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)+20s²(s + 6)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s²(s + 6)(s + 10)). What is the system type N?

G(s) = 20/(s²(s + 6)(s + 10))
N =
2.Easy
r(t) = 2+10(s + 5)(s + 10)C(s)−

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

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

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

G(s) = 2/(s(s + 2))r(t) = 4t
ess =
4.Medium
r(t) = 1 + 1t+50s(s + 2)C(s)−

The input r(t) = 1 + 1t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50/(s(s + 2)). Find the steady-state error ess.

G(s) = 50/(s(s + 2))r(t) = 1 + 1t
ess =
5.Hard
R(s)+K = 810s(s + 8)C(s)−

A proportional controller K = 8 drives the plant G(s) = 10/(s(s + 8)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 10/(s(s + 8))K = 8
tp = s
6.Hard
R(s)+K = 501s(s + 8)C(s)−

A proportional controller K = 50 drives the plant G(s) = 1/(s(s + 8)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 8))K = 50
ts = s

Answer key

  1. 1. N = 2
  2. 2. ess = 1.667
  3. 3. ess = 4
  4. 4. ess = 0.04
  5. 5. tp = 0.3927 s
  6. 6. ts = 1 s

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