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)+20(s + 1)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 1)/(s²(s + 10)). Find the acceleration error constant Ka.

G(s) = 20(s + 1)/(s²(s + 10))
Ka =
2.Easy
R(s)+C(s)G(s)C(s)−

The controller C(s) = 4(s + 3)/s drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. What is the system type N of the loop?

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

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 5)/(s²(s + 10)). Find the steady-state error ess.

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

A unity-feedback loop has G(s) = K/((s + 1)(s + 4)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 1)(s + 4))ess = 2 %
K =
5.Hard
R(s)+K1s(s + 6)C(s)−

A proportional controller K drives the plant G(s) = 1/(s(s + 6)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 1/(s(s + 6))ζ = 0.707
K =
6.Hard
r(t) = 5t+1 + 2/s10(s + 4)(s + 10)C(s)−

A PI controller C(s) = 1 + 2/s drives the plant G(s) = 10/((s + 4)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 1 + 2/sG(s) = 10/((s + 4)(s + 10))r(t) = 5t
ess =

Answer key

  1. 1. Ka = 2
  2. 2. N = 2
  3. 3. ess = 2
  4. 4. K = 196
  5. 5. K = 18.01
  6. 6. ess = 10

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