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)+20s(s + 4)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s(s + 4)(s + 10)). Find the velocity error constant Kv.

G(s) = 20/(s(s + 4)(s + 10))
Kv =
2.Easy
R(s)+10(s + 5)(s + 10)C(s)−

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

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

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 4(s + 2)/(s²(s + 2)). Find the steady-state error ess.

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

The parabolic input r(t) = 5·t²/2 is applied to a unity-feedback loop with G(s) = 4(s + 1)/(s²(s + 10)). Find the steady-state error ess.

G(s) = 4(s + 1)/(s²(s + 10))r(t) = 5·t²/2
ess =
5.Hard
R(s)+K = 45s(s + 5)C(s)−

A proportional controller K = 4 drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 5/(s(s + 5))K = 4
%OS = %
6.Hard
R(s)+5 + Kd​s4s(s + 2)C(s)−

A PD controller C(s) = 5 + Kd·s drives the plant G(s) = 4/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 4/(s(s + 2))Kp = 5ζ = 0.8
Kd = s

Answer key

  1. 1. Kv = 0.5
  2. 2. Kp = 0.2
  3. 3. ess = 2.5
  4. 4. ess = 12.5
  5. 5. %OS = 12.03 %
  6. 6. Kd = 1.289 s

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