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)+50(s + 5)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50/((s + 5)(s + 10)). Find the position error constant Kp.

G(s) = 50/((s + 5)(s + 10))
Kp =
2.Easy
r(t) = 5+K = 205s + 10y(t)−

A proportional controller K = 20 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 10)K = 20r = 5
y(∞) =
3.Medium
r(t) = 10t+Ks(s + 2)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 2)) must follow the ramp r(t) = 10t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 2))r(t) = 10tess = 0.05
K =
4.Medium
r(t) = 1·t²/2+20(s + 5)s²(s + 10)C(s)−

The parabolic input r(t) = 1·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 5)/(s²(s + 10)). Find the steady-state error ess.

G(s) = 20(s + 5)/(s²(s + 10))r(t) = 1·t²/2
ess =
5.Hard
R(s)+K = 14s(s + 2)C(s)−

A proportional controller K = 1 drives the plant G(s) = 4/(s(s + 2)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 4/(s(s + 2))K = 1
%OS = %
6.Hard
R(s)+K = 104s(s + 8)C(s)−

A proportional controller K = 10 drives the plant G(s) = 4/(s(s + 8)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 4/(s(s + 8))K = 10
%OS = %

Answer key

  1. 1. Kp = 1
  2. 2. y(∞) = 4.545
  3. 3. K = 400
  4. 4. ess = 0.1
  5. 5. %OS = 16.3 %
  6. 6. %OS = 7.691 %

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