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+50(s + 1)(s + 2)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 50/((s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 50/((s + 1)(s + 2))r(t) = 1·u(t)
ess =
2.Easy
R(s)+10(s + 1)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 10(s + 1)/(s²(s + 4))
Ka =
3.Medium
r(t) = 5·t²/2+2(s + 4)s²(s + 2)C(s)−

The parabolic input r(t) = 5·t²/2 is applied to a unity-feedback loop with G(s) = 2(s + 4)/(s²(s + 2)). Find the steady-state error ess.

G(s) = 2(s + 4)/(s²(s + 2))r(t) = 5·t²/2
ess =
4.Medium
R(s)+K(s + 4)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 4)(s + 5)). Find the gain K that gives a steady-state error of 20 % for a step input.

G(s) = K/((s + 4)(s + 5))ess = 20 %
K =
5.Hard
R(s)+K = 252s(s + 8)C(s)−

A proportional controller K = 25 drives the plant G(s) = 2/(s(s + 8)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 2/(s(s + 8))K = 25
ts = s
6.Hard
R(s)+Kp​ + Ki​/s1s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.8. Find the integral gain Ki.

G(s) = 1/(s + 2)ωₙ = 5 rad/sζ = 0.8
Ki =

Answer key

  1. 1. ess = 0.03846
  2. 2. Ka = 2.5
  3. 3. ess = 1.25
  4. 4. K = 80
  5. 5. ts = 1 s
  6. 6. Ki = 25

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