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)+2(s + 4)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 4)/(s²(s + 10)). Find the acceleration error constant Ka.

G(s) = 2(s + 4)/(s²(s + 10))
Ka =
2.Easy
R(s)+C(s)G(s)C(s)−

The controller C(s) = 2 drives the plant G(s) = 1/(s(s + 4)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 2G(s) = 1/(s(s + 4))
N =
3.Medium
r(t) = 5·t²/2+10(s + 4)s²(s + 4)C(s)−

The parabolic input r(t) = 5·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 4)/(s²(s + 4)). Find the steady-state error ess.

G(s) = 10(s + 4)/(s²(s + 4))r(t) = 5·t²/2
ess =
4.Medium
r(t) = 5t+Ks(s + 10)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 10)) must follow the ramp r(t) = 5t with a steady-state error of 0.1. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 5tess = 0.1
K =
5.Hard
R(s)+K = 202s(s + 4)C(s)−

A proportional controller K = 20 drives the plant G(s) = 2/(s(s + 4)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 2/(s(s + 4))K = 20
ts = s
6.Hard
r(t) = 5t+1 + 2/s20(s + 2)(s + 10)C(s)−

A PI controller C(s) = 1 + 2/s drives the plant G(s) = 20/((s + 2)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 1 + 2/sG(s) = 20/((s + 2)(s + 10))r(t) = 5t
ess =

Answer key

  1. 1. Ka = 0.8
  2. 2. N = 1
  3. 3. ess = 0.5
  4. 4. K = 500
  5. 5. ts = 2 s
  6. 6. ess = 2.5

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