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(t) = 1+K = 41s + 1y(t)−

A proportional controller K = 4 drives the plant G(s) = 1/(s + 1) in a unity-feedback loop. A step of size 1 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 1)K = 4r = 1
y(∞) =
2.Easy
R(s)+40(s + 2)(s + 1)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 40(s + 2)/((s + 1)(s + 5)). Find the position error constant Kp.

G(s) = 40(s + 2)/((s + 1)(s + 5))
Kp =
3.Medium
r(t) = 5 + 3t+20s(s + 1)(s + 2)C(s)−

The input r(t) = 5 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20/(s(s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 20/(s(s + 1)(s + 2))r(t) = 5 + 3t
ess =
4.Medium
r(t) = 2·t²/2+20(s + 4)s²(s + 2)C(s)−

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 4)/(s²(s + 2)). Find the steady-state error ess.

G(s) = 20(s + 4)/(s²(s + 2))r(t) = 2·t²/2
ess =
5.Hard
R(s)+K = 24s(s + 2)C(s)−

A proportional controller K = 2 drives the plant G(s) = 4/(s(s + 2)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 4/(s(s + 2))K = 2
%OS = %
6.Hard
R(s)+Kp​ + Ki​/s4s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 4/(s + 4)ωₙ = 5 rad/sζ = 0.6
Ki =

Answer key

  1. 1. y(∞) = 0.8
  2. 2. Kp = 16
  3. 3. ess = 0.3
  4. 4. ess = 0.05
  5. 5. %OS = 30.5 %
  6. 6. Ki = 6.25

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