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(t) = 2+4(s + 2)(s + 10)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 4/((s + 2)(s + 10)). Find the steady-state error ess.

G(s) = 4/((s + 2)(s + 10))r(t) = 2·u(t)
ess =
2.Easy
r(t) = 10+K = 11s + 2y(t)−

A proportional controller K = 1 drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 2)K = 1r = 10
y(∞) =
3.Medium
r(t) = 1t+Ks(s + 1)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 1t with a steady-state error of 0.5. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 1tess = 0.5
K =
4.Medium
R(s)+K(s + 4)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 4)(s + 10)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 4)(s + 10))ess = 2 %
K =
5.Hard
R(s)+K = 1010s(s + 8)C(s)−

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

G(s) = 10/(s(s + 8))K = 10
%OS = %
6.Hard
R(s)+K = 41s(s + 2)C(s)−

A proportional controller K = 4 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 = 4
tp = s

Answer key

  1. 1. ess = 1.667
  2. 2. y(∞) = 3.333
  3. 3. K = 2
  4. 4. K = 1960
  5. 5. %OS = 25.38 %
  6. 6. tp = 1.814 s

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