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)+10s(s + 4)C(s)−

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

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

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 3)/(s²(s + 2)). Find the acceleration error constant Ka.

G(s) = 5(s + 3)/(s²(s + 2))
Ka =
3.Medium
r(t) = 10t+Ks(s + 1)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 10t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 10tess = 0.05
K =
4.Medium
r(t) = 1·t²/2+2(s + 5)s²(s + 5)C(s)−

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

G(s) = 2(s + 5)/(s²(s + 5))r(t) = 1·t²/2
ess =
5.Hard
R(s)+K10s(s + 4)C(s)−

A proportional controller K drives the plant G(s) = 10/(s(s + 4)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 10/(s(s + 4))ζ = 0.6
K =
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 = 3 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 5/(s + 4)ωₙ = 3 rad/sζ = 1
Ki =

Answer key

  1. 1. Kv = 2.5
  2. 2. Ka = 7.5
  3. 3. K = 200
  4. 4. ess = 0.5
  5. 5. K = 1.111
  6. 6. Ki = 1.8

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