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

C(s) = 4(s + 5)/sG(s) = 5/(s(s + 4))
N =
2.Easy
R(s)+5(s + 3)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 3)/(s²(s + 10)). Find the acceleration error constant Ka.

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

The parabolic input r(t) = 4·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 2)/(s²(s + 4)). Find the steady-state error ess.

G(s) = 5(s + 2)/(s²(s + 4))r(t) = 4·t²/2
ess =
4.Medium
r(t) = 0.5t+10s(s + 2)(s + 4)C(s)−

A ramp r(t) = 0.5t is applied to a unity-feedback loop with G(s) = 10/(s(s + 2)(s + 4)). Find the steady-state error ess.

G(s) = 10/(s(s + 2)(s + 4))r(t) = 0.5t
ess =
5.Hard
r(t) = 5t+1 + 4/s2(s + 1)(s + 10)C(s)−

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

C(s) = 1 + 4/sG(s) = 2/((s + 1)(s + 10))r(t) = 5t
ess =
6.Hard
R(s)+K10s(s + 6)C(s)−

A proportional controller K drives the plant G(s) = 10/(s(s + 6)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.4.

G(s) = 10/(s(s + 6))ζ = 0.4
K =

Answer key

  1. 1. N = 2
  2. 2. Ka = 1.5
  3. 3. ess = 1.6
  4. 4. ess = 0.4
  5. 5. ess = 6.25
  6. 6. K = 5.625

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