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 = 2010s + 2y(t)−

A proportional controller K = 20 drives the plant G(s) = 10/(s + 2) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 10/(s + 2)K = 20r = 10
y(∞) =
2.Easy
R(s)+10(s + 3)s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 3)/(s²(s + 1)). Find the acceleration error constant Ka.

G(s) = 10(s + 3)/(s²(s + 1))
Ka =
3.Medium
r(t) = 5 + 2t+10s(s + 1)(s + 4)C(s)−

The input r(t) = 5 + 2t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 10/(s(s + 1)(s + 4)). Find the steady-state error ess.

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

The parabolic input r(t) = 5·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 1)/(s²(s + 2)). Find the steady-state error ess.

G(s) = 10(s + 1)/(s²(s + 2))r(t) = 5·t²/2
ess =
5.Hard
R(s)+Kp​ + Ki​/s5s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 2 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 5/(s + 2)ωₙ = 2 rad/sζ = 1
Ki =
6.Hard
R(s)+K5s(s + 5)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 5/(s(s + 5))ζ = 0.707
K =

Answer key

  1. 1. y(∞) = 9.901
  2. 2. Ka = 30
  3. 3. ess = 0.8
  4. 4. ess = 1
  5. 5. Ki = 0.8
  6. 6. K = 2.501

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