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) = 10+5(s + 3)(s + 2)(s + 10)C(s)−

A step of size 10 is applied to a unity-feedback loop with G(s) = 5(s + 3)/((s + 2)(s + 10)). Find the steady-state error ess.

G(s) = 5(s + 3)/((s + 2)(s + 10))r(t) = 10·u(t)
ess =
2.Easy
R(s)+C(s)G(s)C(s)−

The controller C(s) = 10 drives the plant G(s) = 1/(s(s + 5)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 10G(s) = 1/(s(s + 5))
N =
3.Medium
r(t) = 10t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 10t with a steady-state error of 0.5. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 10tess = 0.5
K =
4.Medium
r(t) = 10 + 5t+4(s + 3)s(s + 10)C(s)−

The input r(t) = 10 + 5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4(s + 3)/(s(s + 10)). Find the steady-state error ess.

G(s) = 4(s + 3)/(s(s + 10))r(t) = 10 + 5t
ess =
5.Hard
R(s)+K = 21s(s + 2)C(s)−

A proportional controller K = 2 drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 2))K = 2
ts = s
6.Hard
R(s)+K = 402s(s + 10)C(s)−

A proportional controller K = 40 drives the plant G(s) = 2/(s(s + 10)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 2/(s(s + 10))K = 40
%OS = %

Answer key

  1. 1. ess = 5.714
  2. 2. N = 1
  3. 3. K = 100
  4. 4. ess = 4.167
  5. 5. ts = 4 s
  6. 6. %OS = 12.03 %

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