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)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 2)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 5(s + 2)/(s²(s + 4))
Ka =
2.Easy
R(s)+4(s + 3)s²(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 3)/(s²(s + 2)). Find the acceleration error constant Ka.

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

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 3)/(s²(s + 2)). Find the steady-state error ess.

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

A unity-feedback loop has G(s) = K/((s + 2)(s + 10)). Find the gain K that gives a steady-state error of 25 % for a step input.

G(s) = K/((s + 2)(s + 10))ess = 25 %
K =
5.Hard
R(s)+K = 255s(s + 6)C(s)−

A proportional controller K = 25 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 = 25
%OS = %
6.Hard
R(s)+Kp​ + Ki​/s4s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 4/(s + 5)ωₙ = 3 rad/sζ = 1
Ki =

Answer key

  1. 1. Ka = 2.5
  2. 2. Ka = 6
  3. 3. ess = 1.333
  4. 4. K = 60
  5. 5. %OS = 41.68 %
  6. 6. Ki = 2.25

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