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)+50(s + 1)s(s + 1)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50(s + 1)/(s(s + 1)(s + 10)). Find the velocity error constant Kv.

G(s) = 50(s + 1)/(s(s + 1)(s + 10))
Kv =
2.Easy
r(t) = 2+K = 12s + 10y(t)−

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

G(s) = 2/(s + 10)K = 1r = 2
y(∞) =
3.Medium
r(t) = 1t+Ks(s + 4)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 4)) must follow the ramp r(t) = 1t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 4))r(t) = 1tess = 0.05
K =
4.Medium
r(t) = 2 + 4t+20(s + 1)s(s + 1)(s + 10)C(s)−

The input r(t) = 2 + 4t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20(s + 1)/(s(s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 20(s + 1)/(s(s + 1)(s + 10))r(t) = 2 + 4t
ess =
5.Hard
r(t) = 5t+1 + 1/s5(s + 1)(s + 4)C(s)−

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

C(s) = 1 + 1/sG(s) = 5/((s + 1)(s + 4))r(t) = 5t
ess =
6.Hard
r(t) = 5t+2 + 0.5/s4(s + 1)(s + 2)C(s)−

A PI controller C(s) = 2 + 0.5/s drives the plant G(s) = 4/((s + 1)(s + 2)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 2 + 0.5/sG(s) = 4/((s + 1)(s + 2))r(t) = 5t
ess =

Answer key

  1. 1. Kv = 5
  2. 2. y(∞) = 0.3333
  3. 3. K = 80
  4. 4. ess = 2
  5. 5. ess = 4
  6. 6. ess = 5

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