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 + 1)s(s + 1)C(s)−

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

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

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 2)/(s(s + 2)). What is the system type N?

G(s) = 5(s + 2)/(s(s + 2))
N =
3.Medium
r(t) = 0.5t+50(s + 1)s(s + 5)C(s)−

A ramp r(t) = 0.5t is applied to a unity-feedback loop with G(s) = 50(s + 1)/(s(s + 5)). Find the steady-state error ess.

G(s) = 50(s + 1)/(s(s + 5))r(t) = 0.5t
ess =
4.Medium
R(s)+K(s + 1)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 5)). Find the gain K that gives a steady-state error of 4 % for a step input.

G(s) = K/((s + 1)(s + 5))ess = 4 %
K =
5.Hard
r(t) = 5t+5 + 10/s5(s + 1)(s + 2)C(s)−

A PI controller C(s) = 5 + 10/s drives the plant G(s) = 5/((s + 1)(s + 2)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 5 + 10/sG(s) = 5/((s + 1)(s + 2))r(t) = 5t
ess =
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 = 8 rad/s and ζ = 0.5. Find the proportional gain Kp.

G(s) = 2/(s + 1)ωₙ = 8 rad/sζ = 0.5
Kp =

Answer key

  1. 1. Kv = 4
  2. 2. N = 1
  3. 3. ess = 0.05
  4. 4. K = 120
  5. 5. ess = 0.2
  6. 6. Kp = 3.5

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