
Kv =
3.Medium
R(s)+K = 1510s + 1C(s)−

A process G(s) = 5/(10s + 1) is controlled by a proportional gain K = 1 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 5/(10s + 1)K = 1
ess = %
4.Medium
r(t) = 4t+50s(s + 5)(s + 10)C(s)−

A ramp r(t) = 4t is applied to a unity-feedback loop with G(s) = 50/(s(s + 5)(s + 10)). Find the steady-state error ess.

G(s) = 50/(s(s + 5)(s + 10))r(t) = 4t
ess =
5.Hard
R(s)+K10s(s + 2)C(s)−

A proportional controller K drives the plant G(s) = 10/(s(s + 2)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 10/(s(s + 2))ζ = 0.8
K =
6.Hard
R(s)+Kp​ + Ki​/s2s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 2/(s + 1)ωₙ = 10 rad/sζ = 0.6
Ki =

Answer key

  1. 1. Kv = 2.4
  2. 2. Kv = 0.8
  3. 3. ess = 16.67 %
  4. 4. ess = 4
  5. 5. K = 0.1562
  6. 6. Ki = 50

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