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) = 10+K = 31s + 10y(t)−

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

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

The controller C(s) = 2(s + 5)/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 2(s + 5)/sG(s) = 1/(s + 2)
N =
3.Medium
r(t) = 10t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 10t with a steady-state error of 0.2. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 10tess = 0.2
K =
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 10 % for a step input.

G(s) = K/((s + 4)(s + 10))ess = 10 %
K =
5.Hard
R(s)+K = 44s(s + 6)C(s)−

A proportional controller K = 4 drives the plant G(s) = 4/(s(s + 6)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 4/(s(s + 6))K = 4
%OS = %
6.Hard
R(s)+K = 82s(s + 2)C(s)−

A proportional controller K = 8 drives the plant G(s) = 2/(s(s + 2)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 2/(s(s + 2))K = 8
tp = s

Answer key

  1. 1. y(∞) = 2.308
  2. 2. N = 1
  3. 3. K = 250
  4. 4. K = 360
  5. 5. %OS = 2.838 %
  6. 6. tp = 0.8112 s

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