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)+40(s + 4)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 40/((s + 4)(s + 10)). Find the position error constant Kp.

G(s) = 40/((s + 4)(s + 10))
Kp =
2.Easy
r(t) = 2+50(s + 3)(s + 5)(s + 10)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 50(s + 3)/((s + 5)(s + 10)). Find the steady-state error ess.

G(s) = 50(s + 3)/((s + 5)(s + 10))r(t) = 2·u(t)
ess =
3.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 0.01. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 10tess = 0.01
K =
4.Medium
r(t) = 1t+20s(s + 1)C(s)−

A ramp r(t) = 1t is applied to a unity-feedback loop with G(s) = 20/(s(s + 1)). Find the steady-state error ess.

G(s) = 20/(s(s + 1))r(t) = 1t
ess =
5.Hard
R(s)+Kp​ + Ki​/s10s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 10/(s + 1)ωₙ = 3 rad/sζ = 0.8
Kp =
6.Hard
R(s)+K = 12s(s + 2)C(s)−

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

G(s) = 2/(s(s + 2))K = 1
ts = s

Answer key

  1. 1. Kp = 1
  2. 2. ess = 0.5
  3. 3. K = 10000
  4. 4. ess = 0.05
  5. 5. Kp = 0.38
  6. 6. ts = 4 s

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