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+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 5 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 1)K = 3r = 5
y(∞) =
2.Easy
R(s)+4(s + 3)s(s + 1)(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 3)/(s(s + 1)(s + 2)). Find the velocity error constant Kv.

G(s) = 4(s + 3)/(s(s + 1)(s + 2))
Kv =
3.Medium
r(t) = 4t+2(s + 1)s(s + 1)(s + 10)C(s)−

A ramp r(t) = 4t is applied to a unity-feedback loop with G(s) = 2(s + 1)/(s(s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 2(s + 1)/(s(s + 1)(s + 10))r(t) = 4t
ess =
4.Medium
R(s)+K = 920.5s + 1C(s)−

A process G(s) = 2/(0.5s + 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) = 2/(0.5s + 1)K = 9
ess = %
5.Hard
R(s)+K = 505s(s + 10)C(s)−

A proportional controller K = 50 drives the plant G(s) = 5/(s(s + 10)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 5/(s(s + 10))K = 50
%OS = %
6.Hard
R(s)+K10s(s + 4)C(s)−

A proportional controller K drives the plant G(s) = 10/(s(s + 4)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 10/(s(s + 4))ζ = 0.8
K =

Answer key

  1. 1. y(∞) = 4.286
  2. 2. Kv = 6
  3. 3. ess = 20
  4. 4. ess = 5.263 %
  5. 5. %OS = 35.09 %
  6. 6. K = 0.625

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