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)+C(s)G(s)C(s)−

The controller C(s) = 10(s + 3)/s drives the plant G(s) = 1/(s(s + 10)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 10(s + 3)/sG(s) = 1/(s(s + 10))
N =
2.Easy
R(s)+10(s + 4)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10/((s + 4)(s + 10)). Find the position error constant Kp.

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

A unity-feedback loop with G(s) = K/(s(s + 10)) must follow the ramp r(t) = 2t with a steady-state error of 0.1. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 2tess = 0.1
K =
4.Medium
r(t) = 2 + 2t+2(s + 3)s(s + 10)C(s)−

The input r(t) = 2 + 2t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 2(s + 3)/(s(s + 10)). Find the steady-state error ess.

G(s) = 2(s + 3)/(s(s + 10))r(t) = 2 + 2t
ess =
5.Hard
R(s)+K = 24s(s + 4)C(s)−

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

G(s) = 4/(s(s + 4))K = 2
%OS = %
6.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 = 0.8.

G(s) = 2/(s(s + 5))ζ = 0.8
K =

Answer key

  1. 1. N = 2
  2. 2. Kp = 0.25
  3. 3. K = 200
  4. 4. ess = 3.333
  5. 5. %OS = 4.321 %
  6. 6. K = 4.883

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