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)+40(s + 3)(s + 4)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 40(s + 3)/((s + 4)(s + 5)). Find the position error constant Kp.

G(s) = 40(s + 3)/((s + 4)(s + 5))
Kp =
2.Easy
R(s)+4(s + 3)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 3)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 4(s + 3)/(s²(s + 4))
Ka =
3.Medium
R(s)+K(s + 2)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 2)(s + 5)). Find the gain K that gives a steady-state error of 10 % for a step input.

G(s) = K/((s + 2)(s + 5))ess = 10 %
K =
4.Medium
R(s)+K = 945s + 1C(s)−

A process G(s) = 4/(5s + 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) = 4/(5s + 1)K = 9
ess = %
5.Hard
r(t) = 2t+5 + 5/s4(s + 2)(s + 5)C(s)−

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

C(s) = 5 + 5/sG(s) = 4/((s + 2)(s + 5))r(t) = 2t
ess =
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 = 4 rad/s and ζ = 0.8. Find the integral gain Ki.

G(s) = 2/(s + 1)ωₙ = 4 rad/sζ = 0.8
Ki =

Answer key

  1. 1. Kp = 6
  2. 2. Ka = 3
  3. 3. K = 90
  4. 4. ess = 2.703 %
  5. 5. ess = 1
  6. 6. Ki = 8

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