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) = 5+K = 31s + 4y(t)−

A proportional controller K = 3 drives the plant G(s) = 1/(s + 4) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 4)K = 3r = 5
y(∞) =
2.Easy
R(s)+4s(s + 1)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4/(s(s + 1)(s + 5)). Find the velocity error constant Kv.

G(s) = 4/(s(s + 1)(s + 5))
Kv =
3.Medium
R(s)+K = 145s + 1C(s)−

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

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

G(s) = 10/(s(s + 2)(s + 4))r(t) = 5 + 2t
ess =
5.Hard
R(s)+K = 252s(s + 4)C(s)−

A proportional controller K = 25 drives the plant G(s) = 2/(s(s + 4)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 2/(s(s + 4))K = 25
%OS = %
6.Hard
R(s)+K = 405s(s + 8)C(s)−

A proportional controller K = 40 drives the plant G(s) = 5/(s(s + 8)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 5/(s(s + 8))K = 40
tp = s

Answer key

  1. 1. y(∞) = 2.143
  2. 2. Kv = 0.8
  3. 3. ess = 20 %
  4. 4. ess = 1.6
  5. 5. %OS = 39.6 %
  6. 6. tp = 0.2316 s

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