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)+20(s + 5)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/((s + 5)(s + 10)). Find the position error constant Kp.

G(s) = 20/((s + 5)(s + 10))
Kp =
2.Easy
r(t) = 1+40(s + 2)(s + 5)(s + 10)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 40(s + 2)/((s + 5)(s + 10)). Find the steady-state error ess.

G(s) = 40(s + 2)/((s + 5)(s + 10))r(t) = 1·u(t)
ess =
3.Medium
r(t) = 4t+4(s + 2)s(s + 1)(s + 5)C(s)−

A ramp r(t) = 4t is applied to a unity-feedback loop with G(s) = 4(s + 2)/(s(s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 4(s + 2)/(s(s + 1)(s + 5))r(t) = 4t
ess =
4.Medium
R(s)+K = 921s + 1C(s)−

A process G(s) = 2/(1s + 1) is controlled by a proportional gain K = 9 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 2/(1s + 1)K = 9
ess = %
5.Hard
R(s)+K = 105s(s + 6)C(s)−

A proportional controller K = 10 drives the plant G(s) = 5/(s(s + 6)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 5/(s(s + 6))K = 10
%OS = %
6.Hard
R(s)+Kp​ + Ki​/s2s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 2/(s + 4)ωₙ = 5 rad/sζ = 0.7
Ki =

Answer key

  1. 1. Kp = 0.4
  2. 2. ess = 0.3846
  3. 3. ess = 2.5
  4. 4. ess = 5.263 %
  5. 5. %OS = 22.95 %
  6. 6. Ki = 12.5

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