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 + 1)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2/((s + 1)(s + 5)). Find the position error constant Kp.

G(s) = 2/((s + 1)(s + 5))
Kp =
2.Easy
r(t) = 5+40(s + 2)(s + 5)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 40/((s + 2)(s + 5)). Find the steady-state error ess.

G(s) = 40/((s + 2)(s + 5))r(t) = 5·u(t)
ess =
3.Medium
R(s)+K = 440.5s + 1C(s)−

A process G(s) = 4/(0.5s + 1) is controlled by a proportional gain K = 4 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 4/(0.5s + 1)K = 4
ess = %
4.Medium
r(t) = 1 + 3t+20s(s + 1)(s + 2)C(s)−

The input r(t) = 1 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20/(s(s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 20/(s(s + 1)(s + 2))r(t) = 1 + 3t
ess =
5.Hard
R(s)+K = 254s(s + 10)C(s)−

A proportional controller K = 25 drives the plant G(s) = 4/(s(s + 10)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 4/(s(s + 10))K = 25
tp = s
6.Hard
R(s)+K = 165s(s + 4)C(s)−

A proportional controller K = 16 drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 4))K = 16
ts = s

Answer key

  1. 1. Kp = 0.4
  2. 2. ess = 1
  3. 3. ess = 5.882 %
  4. 4. ess = 0.3
  5. 5. tp = 0.3628 s
  6. 6. ts = 2 s

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