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) = 2+K = 12s + 1y(t)−

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

G(s) = 2/(s + 1)K = 1r = 2
y(∞) =
2.Easy
R(s)+50s(s + 1)C(s)−

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

G(s) = 50/(s(s + 1))
Kv =
3.Medium
R(s)+K = 115s + 1C(s)−

A process G(s) = 1/(5s + 1) is controlled by a proportional gain K = 1 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 1/(5s + 1)K = 1
ess = %
4.Medium
r(t) = 3t+50(s + 4)s(s + 5)C(s)−

A ramp r(t) = 3t is applied to a unity-feedback loop with G(s) = 50(s + 4)/(s(s + 5)). Find the steady-state error ess.

G(s) = 50(s + 4)/(s(s + 5))r(t) = 3t
ess =
5.Hard
r(t) = 5t+5 + 5/s5(s + 2)(s + 4)C(s)−

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

C(s) = 5 + 5/sG(s) = 5/((s + 2)(s + 4))r(t) = 5t
ess =
6.Hard
R(s)+K = 251s(s + 8)C(s)−

A proportional controller K = 25 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 = 25
ts = s

Answer key

  1. 1. y(∞) = 1.333
  2. 2. Kv = 50
  3. 3. ess = 50 %
  4. 4. ess = 0.075
  5. 5. ess = 1.6
  6. 6. ts = 1 s

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