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)+4(s + 3)s²(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 3)/(s²(s + 2)). Find the acceleration error constant Ka.

G(s) = 4(s + 3)/(s²(s + 2))
Ka =
2.Easy
R(s)+4(s + 2)(s + 5)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 2)/((s + 5)(s + 10)). Find the position error constant Kp.

G(s) = 4(s + 2)/((s + 5)(s + 10))
Kp =
3.Medium
R(s)+K = 2051s + 1C(s)−

A process G(s) = 5/(1s + 1) is controlled by a proportional gain K = 20 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 5/(1s + 1)K = 20
ess = %
4.Medium
r(t) = 4·t²/2+10(s + 2)s²(s + 1)C(s)−

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

G(s) = 10(s + 2)/(s²(s + 1))r(t) = 4·t²/2
ess =
5.Hard
R(s)+K = 84s(s + 8)C(s)−

A proportional controller K = 8 drives the plant G(s) = 4/(s(s + 8)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 4/(s(s + 8))K = 8
%OS = %
6.Hard
R(s)+Kp​ + Ki​/s2s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 2/(s + 5)ωₙ = 8 rad/sζ = 0.7
Kp =

Answer key

  1. 1. Ka = 6
  2. 2. Kp = 0.16
  3. 3. ess = 0.9901 %
  4. 4. ess = 0.2
  5. 5. %OS = 4.321 %
  6. 6. Kp = 3.1

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