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)+C(s)G(s)C(s)−

The controller C(s) = 10(s + 3)/s drives the plant G(s) = 2/(s(s + 2)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 10(s + 3)/sG(s) = 2/(s(s + 2))
N =
2.Easy
R(s)+10(s + 4)s(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 4)/(s(s + 2)). Find the velocity error constant Kv.

G(s) = 10(s + 4)/(s(s + 2))
Kv =
3.Medium
R(s)+K = 100.51s + 1C(s)−

A process G(s) = 0.5/(1s + 1) is controlled by a proportional gain K = 10 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 0.5/(1s + 1)K = 10
ess = %
4.Medium
r(t) = 4·t²/2+20(s + 4)s²(s + 1)C(s)−

The parabolic input r(t) = 4·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 4)/(s²(s + 1)). Find the steady-state error ess.

G(s) = 20(s + 4)/(s²(s + 1))r(t) = 4·t²/2
ess =
5.Hard
r(t) = 2t+1 + 4/s10(s + 1)(s + 5)C(s)−

A PI controller C(s) = 1 + 4/s drives the plant G(s) = 10/((s + 1)(s + 5)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 2t.

C(s) = 1 + 4/sG(s) = 10/((s + 1)(s + 5))r(t) = 2t
ess =
6.Hard
R(s)+K = 404s(s + 10)C(s)−

A proportional controller K = 40 drives the plant G(s) = 4/(s(s + 10)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 4/(s(s + 10))K = 40
ts = s

Answer key

  1. 1. N = 2
  2. 2. Kv = 20
  3. 3. ess = 16.67 %
  4. 4. ess = 0.05
  5. 5. ess = 0.25
  6. 6. ts = 0.8 s

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