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)+10(s + 4)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10/((s + 4)(s + 5)). Find the position error constant Kp.

G(s) = 10/((s + 4)(s + 5))
Kp =
2.Easy
R(s)+10s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10/(s(s + 10)). Find the velocity error constant Kv.

G(s) = 10/(s(s + 10))
Kv =
3.Medium
R(s)+K(s + 2)(s + 4)C(s)−

A unity-feedback loop has G(s) = K/((s + 2)(s + 4)). Find the gain K that gives a steady-state error of 4 % for a step input.

G(s) = K/((s + 2)(s + 4))ess = 4 %
K =
4.Medium
r(t) = 2 + 0.5t+10s(s + 5)C(s)−

The input r(t) = 2 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 10/(s(s + 5)). Find the steady-state error ess.

G(s) = 10/(s(s + 5))r(t) = 2 + 0.5t
ess =
5.Hard
R(s)+Kp​ + Ki​/s5s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 4 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 5/(s + 4)ωₙ = 4 rad/sζ = 1
Ki =
6.Hard
R(s)+K = 161s(s + 6)C(s)−

A proportional controller K = 16 drives the plant G(s) = 1/(s(s + 6)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 1/(s(s + 6))K = 16
%OS = %

Answer key

  1. 1. Kp = 0.5
  2. 2. Kv = 1
  3. 3. K = 192
  4. 4. ess = 0.25
  5. 5. Ki = 3.2
  6. 6. %OS = 2.838 %

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