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 + 2)s(s + 3)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 2)/(s(s + 3)(s + 4)). What is the system type N?

G(s) = 10(s + 2)/(s(s + 3)(s + 4))
N =
2.Easy
R(s)+50s(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50/(s(s + 1)). Find the velocity error constant Kv.

G(s) = 50/(s(s + 1))
Kv =
3.Medium
r(t) = 1t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 1t with a steady-state error of 0.01. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 1tess = 0.01
K =
4.Medium
r(t) = 10·t²/2+5(s + 4)s²(s + 5)C(s)−

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

G(s) = 5(s + 4)/(s²(s + 5))r(t) = 10·t²/2
ess =
5.Hard
R(s)+K5s(s + 8)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 8)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 5/(s(s + 8))ζ = 0.6
K =
6.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 = 5 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 1/(s + 5)ωₙ = 5 rad/sζ = 0.6
Ki =

Answer key

  1. 1. N = 1
  2. 2. Kv = 50
  3. 3. K = 500
  4. 4. ess = 2.5
  5. 5. K = 8.889
  6. 6. Ki = 25

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