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)+20(s + 1)s²(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/(s²(s + 5)). Find the acceleration error constant Ka.

G(s) = 20(s + 1)/(s²(s + 5))
Ka =
2.Easy
r(t) = 1+K = 51s + 2y(t)−

A proportional controller K = 5 drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. A step of size 1 is applied. Find the final value of the output y(∞).

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

The input r(t) = 2 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 10/(s(s + 1)). Find the steady-state error ess.

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

A unity-feedback loop with G(s) = K/(s(s + 2)) must follow the ramp r(t) = 5t with a steady-state error of 0.5. Find the gain K.

G(s) = K/(s(s + 2))r(t) = 5tess = 0.5
K =
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​/s2s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 2/(s + 1)ωₙ = 8 rad/sζ = 0.7
Ki =

Answer key

  1. 1. Ka = 4
  2. 2. y(∞) = 0.7143
  3. 3. ess = 0.3
  4. 4. K = 20
  5. 5. K = 1.111
  6. 6. Ki = 32

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