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)+40(s + 1)(s + 1)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 40(s + 1)/((s + 1)(s + 10)). Find the position error constant Kp.

G(s) = 40(s + 1)/((s + 1)(s + 10))
Kp =
2.Easy
r(t) = 2+50(s + 2)(s + 2)(s + 4)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 50(s + 2)/((s + 2)(s + 4)). Find the steady-state error ess.

G(s) = 50(s + 2)/((s + 2)(s + 4))r(t) = 2·u(t)
ess =
3.Medium
r(t) = 1t+5s(s + 4)C(s)−

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

G(s) = 5/(s(s + 4))r(t) = 1t
ess =
4.Medium
r(t) = 2 + 0.5t+20s(s + 4)C(s)−

The input r(t) = 2 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20/(s(s + 4)). Find the steady-state error ess.

G(s) = 20/(s(s + 4))r(t) = 2 + 0.5t
ess =
5.Hard
R(s)+8 + Kd​s5s(s + 2)C(s)−

A PD controller C(s) = 8 + Kd·s drives the plant G(s) = 5/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 5/(s(s + 2))Kp = 8ζ = 0.707
Kd = s
6.Hard
R(s)+K = 251s(s + 5)C(s)−

A proportional controller K = 25 drives the plant G(s) = 1/(s(s + 5)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 1/(s(s + 5))K = 25
tp = s

Answer key

  1. 1. Kp = 4
  2. 2. ess = 0.1481
  3. 3. ess = 0.8
  4. 4. ess = 0.1
  5. 5. Kd = 1.389 s
  6. 6. tp = 0.7255 s

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