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)+2(s + 2)s²(s + 5)C(s)−

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

G(s) = 2(s + 2)/(s²(s + 5))
Ka =
2.Easy
R(s)+50(s + 2)(s + 5)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50(s + 2)/((s + 5)(s + 10)). Find the position error constant Kp.

G(s) = 50(s + 2)/((s + 5)(s + 10))
Kp =
3.Medium
r(t) = 10 + 3t+20s(s + 2)C(s)−

The input r(t) = 10 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20/(s(s + 2)). Find the steady-state error ess.

G(s) = 20/(s(s + 2))r(t) = 10 + 3t
ess =
4.Medium
R(s)+K(s + 5)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 5)(s + 10)). Find the gain K that gives a steady-state error of 20 % for a step input.

G(s) = K/((s + 5)(s + 10))ess = 20 %
K =
5.Hard
r(t) = 5t+1 + 4/s4(s + 2)(s + 4)C(s)−

A PI controller C(s) = 1 + 4/s drives the plant G(s) = 4/((s + 2)(s + 4)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 1 + 4/sG(s) = 4/((s + 2)(s + 4))r(t) = 5t
ess =
6.Hard
R(s)+Kp​ + Ki​/s2s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 5) 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 + 5)ωₙ = 8 rad/sζ = 0.7
Ki =

Answer key

  1. 1. Ka = 0.8
  2. 2. Kp = 2
  3. 3. ess = 0.3
  4. 4. K = 200
  5. 5. ess = 2.5
  6. 6. Ki = 32

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