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)+2(s + 2)s²(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 2)/(s²(s + 2)). Find the acceleration error constant Ka.

G(s) = 2(s + 2)/(s²(s + 2))
Ka =
2.Easy
R(s)+2s(s + 2)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2/(s(s + 2)(s + 4)). Find the velocity error constant Kv.

G(s) = 2/(s(s + 2)(s + 4))
Kv =
3.Medium
r(t) = 4t+2(s + 2)s(s + 1)C(s)−

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

G(s) = 2(s + 2)/(s(s + 1))r(t) = 4t
ess =
4.Medium
r(t) = 4t+50(s + 4)s(s + 4)C(s)−

A ramp r(t) = 4t is applied to a unity-feedback loop with G(s) = 50(s + 4)/(s(s + 4)). Find the steady-state error ess.

G(s) = 50(s + 4)/(s(s + 4))r(t) = 4t
ess =
5.Hard
R(s)+10 + Kd​s5s(s + 4)C(s)−

A PD controller C(s) = 10 + Kd·s drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 5/(s(s + 4))Kp = 10ζ = 0.8
Kd = s
6.Hard
R(s)+K4s(s + 5)C(s)−

A proportional controller K drives the plant G(s) = 4/(s(s + 5)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.4.

G(s) = 4/(s(s + 5))ζ = 0.4
K =

Answer key

  1. 1. Ka = 2
  2. 2. Kv = 0.25
  3. 3. ess = 1
  4. 4. ess = 0.08
  5. 5. Kd = 1.463 s
  6. 6. K = 9.766

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