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 = 110s + 1y(t)−

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

G(s) = 10/(s + 1)K = 1r = 2
y(∞) =
2.Easy
R(s)+4(s + 2)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4/((s + 2)(s + 4)). Find the position error constant Kp.

G(s) = 4/((s + 2)(s + 4))
Kp =
3.Medium
R(s)+K = 1010.5s + 1C(s)−

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

A unity-feedback loop with G(s) = K/(s(s + 2)) must follow the ramp r(t) = 1t with a steady-state error of 0.01. Find the gain K.

G(s) = K/(s(s + 2))r(t) = 1tess = 0.01
K =
5.Hard
R(s)+K = 255s(s + 5)C(s)−

A proportional controller K = 25 drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 5))K = 25
ts = s
6.Hard
R(s)+K = 164s(s + 6)C(s)−

A proportional controller K = 16 drives the plant G(s) = 4/(s(s + 6)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 4/(s(s + 6))K = 16
tp = s

Answer key

  1. 1. y(∞) = 1.818
  2. 2. Kp = 0.5
  3. 3. ess = 9.091 %
  4. 4. K = 200
  5. 5. ts = 1.6 s
  6. 6. tp = 0.4236 s

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