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)+10(s + 1)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 10(s + 1)/(s²(s + 4))
Ka =
2.Easy
r(t) = 10+K = 32s + 1y(t)−

A proportional controller K = 3 drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 1)K = 3r = 10
y(∞) =
3.Medium
R(s)+K = 250.5s + 1C(s)−

A process G(s) = 5/(0.5s + 1) is controlled by a proportional gain K = 2 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 5/(0.5s + 1)K = 2
ess = %
4.Medium
r(t) = 1 + 4t+50(s + 1)s(s + 4)C(s)−

The input r(t) = 1 + 4t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50(s + 1)/(s(s + 4)). Find the steady-state error ess.

G(s) = 50(s + 1)/(s(s + 4))r(t) = 1 + 4t
ess =
5.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 = 6 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 2/(s + 5)ωₙ = 6 rad/sζ = 0.8
Kp =
6.Hard
R(s)+K = 501s(s + 4)C(s)−

A proportional controller K = 50 drives the plant G(s) = 1/(s(s + 4)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 1/(s(s + 4))K = 50
%OS = %

Answer key

  1. 1. Ka = 2.5
  2. 2. y(∞) = 8.571
  3. 3. ess = 9.091 %
  4. 4. ess = 0.32
  5. 5. Kp = 2.3
  6. 6. %OS = 39.6 %

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