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)+4(s + 5)s²(s + 1)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 5)/(s²(s + 1)(s + 4)). What is the system type N?

G(s) = 4(s + 5)/(s²(s + 1)(s + 4))
N =
2.Easy
r(t) = 5+10(s + 1)(s + 5)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 10/((s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 10/((s + 1)(s + 5))r(t) = 5·u(t)
ess =
3.Medium
r(t) = 0.5·t²/2+20(s + 1)s²(s + 4)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 1)/(s²(s + 4)). Find the steady-state error ess.

G(s) = 20(s + 1)/(s²(s + 4))r(t) = 0.5·t²/2
ess =
4.Medium
R(s)+K(s + 4)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 4)(s + 10)). Find the gain K that gives a steady-state error of 5 % for a step input.

G(s) = K/((s + 4)(s + 10))ess = 5 %
K =
5.Hard
R(s)+K5s(s + 5)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.4.

G(s) = 5/(s(s + 5))ζ = 0.4
K =
6.Hard
R(s)+Kp​ + Ki​/s5s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.8. Find the integral gain Ki.

G(s) = 5/(s + 4)ωₙ = 5 rad/sζ = 0.8
Ki =

Answer key

  1. 1. N = 2
  2. 2. ess = 1.667
  3. 3. ess = 0.1
  4. 4. K = 760
  5. 5. K = 7.812
  6. 6. Ki = 5

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