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) = 5+40(s + 2)(s + 4)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 40/((s + 2)(s + 4)). Find the steady-state error ess.

G(s) = 40/((s + 2)(s + 4))r(t) = 5·u(t)
ess =
2.Easy
R(s)+5s³(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/(s³(s + 5)). What is the system type N?

G(s) = 5/(s³(s + 5))
N =
3.Medium
r(t) = 10·t²/2+20(s + 3)s²(s + 4)C(s)−

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 3)/(s²(s + 4)). Find the steady-state error ess.

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

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 1t with a steady-state error of 0.2. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 1tess = 0.2
K =
5.Hard
R(s)+K2s(s + 5)C(s)−

A proportional controller K drives the plant G(s) = 2/(s(s + 5)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 2/(s(s + 5))ζ = 0.707
K =
6.Hard
R(s)+K = 45s(s + 2)C(s)−

A proportional controller K = 4 drives the plant G(s) = 5/(s(s + 2)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 5/(s(s + 2))K = 4
%OS = %

Answer key

  1. 1. ess = 0.8333
  2. 2. N = 3
  3. 3. ess = 0.6667
  4. 4. K = 25
  5. 5. K = 6.252
  6. 6. %OS = 48.64 %

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