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)+C(s)G(s)C(s)−

The controller C(s) = 10(s + 1)/s drives the plant G(s) = 1/(s(s + 10)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 10(s + 1)/sG(s) = 1/(s(s + 10))
N =
2.Easy
R(s)+20(s + 1)s³(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/(s³(s + 2)). What is the system type N?

G(s) = 20(s + 1)/(s³(s + 2))
N =
3.Medium
r(t) = 1t+20(s + 3)s(s + 1)C(s)−

A ramp r(t) = 1t is applied to a unity-feedback loop with G(s) = 20(s + 3)/(s(s + 1)). Find the steady-state error ess.

G(s) = 20(s + 3)/(s(s + 1))r(t) = 1t
ess =
4.Medium
r(t) = 0.5·t²/2+2(s + 5)s²(s + 1)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 2(s + 5)/(s²(s + 1)). Find the steady-state error ess.

G(s) = 2(s + 5)/(s²(s + 1))r(t) = 0.5·t²/2
ess =
5.Hard
R(s)+Kp​ + Ki​/s1s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 1. Find the proportional gain Kp.

G(s) = 1/(s + 5)ωₙ = 10 rad/sζ = 1
Kp =
6.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 = 0.6. Find the proportional gain Kp.

G(s) = 5/(s + 4)ωₙ = 4 rad/sζ = 0.6
Kp =

Answer key

  1. 1. N = 2
  2. 2. N = 3
  3. 3. ess = 0.01667
  4. 4. ess = 0.05
  5. 5. Kp = 15
  6. 6. Kp = 0.16

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