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)+5(s + 2)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/((s + 2)(s + 10)). Find the position error constant Kp.

G(s) = 5/((s + 2)(s + 10))
Kp =
2.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 =
3.Medium
r(t) = 2 + 0.5t+2(s + 3)s(s + 1)C(s)−

The input r(t) = 2 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 2(s + 3)/(s(s + 1)). Find the steady-state error ess.

G(s) = 2(s + 3)/(s(s + 1))r(t) = 2 + 0.5t
ess =
4.Medium
r(t) = 4·t²/2+10(s + 5)s²(s + 5)C(s)−

The parabolic input r(t) = 4·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 5)/(s²(s + 5)). Find the steady-state error ess.

G(s) = 10(s + 5)/(s²(s + 5))r(t) = 4·t²/2
ess =
5.Hard
R(s)+K = 51s(s + 2)C(s)−

A proportional controller K = 5 drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 1/(s(s + 2))K = 5
tp = s
6.Hard
R(s)+Kp​ + Ki​/s4s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 4) 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) = 4/(s + 4)ωₙ = 5 rad/sζ = 0.6
Ki =

Answer key

  1. 1. Kp = 0.25
  2. 2. Kv = 0.2
  3. 3. ess = 0.08333
  4. 4. ess = 0.4
  5. 5. tp = 1.571 s
  6. 6. Ki = 6.25

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