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 + 2)(s + 1)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 2)/((s + 1)(s + 10)). Find the position error constant Kp.

G(s) = 20(s + 2)/((s + 1)(s + 10))
Kp =
2.Easy
R(s)+20(s + 1)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/(s²(s + 10)). Find the acceleration error constant Ka.

G(s) = 20(s + 1)/(s²(s + 10))
Ka =
3.Medium
R(s)+K = 951s + 1C(s)−

A process G(s) = 5/(1s + 1) is controlled by a proportional gain K = 9 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 5/(1s + 1)K = 9
ess = %
4.Medium
r(t) = 4t+10(s + 3)s(s + 10)C(s)−

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

G(s) = 10(s + 3)/(s(s + 10))r(t) = 4t
ess =
5.Hard
R(s)+Kp​ + Ki​/s5s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 5/(s + 5)ωₙ = 8 rad/sζ = 0.6
Kp =
6.Hard
r(t) = 2t+5 + 1/s4(s + 1)(s + 5)C(s)−

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

C(s) = 5 + 1/sG(s) = 4/((s + 1)(s + 5))r(t) = 2t
ess =

Answer key

  1. 1. Kp = 4
  2. 2. Ka = 2
  3. 3. ess = 2.174 %
  4. 4. ess = 1.333
  5. 5. Kp = 0.92
  6. 6. ess = 2.5

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