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) = 1+K = 21s + 5y(t)−

A proportional controller K = 2 drives the plant G(s) = 1/(s + 5) in a unity-feedback loop. A step of size 1 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 5)K = 2r = 1
y(∞) =
2.Easy
R(s)+20(s + 1)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/(s²(s + 10)). Find the acceleration error constant Ka.

G(s) = 20(s + 1)/(s²(s + 10))
Ka =
3.Medium
R(s)+K = 100.55s + 1C(s)−

A process G(s) = 0.5/(5s + 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/(5s + 1)K = 10
ess = %
4.Medium
r(t) = 5 + 0.5t+50(s + 4)s(s + 2)C(s)−

The input r(t) = 5 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50(s + 4)/(s(s + 2)). Find the steady-state error ess.

G(s) = 50(s + 4)/(s(s + 2))r(t) = 5 + 0.5t
ess =
5.Hard
R(s)+K = 161s(s + 2)C(s)−

A proportional controller K = 16 drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 1/(s(s + 2))K = 16
tp = s
6.Hard
R(s)+K = 162s(s + 8)C(s)−

A proportional controller K = 16 drives the plant G(s) = 2/(s(s + 8)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 2/(s(s + 8))K = 16
tp = s

Answer key

  1. 1. y(∞) = 0.2857
  2. 2. Ka = 2
  3. 3. ess = 16.67 %
  4. 4. ess = 0.005
  5. 5. tp = 0.8112 s
  6. 6. tp = 0.7854 s

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