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)+2s(s + 1)(s + 2)C(s)−

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

G(s) = 2/(s(s + 1)(s + 2))
Kv =
2.Easy
r(t) = 5+K = 310s + 4y(t)−

A proportional controller K = 3 drives the plant G(s) = 10/(s + 4) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 10/(s + 4)K = 3r = 5
y(∞) =
3.Medium
r(t) = 2t+Ks(s + 2)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 2)) must follow the ramp r(t) = 2t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 2))r(t) = 2tess = 0.05
K =
4.Medium
r(t) = 10t+Ks(s + 10)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 10)) must follow the ramp r(t) = 10t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 10tess = 0.05
K =
5.Hard
R(s)+Kp​ + Ki​/s1s + 2C(s)−

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

G(s) = 1/(s + 2)ωₙ = 2 rad/sζ = 0.6
Ki =
6.Hard
R(s)+Kp​ + Ki​/s10s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 0.5. Find the proportional gain Kp.

G(s) = 10/(s + 4)ωₙ = 6 rad/sζ = 0.5
Kp =

Answer key

  1. 1. Kv = 1
  2. 2. y(∞) = 4.412
  3. 3. K = 80
  4. 4. K = 2000
  5. 5. Ki = 4
  6. 6. Kp = 0.2

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