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)+10(s + 3)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 3)/(s²(s + 10)). Find the acceleration error constant Ka.

G(s) = 10(s + 3)/(s²(s + 10))
Ka =
2.Easy
r(t) = 2+50(s + 1)(s + 1)(s + 5)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 50(s + 1)/((s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 50(s + 1)/((s + 1)(s + 5))r(t) = 2·u(t)
ess =
3.Medium
r(t) = 10t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 10t with a steady-state error of 0.2. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 10tess = 0.2
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​/s2s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 1. Find the proportional gain Kp.

G(s) = 2/(s + 5)ωₙ = 6 rad/sζ = 1
Kp =
6.Hard
R(s)+Kp​ + Ki​/s5s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 5/(s + 2)ωₙ = 8 rad/sζ = 0.6
Kp =

Answer key

  1. 1. Ka = 3
  2. 2. ess = 0.1818
  3. 3. K = 250
  4. 4. K = 2000
  5. 5. Kp = 3.5
  6. 6. Kp = 1.52

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