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)+20(s + 4)s(s + 1)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 4)/(s(s + 1)(s + 5)). Find the velocity error constant Kv.

G(s) = 20(s + 4)/(s(s + 1)(s + 5))
Kv =
2.Easy
r(t) = 2+20(s + 3)(s + 2)(s + 5)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 20(s + 3)/((s + 2)(s + 5)). Find the steady-state error ess.

G(s) = 20(s + 3)/((s + 2)(s + 5))r(t) = 2·u(t)
ess =
3.Medium
r(t) = 5 + 1t+50(s + 3)s(s + 1)(s + 5)C(s)−

The input r(t) = 5 + 1t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50(s + 3)/(s(s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 50(s + 3)/(s(s + 1)(s + 5))r(t) = 5 + 1t
ess =
4.Medium
R(s)+K(s + 1)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 5)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 1)(s + 5))ess = 2 %
K =
5.Hard
R(s)+Kp​ + Ki​/s10s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 10/(s + 1)ωₙ = 8 rad/sζ = 0.8
Kp =
6.Hard
R(s)+K = 1610s(s + 5)C(s)−

A proportional controller K = 16 drives the plant G(s) = 10/(s(s + 5)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 10/(s(s + 5))K = 16
%OS = %

Answer key

  1. 1. Kv = 16
  2. 2. ess = 0.2857
  3. 3. ess = 0.03333
  4. 4. K = 245
  5. 5. Kp = 1.18
  6. 6. %OS = 53.08 %

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