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)+4s(s + 4)(s + 5)C(s)−

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

G(s) = 4/(s(s + 4)(s + 5))
Kv =
2.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 =
3.Medium
r(t) = 10 + 3t+4s(s + 4)C(s)−

The input r(t) = 10 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4/(s(s + 4)). Find the steady-state error ess.

G(s) = 4/(s(s + 4))r(t) = 10 + 3t
ess =
4.Medium
R(s)+K(s + 4)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 4)(s + 5)). Find the gain K that gives a steady-state error of 10 % for a step input.

G(s) = K/((s + 4)(s + 5))ess = 10 %
K =
5.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 = 4 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 2/(s + 1)ωₙ = 4 rad/sζ = 0.7
Ki =
6.Hard
R(s)+Kp​ + Ki​/s5s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 5/(s + 1)ωₙ = 5 rad/sζ = 0.6
Ki =

Answer key

  1. 1. Kv = 0.2
  2. 2. Kv = 4
  3. 3. ess = 3
  4. 4. K = 180
  5. 5. Ki = 8
  6. 6. Ki = 5

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