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)+4(s + 4)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 4)/(s²(s + 10)). Find the acceleration error constant Ka.

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

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 4)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 10(s + 4)/(s²(s + 4))
Ka =
3.Medium
r(t) = 2t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 2t with a steady-state error of 0.1. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 2tess = 0.1
K =
4.Medium
R(s)+K(s + 1)(s + 2)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 2)). Find the gain K that gives a steady-state error of 10 % for a step input.

G(s) = K/((s + 1)(s + 2))ess = 10 %
K =
5.Hard
R(s)+Kp​ + Ki​/s4s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.8. Find the integral gain Ki.

G(s) = 4/(s + 5)ωₙ = 5 rad/sζ = 0.8
Ki =
6.Hard
r(t) = 1t+2 + 0.5/s4(s + 1)(s + 10)C(s)−

A PI controller C(s) = 2 + 0.5/s drives the plant G(s) = 4/((s + 1)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 1t.

C(s) = 2 + 0.5/sG(s) = 4/((s + 1)(s + 10))r(t) = 1t
ess =

Answer key

  1. 1. Ka = 1.6
  2. 2. Ka = 10
  3. 3. K = 100
  4. 4. K = 18
  5. 5. Ki = 6.25
  6. 6. ess = 5

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