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) = 5+40(s + 1)(s + 4)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 40/((s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 40/((s + 1)(s + 4))r(t) = 5·u(t)
ess =
2.Easy
R(s)+10s(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10/(s(s + 2)). Find the velocity error constant Kv.

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

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 4)/(s²(s + 5)). Find the steady-state error ess.

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

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 4)/(s²(s + 10)). Find the steady-state error ess.

G(s) = 10(s + 4)/(s²(s + 10))r(t) = 2·t²/2
ess =
5.Hard
R(s)+Kp​ + Ki​/s10s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 0.5. Find the integral gain Ki.

G(s) = 10/(s + 5)ωₙ = 6 rad/sζ = 0.5
Ki =
6.Hard
R(s)+8 + Kd​s2s(s + 2)C(s)−

A PD controller C(s) = 8 + Kd·s drives the plant G(s) = 2/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.9.

G(s) = 2/(s(s + 2))Kp = 8ζ = 0.9
Kd = s

Answer key

  1. 1. ess = 0.4545
  2. 2. Kv = 5
  3. 3. ess = 0.25
  4. 4. ess = 0.5
  5. 5. Ki = 3.6
  6. 6. Kd = 2.6 s

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