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)C(s)−

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

G(s) = 20/(s(s + 4))
Kv =
2.Easy
r(t) = 5+K = 510s + 10y(t)−

A proportional controller K = 5 drives the plant G(s) = 10/(s + 10) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 10/(s + 10)K = 5r = 5
y(∞) =
3.Medium
r(t) = 1 + 3t+2s(s + 1)C(s)−

The input r(t) = 1 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 2/(s(s + 1)). Find the steady-state error ess.

G(s) = 2/(s(s + 1))r(t) = 1 + 3t
ess =
4.Medium
r(t) = 10 + 3t+4s(s + 1)(s + 5)C(s)−

The input r(t) = 10 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4/(s(s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 4/(s(s + 1)(s + 5))r(t) = 10 + 3t
ess =
5.Hard
R(s)+Kp​ + Ki​/s4s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.5. Find the integral gain Ki.

G(s) = 4/(s + 1)ωₙ = 3 rad/sζ = 0.5
Ki =
6.Hard
R(s)+K5s(s + 2)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 2)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.4.

G(s) = 5/(s(s + 2))ζ = 0.4
K =

Answer key

  1. 1. Kv = 5
  2. 2. y(∞) = 4.167
  3. 3. ess = 1.5
  4. 4. ess = 3.75
  5. 5. Ki = 2.25
  6. 6. K = 1.25

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